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# Symmetric Functions

For a comprehensive tutorial on how to use symmetric functions in Sage

See also

`SymmetricFunctions()`

We define the algebra of symmetric functions in the Schur and elementary bases:

    sage: s = SymmetricFunctions(QQ).schur()
    sage: e = SymmetricFunctions(QQ).elementary()

Each is actually a graded Hopf algebra whose basis is indexed by integer partitions:

    sage: s.category()
    Category of bases of Symmetric Functions over Rational Field
    sage: s.basis().keys()
    Partitions

Let us compute with some elements in different bases:

    sage: f1 = s([2,1]); f1
    s[2, 1]
    sage: f2 = e(f1); f2 # basis conversion
    e[2, 1] - e[3]
    sage: f1 == f2
    True
    sage: f1.expand(3, alphabet=['x','y','z'])
    x^2*y + x*y^2 + x^2*z + 2*x*y*z + y^2*z + x*z^2 + y*z^2
    sage: f2.expand(3, alphabet=['x','y','z'])
    x^2*y + x*y^2 + x^2*z + 2*x*y*z + y^2*z + x*z^2 + y*z^2

    sage: m = SymmetricFunctions(QQ).monomial()
    sage: m([3,1])
    m[3, 1]
    sage: m(4) # This is the constant 4, not the partition 4.
    4*m[]
    sage: m([4]) # This is the partition 4.
    m[4]
    sage: 3*m([3,1])-1/2*m([4])
    3*m[3, 1] - 1/2*m[4]

    sage: p = SymmetricFunctions(QQ).power()
    sage: f = p(3)
    sage: f
    3*p[]
    sage: f.parent()
    Symmetric Functions over Rational Field in the powersum basis
    sage: f + p([3,2])
    3*p[] + p[3, 2]

One can convert symmetric functions to symmetric polynomials and vice versa:

    sage: Sym = SymmetricFunctions(QQ)
    sage: p = Sym.powersum()
    sage: h = Sym.homogeneous()
    sage: f = h[2,1] + 2*p[3,1]
    sage: poly = f.expand(3); poly
    2*x0^4 + 2*x0^3*x1 + 2*x0*x1^3 + 2*x1^4 + 2*x0^3*x2 + 2*x1^3*x2 + 2*x0*x2^3 + 2*x1*x2^3 + 2*x2^4
    + x0^3 + 2*x0^2*x1 + 2*x0*x1^2 + x1^3 + 2*x0^2*x2 + 3*x0*x1*x2 + 2*x1^2*x2 + 2*x0*x2^2 + 2*x1*x2^2 + x2^3
    sage: Sym.from_polynomial(poly)
    3*m[1, 1, 1] + 2*m[2, 1] + m[3] + 2*m[3, 1] + 2*m[4]
    sage: Sym.from_polynomial(poly) == f
    True
    sage: g = h[1,1,1,1]
    sage: poly = g.expand(3)
    sage: Sym.from_polynomial(poly) == g
    False

    sage: Sym = SymmetricFunctions(QQ)
    sage: s = Sym.s()
    sage: h = Sym.h()
    sage: p = Sym.p()
    sage: e = Sym.e()
    sage: m = Sym.m()
    sage: a = s([3,1])
    sage: s(a)
    s[3, 1]
    sage: h(a)
    h[3, 1] - h[4]
    sage: p(a)
    1/8*p[1, 1, 1, 1] + 1/4*p[2, 1, 1] - 1/8*p[2, 2] - 1/4*p[4]
    sage: e(a)
    e[2, 1, 1] - e[2, 2] - e[3, 1] + e[4]
    sage: m(a)
    3*m[1, 1, 1, 1] + 2*m[2, 1, 1] + m[2, 2] + m[3, 1]
    sage: a.expand(4)
    x0^3*x1 + x0^2*x1^2 + x0*x1^3 + x0^3*x2 + 2*x0^2*x1*x2 + 2*x0*x1^2*x2 + x1^3*x2 + x0^2*x2^2 + 2*x0*x1*x2^2 + x1^2*x2^2 + x0*x2^3 + x1*x2^3 + x0^3*x3 + 2*x0^2*x1*x3 + 2*x0*x1^2*x3 + x1^3*x3 + 2*x0^2*x2*x3 + 3*x0*x1*x2*x3 + 2*x1^2*x2*x3 + 2*x0*x2^2*x3 + 2*x1*x2^2*x3 + x2^3*x3 + x0^2*x3^2 + 2*x0*x1*x3^2 + x1^2*x3^2 + 2*x0*x2*x3^2 + 2*x1*x2*x3^2 + x2^2*x3^2 + x0*x3^3 + x1*x3^3 + x2*x3^3

Here are further examples:

    sage: h(m([1]))
    h[1]
    sage: h( m([2]) +m([1,1]) )
    h[2]
    sage: h( m([3]) + m([2,1]) + m([1,1,1]) )
    h[3]
    sage: h( m([4]) + m([3,1]) + m([2,2]) + m([2,1,1]) + m([1,1,1,1]) )
    h[4]
    sage: k = 5
    sage: h( sum([ m(part) for part in Partitions(k)]) )
    h[5]
    sage: k = 10
    sage: h( sum([ m(part) for part in Partitions(k)]) )
    h[10]

    sage: P3 = Partitions(3)
    sage: P3.list()
    [[3], [2, 1], [1, 1, 1]]
    sage: m = SymmetricFunctions(QQ).monomial()
    sage: f = sum([m(p) for p in P3])
    sage: m.get_print_style()
    'lex'
    sage: f
    m[1, 1, 1] + m[2, 1] + m[3]
    sage: m.set_print_style('length')
    sage: f
    m[3] + m[2, 1] + m[1, 1, 1]
    sage: m.set_print_style('maximal_part')
    sage: f
    m[1, 1, 1] + m[2, 1] + m[3]
    sage: m.set_print_style('lex')

    sage: Sym = SymmetricFunctions(QQ)
    sage: s = Sym.s()
    sage: m = Sym.m()
    sage: m([3])*s([2,1])
    2*m[3, 1, 1, 1] + m[3, 2, 1] + 2*m[4, 1, 1] + m[4, 2] + m[5, 1]
    sage: s(m([3])*s([2,1]))
    s[2, 1, 1, 1, 1] - s[2, 2, 2] - s[3, 3] + s[5, 1]
    sage: s(s([2,1])*m([3]))
    s[2, 1, 1, 1, 1] - s[2, 2, 2] - s[3, 3] + s[5, 1]
    sage: e = Sym.e()
    sage: e([4])*e([3])*e([1])
    e[4, 3, 1]

    sage: s = SymmetricFunctions(QQ).s()
    sage: z = s([2,1]) + s([1,1,1])
    sage: z.coefficient([2,1])
    1
    sage: z.length()
    2
    sage: z.support()
    [[1, 1, 1], [2, 1]]
    sage: z.degree()
    3

TESTS:

Check that we can handle large integers properly (trac ticket #13413):

    sage: s = SymmetricFunctions(QQ).s()
    sage: p = SymmetricFunctions(QQ).p()
    sage: max(s(p([1]*36)).coefficients())  # long time (4s on sage.math, 2013)
    40971642983700000000

BACKWARD INCOMPATIBLE CHANGES (trac ticket #5457):

The symmetric functions code has been refactored to take advantage of the coercion systems. This introduced a couple of glitches:

- On some bases changes, coefficients in Jack polynomials are not normalized

- Except in a few cases, conversions and coercions are only defined between symmetric functions over the same coefficient ring. E.g. the following does not work anymore:

      sage: s  = SymmetricFunctions(QQ)
      sage: s2 = SymmetricFunctions(QQ['t'])
      sage: s([1]) + s2([2]) # todo: not implemented

  This feature will probably come back at some point through improvements to the Sage coercion system.

Backward compatibility should be essentially retained.

AUTHORS:

- Mike Hansen (2007-06-15)
- Nicolas M. Thiery (partial refactoring)
- Mike Zabrocki, Anne Schilling (2012)
- Darij Grinberg (2013) Sym over rings that are not characteristic 0

`sage.combinat.sf.sfa.``SymmetricFunctionAlgebra`(R, basis='schur')

  This is deprecated in trac ticket #15473. Use instead `SymmetricFunctions` as `SymmetricFunctions(R).basis()`

  INPUT:

  - `R` – ring with identity basis
  - `basis` – a string for the name of the basis, must be one of ‘schur’, ‘elementary’, ‘homogeneous’, ‘power’, ‘monomial’ or their abbreviations ‘s’, ‘e’, ‘h’, ‘p’, ‘m’

  OUTPUT: A SymmetricFunctionAlgebra

  EXAMPLES:

      sage: SymmetricFunctionAlgebra(QQ)
      doctest:...: DeprecationWarning: this function is deprecated. Use SymmetricFunctions(R).basis()
      See http://trac.sagemath.org/15473 for details.
      Symmetric Functions over Rational Field in the Schur basis

class `sage.combinat.sf.sfa.``SymmetricFunctionAlgebra_generic`(Sym, basis_name=None, prefix=None)

  Bases: `sage.combinat.free_module.CombinatorialFreeModule`

  Abstract base class for symmetric function algebras.

  Todo

  Most of the methods in this class are generic (manipulations of morphisms, ...) and should be generalized (or removed)

  TESTS:

      sage: s = SymmetricFunctions(QQ).s()
      sage: m = SymmetricFunctions(ZZ).m()
      sage: s(m([2,1]))
      -2*s[1, 1, 1] + s[2, 1]

  `Element`

    alias of `SymmetricFunctionAlgebra_generic_Element`

  `basis_name`()

    Return the name of the basis of `self`.

    This is used for output and, for the classical bases of symmetric functions, to connect this basis with Symmetrica.

    EXAMPLES:

        sage: Sym = SymmetricFunctions(QQ)
        sage: s = Sym.s()
        sage: s.basis_name()
        'Schur'
        sage: p = Sym.p()
        sage: p.basis_name()
        'powersum'
        sage: h = Sym.h()
        sage: h.basis_name()
        'homogeneous'
        sage: e = Sym.e()
        sage: e.basis_name()
        'elementary'
        sage: m = Sym.m()
        sage: m.basis_name()
        'monomial'
        sage: f = Sym.f()
        sage: f.basis_name()
        'forgotten'

  `coproduct_by_coercion`(elt)

    Return the coproduct of the element `elt` by coercion to the Schur basis.

    INPUT:

    - `elt` – an instance of this basis

    OUTPUT:

    - The image of `elt` under the comultiplication (=coproduct) of the coalgebra of symmetric functions. The result is an element of the tensor squared of the basis `self`.

    EXAMPLES:

        sage: m = SymmetricFunctions(QQ).m()
        sage: m[3,1,1].coproduct()
        m[] # m[3, 1, 1] + m[1] # m[3, 1] + m[1, 1] # m[3] + m[3] # m[1, 1] + m[3, 1] # m[1] + m[3, 1, 1] # m[]
        sage: m.coproduct_by_coercion(m[2,1])
        m[] # m[2, 1] + m[1] # m[2] + m[2] # m[1] + m[2, 1] # m[]
        sage: m.coproduct_by_coercion(m[2,1]) == m([2,1]).coproduct()
        True
        sage: McdH = SymmetricFunctions(QQ['q','t'].fraction_field()).macdonald().H()
        sage: McdH[2,1].coproduct()
        McdH[] # McdH[2, 1] + ((q^2*t-1)/(q*t-1))*McdH[1] # McdH[1, 1] + ((q*t^2-1)/(q*t-1))*McdH[1] # McdH[2] + ((q^2*t-1)/(q*t-1))*McdH[1, 1] # McdH[1] + ((q*t^2-1)/(q*t-1))*McdH[2] # McdH[1] + McdH[2, 1] # McdH[]
        sage: HLQp = SymmetricFunctions(QQ['t'].fraction_field()).hall_littlewood().Qp()
        sage: HLQp[2,1].coproduct()
        HLQp[] # HLQp[2, 1] + HLQp[1] # HLQp[1, 1] + HLQp[1] # HLQp[2] + HLQp[1, 1] # HLQp[1] + HLQp[2] # HLQp[1] + HLQp[2, 1] # HLQp[]
        sage: Sym = SymmetricFunctions(FractionField(QQ['t']))
        sage: LLT = Sym.llt(3)
        sage: LLT.cospin([3,2,1]).coproduct()
        (t+1)*m[] # m[1, 1] + m[] # m[2] + (t+1)*m[1] # m[1] + (t+1)*m[1, 1] # m[] + m[2] # m[]
        sage: f = SymmetricFunctions(ZZ).f()
        sage: f[3].coproduct()
        f[] # f[3] + f[3] # f[]
        sage: f[3,2,1].coproduct()
        f[] # f[3, 2, 1] + f[1] # f[3, 2] + f[2] # f[3, 1] + f[2, 1] # f[3] + f[3] # f[2, 1] + f[3, 1] # f[2] + f[3, 2] # f[1] + f[3, 2, 1] # f[]

  `dual_basis`(scalar=None, scalar_name='', basis_name=None, prefix=None)

    Return the dual basis of `self` with respect to the scalar product `scalar`.

    INPUT:

    - `scalar` – A function `zee` from partitions to the base ring which specifies the scalar product by \(\langle p_{\lambda}, p_{\lambda} \rangle = \mathrm{zee}(\lambda)\). (Independently on the function chosen, the power sum basis will always be orthogonal; the function `scalar` only determines the norms of the basis elements.) If `scalar` is None, then the standard (Hall) scalar product is used.
    - `scalar_name` – name of the scalar function
    - `prefix` – prefix used to display the basis

    EXAMPLES:

    The duals of the elementary symmetric functions with respect to the Hall scalar product are the forgotten symmetric functions.

        sage: e = SymmetricFunctions(QQ).e()
        sage: f = e.dual_basis(prefix='f'); f
        Dual basis to Symmetric Functions over Rational Field in the elementary basis with respect to the Hall scalar product
        sage: f([2,1])^2
        4*f[2, 2, 1, 1] + 6*f[2, 2, 2] + 2*f[3, 2, 1] + 2*f[3, 3] + 2*f[4, 1, 1] + f[4, 2]
        sage: f([2,1]).scalar(e([2,1]))
        1
        sage: f([2,1]).scalar(e([1,1,1]))
        0

    Since the power-sum symmetric functions are orthogonal, their duals with respect to the Hall scalar product are scalar multiples of themselves.

        sage: p = SymmetricFunctions(QQ).p()
        sage: q = p.dual_basis(prefix='q'); q
        Dual basis to Symmetric Functions over Rational Field in the powersum basis with respect to the Hall scalar product
        sage: q([2,1])^2
        4*q[2, 2, 1, 1]
        sage: p([2,1]).scalar(q([2,1]))
        1
        sage: p([2,1]).scalar(q([1,1,1]))
        0

  `from_polynomial`(poly, check=True)

    Convert polynomial to a symmetric function in the monomial basis and then to the basis `self`.

    INPUT:

    - `poly` – a symmetric polynomial
    - `check` – (default: `True`) boolean, specifies whether the computation checks that the polynomial is indeed symmetric

    EXAMPLES:

        sage: Sym = SymmetricFunctions(QQ)
        sage: h = Sym.homogeneous()
        sage: f = (h([]) + h([2,1]) + h([3])).expand(3)
        sage: h.from_polynomial(f)
        h[] + h[2, 1] + h[3]
        sage: s = Sym.s()
        sage: g = (s([]) + s([2,1])).expand(3); g
        x0^2*x1 + x0*x1^2 + x0^2*x2 + 2*x0*x1*x2 + x1^2*x2 + x0*x2^2 + x1*x2^2 + 1
        sage: s.from_polynomial(g)
        s[] + s[2, 1]

  `get_print_style`()

    Return the value of the current print style for `self`.

    EXAMPLES:

        sage: s = SymmetricFunctions(QQ).s()
        sage: s.get_print_style()
        'lex'
        sage: s.set_print_style('length')
        sage: s.get_print_style()
        'length'
        sage: s.set_print_style('lex')

  `prefix`()

    Return the prefix on the elements of `self`.

    EXAMPLES:

        sage: schur = SymmetricFunctions(QQ).schur()
        sage: schur([3,2,1])
        s[3, 2, 1]
        sage: schur.prefix()
        's'

  `set_print_style`(ps)

    Set the value of the current print style to `ps`.

    INPUT:

    - `ps` – a string specifying the printing style

    EXAMPLES:

        sage: s = SymmetricFunctions(QQ).s()
        sage: s.get_print_style()
        'lex'
        sage: s.set_print_style('length')
        sage: s.get_print_style()
        'length'
        sage: s.set_print_style('lex')

  `symmetric_function_ring`()

    Return the family of symmetric functions associated to the basis `self`.

    OUTPUT:

    - returns an instance of the ring of symmetric functions

    EXAMPLES:

        sage: schur = SymmetricFunctions(QQ).schur()
        sage: schur.symmetric_function_ring()
        Symmetric Functions over Rational Field
        sage: power = SymmetricFunctions(QQ['t']).power()
        sage: power.symmetric_function_ring()
        Symmetric Functions over Univariate Polynomial Ring in t over Rational Field

  `transition_matrix`(basis, n)

    Return the transition matrix between `self` and `basis` for the homogeneous component of degree `n`.

    INPUT:

    - `basis` – a basis of the ring of symmetric functions
    - `n` – a nonnegative integer

    OUTPUT:

    - a matrix of coefficients giving the expansion of the homogeneous degree-\(n\) elements of `self` in the degree-\(n\) elements of `basis`

    EXAMPLES:

        sage: s = SymmetricFunctions(QQ).s()
        sage: m = SymmetricFunctions(QQ).m()
        sage: s.transition_matrix(m,5)
        [1 1 1 1 1 1 1]
        [0 1 1 2 2 3 4]
        [0 0 1 1 2 3 5]
        [0 0 0 1 1 3 6]
        [0 0 0 0 1 2 5]
        [0 0 0 0 0 1 4]
        [0 0 0 0 0 0 1]
        sage: s.transition_matrix(m,1)
        [1]
        sage: s.transition_matrix(m,0)
        [1]

        sage: p = SymmetricFunctions(QQ).p()
        sage: s.transition_matrix(p, 4)
        [ 1/4  1/3  1/8  1/4 1/24]
        [-1/4    0 -1/8  1/4  1/8]
        [   0 -1/3  1/4    0 1/12]
        [ 1/4    0 -1/8 -1/4  1/8]
        [-1/4  1/3  1/8 -1/4 1/24]
        sage: StoP = s.transition_matrix(p,4)
        sage: a = s([3,1])+5*s([1,1,1,1])-s([4])
        sage: a
        5*s[1, 1, 1, 1] + s[3, 1] - s[4]
        sage: mon = a.support()
        sage: coeffs = a.coefficients()
        sage: coeffs
        [5, 1, -1]
        sage: mon
        [[1, 1, 1, 1], [3, 1], [4]]
        sage: cm = matrix([[-1,1,0,0,5]])
        sage: cm * StoP
        [-7/4  4/3  3/8 -5/4 7/24]
        sage: p(a)
        7/24*p[1, 1, 1, 1] - 5/4*p[2, 1, 1] + 3/8*p[2, 2] + 4/3*p[3, 1] - 7/4*p[4]

        sage: h = SymmetricFunctions(QQ).h()
        sage: e = SymmetricFunctions(QQ).e()
        sage: s.transition_matrix(m,7) == h.transition_matrix(s,7).transpose()
        True

        sage: h.transition_matrix(m, 7) == h.transition_matrix(m, 7).transpose()
        True

        sage: h.transition_matrix(e, 7) == e.transition_matrix(h, 7)
        True

        sage: p.transition_matrix(s, 5)
        [ 1 -1  0  1  0 -1  1]
        [ 1  0 -1  0  1  0 -1]
        [ 1 -1  1  0 -1  1 -1]
        [ 1  1 -1  0 -1  1  1]
        [ 1  0  1 -2  1  0  1]
        [ 1  2  1  0 -1 -2 -1]
        [ 1  4  5  6  5  4  1]

        sage: e.transition_matrix(m,7) == e.transition_matrix(m,7).transpose()
        True

class `sage.combinat.sf.sfa.``SymmetricFunctionAlgebra_generic_Element`(M, x)

  Bases: `sage.combinat.free_module.CombinatorialFreeModuleElement`

  Class of generic elements for the symmetric function algebra.

  TESTS:

      sage: m = SymmetricFunctions(QQ).m()
      sage: f = sum([m(p) for p in Partitions(3)])
      sage: m.set_print_style('lex')
      sage: f
      m[1, 1, 1] + m[2, 1] + m[3]
      sage: m.set_print_style('length')
      sage: f
      m[3] + m[2, 1] + m[1, 1, 1]
      sage: m.set_print_style('maximal_part')
      sage: f
      m[1, 1, 1] + m[2, 1] + m[3]
      sage: m.set_print_style('lex')

  `adams_operation`(n)

    Return the image of the symmetric function `self` under the \(n\)-th Frobenius operator.

    The \(n\)-th Frobenius operator \(\mathbf{f}_n\) is defined to be the map from the ring of symmetric functions to itself that sends every symmetric function \(P(x_1, x_2, x_3, \ldots)\) to \(P(x_1^n, x_2^n, x_3^n, \ldots)\). This operator \(\mathbf{f}_n\) is a Hopf algebra endomorphism, and satisfies

    \[\mathbf{f}_n m_{(\lambda_1, \lambda_2, \lambda_3, \ldots)} = m_{(n\lambda_1, n\lambda_2, n\lambda_3, \ldots)}\]

    for every partition \((\lambda_1, \lambda_2, \lambda_3, \ldots)\) (where \(m\) means the monomial basis). Moreover, \(\mathbf{f}_n (p_r) = p_{nr}\) for every positive integer \(r\) (where \(p_k\) denotes the \(k\)-th powersum symmetric function).

    The \(n\)-th Frobenius operator is also called the \(n\)-th Frobenius endomorphism. It is not related to the Frobenius map which connects the ring of symmetric functions with the representation theory of the symmetric group.

    The \(n\)-th Frobenius operator is also the \(n\)-th Adams operator of the \(\Lambda\)-ring of symmetric functions over the integers.

    The \(n\)-th Frobenius operator can also be described via plethysm: Every symmetric function \(P\) satisfies \(\mathbf{f}_n(P) = p_n \circ P = P \circ p_n\), where \(p_n\) is the \(n\)-th powersum symmetric function, and \(\circ\) denotes (outer) plethysm.

    `adams_operation()` serves as alias for `frobenius()`, since the Frobenius operators are the Adams operations of the \(\Lambda\)-ring of symmetric functions.

    INPUT:

    - `n` – a positive integer

    OUTPUT:

    The result of applying the \(n\)-th Frobenius operator (on the ring of symmetric functions) to `self`.

    EXAMPLES:

        sage: Sym = SymmetricFunctions(ZZ)
        sage: p = Sym.p()
        sage: h = Sym.h()
        sage: s = Sym.s()
        sage: m = Sym.m()
        sage: s[3].frobenius(2)
        -s[3, 3] + s[4, 2] - s[5, 1] + s[6]
        sage: m[4,2,1].frobenius(3)
        m[12, 6, 3]
        sage: p[4,2,1].frobenius(3)
        p[12, 6, 3]
        sage: h[4].frobenius(2)
        h[4, 4] - 2*h[5, 3] + 2*h[6, 2] - 2*h[7, 1] + 2*h[8]

    The Frobenius endomorphisms are multiplicative:

        sage: all( all( s(lam).frobenius(3) * s(mu).frobenius(3)
        ....:           == (s(lam) * s(mu)).frobenius(3)
        ....:           for mu in Partitions(3) )
        ....:      for lam in Partitions(3) )
        True
        sage: all( all( m(lam).frobenius(2) * m(mu).frobenius(2)
        ....:           == (m(lam) * m(mu)).frobenius(2)
        ....:           for mu in Partitions(4) )
        ....:      for lam in Partitions(4) )
        True
        sage: all( all( p(lam).frobenius(2) * p(mu).frobenius(2)
        ....:           == (p(lam) * p(mu)).frobenius(2)
        ....:           for mu in Partitions(3) )
        ....:      for lam in Partitions(4) )
        True

    Being Hopf algebra endomorphisms, the Frobenius operators commute with the antipode:

        sage: all( p(lam).frobenius(4).antipode()
        ....:      == p(lam).antipode().frobenius(4)
        ....:      for lam in Partitions(3) )
        True

    Testing the \(\mathbf{f}_n(P) = p_n \circ P = P \circ p_n\) equality (over \(\QQ\), since plethysm is currently not defined over \(\ZZ\) in Sage):

        sage: Sym = SymmetricFunctions(QQ)
        sage: s = Sym.s()
        sage: p = Sym.p()
        sage: all( s(lam).frobenius(3) == s(lam).plethysm(p[3])
        ....:      == s(p[3].plethysm(s(lam)))
        ....:      for lam in Partitions(4) )
        True

    By Exercise 7.61 in Stanley’s EC2 [STA] (see the errata on his website), \(\mathbf{f}_n(h_m)\) is a linear combination of Schur polynomials (of straight shapes) using coefficients \(0\), \(1\) and \(-1\) only; moreover, all partitions whose Schur polynomials occur with coefficient \(\neq 0\) in this combination have empty \(n\)-cores. Let us check this on examples:

        sage: all( all( all( (coeff == -1 or coeff == 1)
        ....:                and lam.core(n) == Partition([])
        ....:                for lam, coeff in s([m]).frobenius(n) )
        ....:           for n in range(2, 4) )
        ....:      for m in range(4) )
        True

    See also

    `plethysm()`

    Todo

    This method is fast on the monomial and the powersum bases, while all other bases get converted to the monomial basis. For most bases, this is probably the quickest way to do, but at least the Schur basis should have a better option. (Quoting from Stanley’s EC2 [STA]: “D. G. Duncan, J. London Math. Soc. 27 (1952), 235-236, or Y. M. Chen, A. M. Garsia, and J. B. Remmel, Contemp. Math. 34 (1984), 109-153”.)

  `arithmetic_product`(x)

    Return the arithmetic product of `self` and `x` in the basis of `self`.

    The arithmetic product is a binary operation \(\boxdot\) on the ring of symmetric functions which is bilinear in its two arguments and satisfies

    \[p_{\lambda} \boxdot p_{\mu} = \prod\limits_{i \geq 1, j \geq 1} p_{\mathrm{lcm}(\lambda_i, \mu_j)}^{\mathrm{gcd}(\lambda_i, \mu_j)}\]

    for any two partitions \(\lambda = (\lambda_1, \lambda_2, \lambda_3, \dots )\) and \(\mu = (\mu_1, \mu_2, \mu_3, \dots )\) (where \(p_{\nu}\) denotes the power-sum symmetric function indexed by the partition \(\nu\), and \(p_i\) denotes the \(i\)-th power-sum symmetric function). This is enough to define the arithmetic product if the base ring is torsion-free as a \(\ZZ\)-module; for all other cases the arithmetic product is uniquely determined by requiring it to be functorial in the base ring. See http://mathoverflow.net/questions/138148/ for a discussion of this arithmetic product.

    If \(f\) and \(g\) are two symmetric functions which are homogeneous of degrees \(a\) and \(b\), respectively, then \(f \boxdot g\) is homogeneous of degree \(ab\).

    The arithmetic product is commutative and associative and has unity \(e_1 = p_1 = h_1\).

    INPUT:

    - `x` – element of the ring of symmetric functions over the same base ring as `self`

    OUTPUT:

    Arithmetic product of `self` with `x`; this is a symmetric function over the same base ring as `self`.

    EXAMPLES:

        sage: s = SymmetricFunctions(QQ).s()
        sage: s([2]).arithmetic_product(s([2]))
        s[1, 1, 1, 1] + 2*s[2, 2] + s[4]
        sage: s([2]).arithmetic_product(s([1,1]))
        s[2, 1, 1] + s[3, 1]

    The symmetric function `e[1]` is the unity for the arithmetic product:

        sage: e = SymmetricFunctions(ZZ).e()
        sage: all( e([1]).arithmetic_product(e(q)) == e(q) for q in Partitions(4) )
        True

    The arithmetic product is commutative:

        sage: e = SymmetricFunctions(FiniteField(19)).e()
        sage: m = SymmetricFunctions(FiniteField(19)).m()
        sage: all( all( e(p).arithmetic_product(m(q)) == m(q).arithmetic_product(e(p))  # long time (26s on sage.math, 2013)
        ....:           for q in Partitions(4) )
        ....:      for p in Partitions(4) )
        True

    Note

    The currently existing implementation of this function is technically unsatisfactory. It distinguishes the case when the base ring is a \(\QQ\)-algebra (in which case the arithmetic product can be easily computed using the power sum basis) from the case where it isn’t. In the latter, it does a computation using universal coefficients, again distinguishing the case when it is able to compute the “corresponding” basis of the symmetric function algebra over \(\QQ\) (using the `corresponding_basis_over` hack) from the case when it isn’t (in which case it transforms everything into the Schur basis, which is slow).

  `bernstein_creation_operator`(n)

    Return the image of `self` under the \(n\)-th Bernstein creation operator.

    Let \(n\) be an integer. The \(n\)-th Bernstein creation operator \(\mathbf{B}_n\) is defined as the endomorphism of the space \(Sym\) of symmetric functions which sends every \(f\) to

    \[\sum_{i \geq 0} (-1)^i h_{n+i} e_i^\perp,\]

    where usual notations are in place (\(h\) stands for the complete homogeneous symmetric functions, \(e\) for the elementary ones, and \(e_i^\perp\) means skewing (`skew_by()`) by \(e_i\)).

    This has been studied in [BBSSZ2012], section 2.2, where the following rule is given for computing \(\mathbf{B}_n\) on a Schur function: If \((\alpha_1, \alpha_2, \ldots, \alpha_n)\) is an \(n\)-tuple of integers (positive or not), then

    \[\mathbf{B}_n s_{(\alpha_1, \alpha_2, \ldots, \alpha_n)} = s_{(n, \alpha_1, \alpha_2, \ldots, \alpha_n)}.\]

    Here, \(s_{(\alpha_1, \alpha_2, \ldots, \alpha_n)}\) is the “Schur function” associated to the \(n\)-tuple \((\alpha_1, \alpha_2, \ldots, \alpha_n)\), and defined by literally applying the Jacobi-Trudi identity, i.e., by

    \[s_{(\alpha_1, \alpha_2, \ldots, \alpha_n)} = \det \left( (h_{\alpha_i - i + j})_{i, j = 1, 2, \ldots, n} \right).\]

    This notion of a Schur function clearly extends the classical notion of Schur function corresponding to a partition, but is easily reduced to the latter (in fact, for any \(n\)-tuple \(\alpha\) of integers, one easily sees that \(s_\alpha\) is either \(0\) or minus-plus a Schur function corresponding to a partition; and it is easy to determine which of these is the case and find the partition by a combinatorial algorithm).

    EXAMPLES:

    Let us check that what this method computes agrees with the definition:

        sage: Sym = SymmetricFunctions(ZZ)
        sage: e = Sym.e()
        sage: h = Sym.h()
        sage: s = Sym.s()
        sage: def bernstein_creation_by_def(n, f):
        ....:     # `n`-th Bernstein creation operator applied to `f`
        ....:     # computed according to its definition.
        ....:     res = f.parent().zero()
        ....:     if not f:
        ....:         return res
        ....:     max_degree = max(sum(m) for m, c in f)
        ....:     for i in range(max_degree + 1):
        ....:         if n + i >= 0:
        ....:             res += (-1) ** i * h[n + i] * f.skew_by(e[i])
        ....:     return res
        sage: all( bernstein_creation_by_def(n, s[l]) == s[l].bernstein_creation_operator(n)
        ....:      for n in range(-2, 3) for l in Partitions(4) )
        True
        sage: all( bernstein_creation_by_def(n, s[l]) == s[l].bernstein_creation_operator(n)
        ....:      for n in range(-3, 4) for l in Partitions(3) )
        True
        sage: all( bernstein_creation_by_def(n, e[l]) == e[l].bernstein_creation_operator(n)
        ....:      for n in range(-3, 4) for k in range(3) for l in Partitions(k) )
        True

    Some examples:

        sage: s[3,2].bernstein_creation_operator(3)
        s[3, 3, 2]
        sage: s[3,2].bernstein_creation_operator(1)
        -s[2, 2, 2]
        sage: h[3,2].bernstein_creation_operator(-2)
        h[2, 1]
        sage: h[3,2].bernstein_creation_operator(-1)
        h[2, 1, 1] - h[2, 2] - h[3, 1]
        sage: h[3,2].bernstein_creation_operator(0)
        -h[3, 1, 1] + h[3, 2]
        sage: h[3,2].bernstein_creation_operator(1)
        -h[2, 2, 2] + h[3, 2, 1]
        sage: h[3,2].bernstein_creation_operator(2)
        -h[3, 3, 1] + h[4, 2, 1]

  `degree`()

    Return the degree of `self` (which is defined to be \(0\) for the zero element).

    EXAMPLES:

        sage: s = SymmetricFunctions(QQ).s()
        sage: z = s([4]) + s([2,1]) + s([1,1,1]) + s([1]) + 3
        sage: z.degree()
        4
        sage: s(1).degree()
        0
        sage: s(0).degree()
        0

  `derivative_with_respect_to_p1`(n=1)

    Return the symmetric function obtained by taking the derivative of `self` with respect to the power-sum symmetric function \(p_1\) when the expansion of `self` in the power-sum basis is considered as a polynomial in \(p_k\)‘s (with \(k \geq 1\)).

    This is the same as skewing `self` by the first power-sum symmetric function \(p_1\).

    INPUT:

    - `n` – (default: 1) nonnegative integer which determines which power of the derivative is taken

    EXAMPLES:

        sage: p = SymmetricFunctions(QQ).p()
        sage: a = p([1,1,1])
        sage: a.derivative_with_respect_to_p1()
        3*p[1, 1]
        sage: a.derivative_with_respect_to_p1(1)
        3*p[1, 1]
        sage: a.derivative_with_respect_to_p1(2)
        6*p[1]
        sage: a.derivative_with_respect_to_p1(3)
        6*p[]

        sage: s = SymmetricFunctions(QQ).s()
        sage: s([3]).derivative_with_respect_to_p1()
        s[2]
        sage: s([2,1]).derivative_with_respect_to_p1()
        s[1, 1] + s[2]
        sage: s([1,1,1]).derivative_with_respect_to_p1()
        s[1, 1]
        sage: s(0).derivative_with_respect_to_p1()
        0
        sage: s(1).derivative_with_respect_to_p1()
        0
        sage: s([1]).derivative_with_respect_to_p1()
        s[]

    Let us check that taking the derivative with respect to `p[1]` is equivalent to skewing by `p[1]`:

        sage: p1 = s([1])
        sage: all( s(lam).derivative_with_respect_to_p1()
        ....:      == s(lam).skew_by(p1) for lam in Partitions(4) )
        True

  `expand`(n, alphabet='x')

    Expand the symmetric function as a symmetric polynomial in `n` variables.

    INPUT:

    - `n` – a nonnegative integer
    - `alphabet` – (default: `'x'`) a variable for the expansion

    OUTPUT:

    A monomial expansion of an instance of `self` in \(n\) variables.

    EXAMPLES:

        sage: J = SymmetricFunctions(QQ).jack(t=2).J()
        sage: J([2,1]).expand(3)
        4*x0^2*x1 + 4*x0*x1^2 + 4*x0^2*x2 + 6*x0*x1*x2 + 4*x1^2*x2 + 4*x0*x2^2 + 4*x1*x2^2

  `frobenius`(n)

    Return the image of the symmetric function `self` under the \(n\)-th Frobenius operator.

    The \(n\)-th Frobenius operator \(\mathbf{f}_n\) is defined to be the map from the ring of symmetric functions to itself that sends every symmetric function \(P(x_1, x_2, x_3, \ldots)\) to \(P(x_1^n, x_2^n, x_3^n, \ldots)\). This operator \(\mathbf{f}_n\) is a Hopf algebra endomorphism, and satisfies

    \[\mathbf{f}_n m_{(\lambda_1, \lambda_2, \lambda_3, \ldots)} = m_{(n\lambda_1, n\lambda_2, n\lambda_3, \ldots)}\]

    for every partition \((\lambda_1, \lambda_2, \lambda_3, \ldots)\) (where \(m\) means the monomial basis). Moreover, \(\mathbf{f}_n (p_r) = p_{nr}\) for every positive integer \(r\) (where \(p_k\) denotes the \(k\)-th powersum symmetric function).

    The \(n\)-th Frobenius operator is also called the \(n\)-th Frobenius endomorphism. It is not related to the Frobenius map which connects the ring of symmetric functions with the representation theory of the symmetric group.

    The \(n\)-th Frobenius operator is also the \(n\)-th Adams operator of the \(\Lambda\)-ring of symmetric functions over the integers.

    The \(n\)-th Frobenius operator can also be described via plethysm: Every symmetric function \(P\) satisfies \(\mathbf{f}_n(P) = p_n \circ P = P \circ p_n\), where \(p_n\) is the \(n\)-th powersum symmetric function, and \(\circ\) denotes (outer) plethysm.

    `adams_operation()` serves as alias for `frobenius()`, since the Frobenius operators are the Adams operations of the \(\Lambda\)-ring of symmetric functions.

    INPUT:

    - `n` – a positive integer

    OUTPUT:

    The result of applying the \(n\)-th Frobenius operator (on the ring of symmetric functions) to `self`.

    EXAMPLES:

        sage: Sym = SymmetricFunctions(ZZ)
        sage: p = Sym.p()
        sage: h = Sym.h()
        sage: s = Sym.s()
        sage: m = Sym.m()
        sage: s[3].frobenius(2)
        -s[3, 3] + s[4, 2] - s[5, 1] + s[6]
        sage: m[4,2,1].frobenius(3)
        m[12, 6, 3]
        sage: p[4,2,1].frobenius(3)
        p[12, 6, 3]
        sage: h[4].frobenius(2)
        h[4, 4] - 2*h[5, 3] + 2*h[6, 2] - 2*h[7, 1] + 2*h[8]

    The Frobenius endomorphisms are multiplicative:

        sage: all( all( s(lam).frobenius(3) * s(mu).frobenius(3)
        ....:           == (s(lam) * s(mu)).frobenius(3)
        ....:           for mu in Partitions(3) )
        ....:      for lam in Partitions(3) )
        True
        sage: all( all( m(lam).frobenius(2) * m(mu).frobenius(2)
        ....:           == (m(lam) * m(mu)).frobenius(2)
        ....:           for mu in Partitions(4) )
        ....:      for lam in Partitions(4) )
        True
        sage: all( all( p(lam).frobenius(2) * p(mu).frobenius(2)
        ....:           == (p(lam) * p(mu)).frobenius(2)
        ....:           for mu in Partitions(3) )
        ....:      for lam in Partitions(4) )
        True

    Being Hopf algebra endomorphisms, the Frobenius operators commute with the antipode:

        sage: all( p(lam).frobenius(4).antipode()
        ....:      == p(lam).antipode().frobenius(4)
        ....:      for lam in Partitions(3) )
        True

    Testing the \(\mathbf{f}_n(P) = p_n \circ P = P \circ p_n\) equality (over \(\QQ\), since plethysm is currently not defined over \(\ZZ\) in Sage):

        sage: Sym = SymmetricFunctions(QQ)
        sage: s = Sym.s()
        sage: p = Sym.p()
        sage: all( s(lam).frobenius(3) == s(lam).plethysm(p[3])
        ....:      == s(p[3].plethysm(s(lam)))
        ....:      for lam in Partitions(4) )
        True

    By Exercise 7.61 in Stanley’s EC2 [STA] (see the errata on his website), \(\mathbf{f}_n(h_m)\) is a linear combination of Schur polynomials (of straight shapes) using coefficients \(0\), \(1\) and \(-1\) only; moreover, all partitions whose Schur polynomials occur with coefficient \(\neq 0\) in this combination have empty \(n\)-cores. Let us check this on examples:

        sage: all( all( all( (coeff == -1 or coeff == 1)
        ....:                and lam.core(n) == Partition([])
        ....:                for lam, coeff in s([m]).frobenius(n) )
        ....:           for n in range(2, 4) )
        ....:      for m in range(4) )
        True

    See also

    `plethysm()`

    Todo

    This method is fast on the monomial and the powersum bases, while all other bases get converted to the monomial basis. For most bases, this is probably the quickest way to do, but at least the Schur basis should have a better option. (Quoting from Stanley’s EC2 [STA]: “D. G. Duncan, J. London Math. Soc. 27 (1952), 235-236, or Y. M. Chen, A. M. Garsia, and J. B. Remmel, Contemp. Math. 34 (1984), 109-153”.)

  `hl_creation_operator`(nu, t=None)

    This is the vertex operator that generalizes Jing’s operator.

    It is a linear operator that raises the degree by \(|\nu|\). This creation operator is a t-analogue of multiplication by `s(nu)` .

    See also

    Proposition 5 in [SZ2001].

    INPUT:

    - `nu` – a partition
    - `t` – (default: `None`, in which case `t` is used) a parameter

    REFERENCES:

    [SZ2001] | M. Shimozono, M. Zabrocki, Hall-Littlewood vertex operators and generalized Kostka polynomials. Adv. Math. 158 (2001), no. 1, 66-85.

    EXAMPLES:

        sage: s = SymmetricFunctions(QQ['t']).s()
        sage: s([2]).hl_creation_operator([3,2])
        s[3, 2, 2] + t*s[3, 3, 1] + t*s[4, 2, 1] + t^2*s[4, 3] + t^2*s[5, 2]

        sage: Sym = SymmetricFunctions(FractionField(QQ['t']))
        sage: HLQp = Sym.hall_littlewood().Qp()
        sage: s = Sym.s()
        sage: HLQp(s([2]).hl_creation_operator([2]).hl_creation_operator([3]))
        HLQp[3, 2, 2]
        sage: s([2,2]).hl_creation_operator([2,1])
        t*s[2, 2, 2, 1] + t^2*s[3, 2, 1, 1] + t^2*s[3, 2, 2] + t^3*s[3, 3, 1] + t^3*s[4, 2, 1] + t^4*s[4, 3]
        sage: s(1).hl_creation_operator([2,1,1])
        s[2, 1, 1]
        sage: s(0).hl_creation_operator([2,1,1])
        0
        sage: s([3,2]).hl_creation_operator([2,1,1])
        (t^2-t)*s[2, 2, 2, 2, 1] + t^3*s[3, 2, 2, 1, 1] + (t^3-t^2)*s[3, 2, 2, 2] + t^3*s[3, 3, 1, 1, 1] + t^4*s[3, 3, 2, 1] + t^3*s[4, 2, 1, 1, 1] + t^4*s[4, 2, 2, 1] + 2*t^4*s[4, 3, 1, 1] + t^5*s[4, 3, 2] + t^5*s[4, 4, 1] + t^4*s[5, 2, 1, 1] + t^5*s[5, 3, 1]

    TESTS:

        sage: s(0).hl_creation_operator([1])
        0

  `inner_plethysm`(x)

    Return the inner plethysm of `self` with `x`.

    Whenever \(R\) is a \(\QQ\)-algebra, and \(f\) and \(g\) are two symmetric functions over \(R\) such that the constant term of \(f\) is zero, the inner plethysm of \(f\) with \(g\) is a symmetric function over \(R\), and the degree of this symmetric function is the same as the degree of \(g\). We will denote the inner plethysm of \(f\) with \(g\) by \(f \{ g \}\) (in contrast to the notation of outer plethysm which is generally denoted \(f [ g ]\)); in Sage syntax, it is `f.inner_plethysm(g)`.

    First we describe the axiomatic definition of the operation; see below for a representation-theoretic interpretation. In the following equations, we denote the outer product (i.e., the standard product on the ring of symmetric functions, `product()`) by \(\cdot\) and the Kronecker product (`itensor()`) by \(\ast\)).

    \[(f + g) \{ h \} = f \{ h \} + g \{ h \}\]\[(f \cdot g) \{ h \} = (f \{ h \}) \ast (g \{ h \})\]\[p_k \{ f + g \} = p_k \{ f \} + p_k \{ g \}\]

    where \(p_k\) is the \(k\)-th power-sum symmetric function for every \(k > 0\).

    Let \(\sigma\) be a permutation of cycle type \(\mu\) and let \(\mu^k\) be the cycle type of \(\sigma^k\). Then,

    \[p_k \{ p_\mu/z_\mu \} = \sum_{\nu : \nu^k = \mu } p_{\nu}/z_{\nu}\]

    Since \((p_\mu/z_\mu)_{\mu}\) is a basis for the symmetric functions, these four formulas define the symmetric function operation \(f \{ g \}\) for any symmetric functions \(f\) and \(g\) (where \(f\) has constant term \(0\)) by expanding \(f\) in the power sum basis and \(g\) in the dual basis \(p_\mu/z_\mu\).

    See also

    `itensor()`, `partition_power()`, `plethysm()`

    This operation admits a representation-theoretic interpretation in the case where \(f\) is a Schur function \(s_\lambda\) and \(g\) is a homogeneous degree \(n\) symmetric function with nonnegative integral coefficients in the Schur basis. The symmetric function \(f \{ g \}\) is the Frobenius image of the \(S_n\)-representation constructed as follows.

    The assumptions on \(g\) imply that \(g\) is the Frobenius image of a representation \(\rho\) of the symmetric group \(S_n\):

    \[\rho : S_n \to GL_N.\]

    If the degree \(N\) of this representation is greater than or equal to the number of parts of \(\lambda\), then \(f\), which denotes \(s_\lambda\), corresponds to the character of some irreducible \(GL_N\)-representation, say

    \[\sigma : GL_N \to GL_M.\]

    The composition \(\sigma \circ \rho : S_n \to GL_M\) is a representation of \(S_n\) whose Frobenius image is precisely \(f \{ g \}\).

    If \(N\) is less than the number of parts of \(\lambda\), then \(f \{ g \}\) is \(0\) by definition.

    When \(f\) is a symmetric function with constant term \(\neq 0\), the inner plethysm \(f \{ g \}\) isn’t well-defined in the ring of symmetric functions. Indeed, it is not clear how to define \(1 \{ g \}\). The most sensible way to get around this probably is defining it as the infinite sum \(h_0 + h_1 + h_2 + \cdots\) (where \(h_i\) means the \(i\)-th complete homogeneous symmetric function) in the completion of this ring with respect to its grading. This is how [SchaThi1994] defines \(1 \{ g \}\). The present method, however, sets it to be the sum of \(h_i\) over all \(i\) for which the \(i\)-th homogeneous component of \(g\) is nonzero. This is rather a hack than a reasonable definition. Use with caution!

    Note

    If a symmetric function \(g\) is written in the form \(g = g_0 + g_1 + g_2 + \cdots\) with each \(g_i\) homogeneous of degree \(i\), then \(f \{ g \} = f \{ g_0 \} + f \{ g_1 \} + f \{ g_2 \} + \cdots\) for every \(f\) with constant term \(0\). But in general, inner plethysm is not linear in the second variable.

    REFERENCES:

    [King] | King, R. Branching rules for \(GL_m \supset \Sigma_n\) and the evaluation of inner plethysms. J. Math. Phys. 15, 258 (1974) doi:10.1063/1.1666632

    [SchaThi1994] | Thomas Scharf, Jean-Yves Thibon. A Hopf-algebra approach to inner plethysm. Advances in Mathematics 104 (1994), pp. 30-58. ftp://ftp.mathe2.uni-bayreuth.de/axel/papers/scharf:a_hopf_algebra_approach_to_inner_plethysm.ps.gz

    INPUT:

    - `x` – element of the ring of symmetric functions over the same base ring as `self`

    OUTPUT:

    - an element of symmetric functions in the parent of `self`

    EXAMPLES:

        sage: Sym = SymmetricFunctions(QQ)
        sage: s = Sym.schur()
        sage: p = Sym.power()
        sage: h = Sym.complete()
        sage: s([2,1]).inner_plethysm(s([1,1,1]))
        0
        sage: s([2]).inner_plethysm(s([2,1]))
        s[2, 1] + s[3]
        sage: s([1,1]).inner_plethysm(s([2,1]))
        s[1, 1, 1]
        sage: s[2,1].inner_tensor(s[2,1])
        s[1, 1, 1] + s[2, 1] + s[3]

        sage: f = s([2,1]) + 2*s([3,1])
        sage: f.itensor(f)
        s[1, 1, 1] + s[2, 1] + 4*s[2, 1, 1] + 4*s[2, 2] + s[3] + 4*s[3, 1] + 4*s[4]
        sage: s( h([1,1]).inner_plethysm(f) )
        s[1, 1, 1] + s[2, 1] + 4*s[2, 1, 1] + 4*s[2, 2] + s[3] + 4*s[3, 1] + 4*s[4]

        sage: s([]).inner_plethysm(s([1,1]) + 2*s([2,1])+s([3]))
        s[2] + s[3]
        sage: [s([]).inner_plethysm(s(la)) for la in Partitions(4)]
        [s[4], s[4], s[4], s[4], s[4]]
        sage: s([3]).inner_plethysm(s([]))
        s[]
        sage: s[1,1,1,1].inner_plethysm(s[2,1])
        0
        sage: s[1,1,1,1].inner_plethysm(2*s[2,1])
        s[3]

        sage: p[3].inner_plethysm(p[3])
        0
        sage: p[3,3].inner_plethysm(p[3])
        0
        sage: p[3].inner_plethysm(p[1,1,1])
        p[1, 1, 1] + 2*p[3]
        sage: p[4].inner_plethysm(p[1,1,1,1]/24)
        1/24*p[1, 1, 1, 1] + 1/4*p[2, 1, 1] + 1/8*p[2, 2] + 1/4*p[4]
        sage: p[3,3].inner_plethysm(p[1,1,1])
        6*p[1, 1, 1] + 12*p[3]

    TESTS:

        sage: s(0).inner_plethysm(s(0))
        0
        sage: s(1).inner_plethysm(s(0))
        0
        sage: s(0).inner_plethysm(s(1))
        0
        sage: s(1).inner_plethysm(s(1))
        s[]
        sage: s(2).inner_plethysm(s(1))
        2*s[]
        sage: s(1).inner_plethysm(s(2))
        s[]

  `inner_tensor`(x)

    Return the internal (tensor) product of `self` and `x` in the basis of `self`.

    The internal tensor product can be defined as the linear extension of the definition on power sums \(p_{\lambda} \ast p_{\mu} = \delta_{\lambda,\mu} z_{\lambda} p_{\lambda}\), where \(z_{\lambda} = (1^{r_1} r_1!) (2^{r_2} r_2!) \cdots\) for \(\lambda = (1^{r_1} 2^{r_2} \cdots )\) and where \(\ast\) denotes the internal tensor product. The internal tensor product is also known as the Kronecker product, or as the second multiplication on the ring of symmetric functions.

    Note that the internal product of any two homogeneous symmetric functions of equal degrees is a homogeneous symmetric function of the same degree. On the other hand, the internal product of two homogeneous symmetric functions of distinct degrees is \(0\).

    Note

    The internal product is sometimes referred to as “inner product” in the literature, but unfortunately this name is shared by a different operation, namely the Hall inner product (see `scalar()`).

    INPUT:

    - `x` – element of the ring of symmetric functions over the same base ring as `self`

    OUTPUT:

    - the internal product of `self` with `x` (an element of the ring of symmetric functions in the same basis as `self`)

    The methods `itensor()`, `internal_product()`, `kronecker_product()`, `inner_tensor()` are all synonyms.

    EXAMPLES:

        sage: s = SymmetricFunctions(QQ).s()
        sage: a = s([2,1])
        sage: b = s([3])
        sage: a.itensor(b)
        s[2, 1]
        sage: c = s([3,2,1])
        sage: c.itensor(c)
        s[1, 1, 1, 1, 1, 1] + 2*s[2, 1, 1, 1, 1] + 3*s[2, 2, 1, 1] + 2*s[2, 2, 2]
         + 4*s[3, 1, 1, 1] + 5*s[3, 2, 1] + 2*s[3, 3] + 4*s[4, 1, 1]
         + 3*s[4, 2] + 2*s[5, 1] + s[6]

    There are few quantitative results pertaining to Kronecker products in general, which makes their computation so difficult. Let us test a few of them in different bases.

    The Kronecker product of any homogeneous symmetric function \(f\) of degree \(n\) with the \(n\)-th complete homogeneous symmetric function `h[n]` (a.k.a. `s[n]`) is \(f\):

        sage: h = SymmetricFunctions(ZZ).h()
        sage: all( h([5]).itensor(h(p)) == h(p) for p in Partitions(5) )
        True

    The Kronecker product of a Schur function \(s_{\lambda}\) with the \(n\)-th elementary symmetric function `e[n]`, where \(n = \left| \lambda \right|\), is \(s_{\lambda'}\) (where \(\lambda'\) is the conjugate partition of \(\lambda\)):

        sage: F = CyclotomicField(12)
        sage: s = SymmetricFunctions(F).s()
        sage: e = SymmetricFunctions(F).e()
        sage: all( e([5]).itensor(s(p)) == s(p.conjugate()) for p in Partitions(5) )
        True

    The Kronecker product is commutative:

        sage: e = SymmetricFunctions(FiniteField(19)).e()
        sage: m = SymmetricFunctions(FiniteField(19)).m()
        sage: all( all( e(p).itensor(m(q)) == m(q).itensor(e(p)) for q in Partitions(4) )
        ....:      for p in Partitions(4) )
        True

        sage: F = FractionField(QQ['q','t'])
        sage: mq = SymmetricFunctions(F).macdonald().Q()
        sage: mh = SymmetricFunctions(F).macdonald().H()
        sage: all( all( mq(p).itensor(mh(r)) == mh(r).itensor(mq(p)) for r in Partitions(4) )
        ....:      for p in Partitions(3) )   # long time
        True

    Let us check (on examples) Proposition 5.2 of Gelfand, Krob, Lascoux, Leclerc, Retakh, Thibon, “Noncommutative symmetric functions”, Arxiv hep-th/9407124, for \(r = 2\):

        sage: e = SymmetricFunctions(FiniteField(29)).e()
        sage: s = SymmetricFunctions(FiniteField(29)).s()
        sage: m = SymmetricFunctions(FiniteField(29)).m()
        sage: def tensor_copr(u, v, w):  # computes \mu ((u \otimes v) * \Delta(w)) with
        ....:                            # * meaning Kronecker product and \mu meaning the
        ....:                            # usual multiplication.
        ....:     result = w.parent().zero()
        ....:     for partition_pair, coeff in w.coproduct():
        ....:         result += coeff * w.parent()(u).itensor(partition_pair[0]) * w.parent()(v).itensor(partition_pair[1])
        ....:     return result
        sage: all( all( all( tensor_copr(e[u], s[v], m[w]) == (e[u] * s[v]).itensor(m[w])
        ....:                for w in Partitions(5) )
        ....:           for v in Partitions(2) )
        ....:      for u in Partitions(3) )   # long time
        True

    Some examples from Briand, Orellana, Rosas, “The stability of the Kronecker products of Schur functions.” Arxiv 0907.4652:

        sage: s = SymmetricFunctions(ZZ).s()
        sage: s[2,2].itensor(s[2,2])
        s[1, 1, 1, 1] + s[2, 2] + s[4]
        sage: s[3,2].itensor(s[3,2])
        s[2, 1, 1, 1] + s[2, 2, 1] + s[3, 1, 1] + s[3, 2] + s[4, 1] + s[5]
        sage: s[4,2].itensor(s[4,2])
        s[2, 2, 2] + s[3, 1, 1, 1] + 2*s[3, 2, 1] + s[4, 1, 1] + 2*s[4, 2] + s[5, 1] + s[6]

    An example from p. 220 of Thibon, “Hopf algebras of symmetric functions and tensor products of symmetric group representations”, International Journal of Algebra and Computation, 1991:

        sage: s = SymmetricFunctions(QQbar).s()
        sage: s[2,1].itensor(s[2,1])
        s[1, 1, 1] + s[2, 1] + s[3]

    TESTS:

        sage: s = SymmetricFunctions(QQ).s()
        sage: a = s([8,8])
        sage: a.itensor(a)
        s[4, 4, 4, 4] + s[5, 5, 3, 3] + s[5, 5, 5, 1] + s[6, 4, 4, 2]
         + s[6, 6, 2, 2] + s[6, 6, 4] + s[7, 3, 3, 3] + s[7, 5, 3, 1]
         + s[7, 7, 1, 1] + s[8, 4, 2, 2] + s[8, 4, 4] + s[8, 6, 2]
         + s[8, 8] + s[9, 3, 3, 1] + s[9, 5, 1, 1] + s[10, 2, 2, 2]
         + s[10, 4, 2] + s[10, 6] + s[11, 3, 1, 1] + s[12, 2, 2]
         + s[12, 4] + s[13, 1, 1, 1] + s[14, 2] + s[16]
        sage: s[8].itensor(s[7])
        0
        sage: s(0).itensor(s(0))
        0
        sage: s(1).itensor(s(0))
        0
        sage: s(0).itensor(s(1))
        0
        sage: s(1).itensor(s(1))
        s[]

    Same over the ring of integers:

        sage: s = SymmetricFunctions(ZZ).s()
        sage: a = s([8,8])
        sage: a.itensor(a)
        s[4, 4, 4, 4] + s[5, 5, 3, 3] + s[5, 5, 5, 1] + s[6, 4, 4, 2]
         + s[6, 6, 2, 2] + s[6, 6, 4] + s[7, 3, 3, 3] + s[7, 5, 3, 1]
         + s[7, 7, 1, 1] + s[8, 4, 2, 2] + s[8, 4, 4] + s[8, 6, 2]
         + s[8, 8] + s[9, 3, 3, 1] + s[9, 5, 1, 1] + s[10, 2, 2, 2]
         + s[10, 4, 2] + s[10, 6] + s[11, 3, 1, 1] + s[12, 2, 2]
         + s[12, 4] + s[13, 1, 1, 1] + s[14, 2] + s[16]
        sage: s[8].itensor(s[7])
        0
        sage: s(0).itensor(s(0))
        0
        sage: s(1).itensor(s(0))
        0
        sage: s(0).itensor(s(1))
        0
        sage: s(1).itensor(s(1))
        s[]

    Theorem 2.1 in Bessenrodt, van Willigenburg, Arxiv 1105.3170v2:

        sage: s = SymmetricFunctions(ZZ).s()
        sage: all( all( max( r[0] for r in s(p).itensor(s(q)).monomial_coefficients().keys() )
        ....:           == sum( min(p[i], q.get_part(i)) for i in range(len(p)) )
        ....:           for p in Partitions(4) )
        ....:      for q in Partitions(4) )
        True
        sage: all( all( max( len(r) for r in s(p).itensor(s(q)).monomial_coefficients().keys() )
        ....:           == sum( min(p[i], q.conjugate().get_part(i)) for i in range(len(p)) )
        ....:           for p in Partitions(4) )
        ....:      for q in Partitions(4) )
        True

    Check that the basis and ground ring of `self` are preserved:

        sage: F = CyclotomicField(12)
        sage: s = SymmetricFunctions(F).s()
        sage: e = SymmetricFunctions(F).e()
        sage: e[3].itensor(s[3])
        e[3]
        sage: s[3].itensor(e[3])
        s[1, 1, 1]
        sage: parent(e[3].itensor(s[3]))
        Symmetric Functions over Cyclotomic Field of order 12 and degree 4 in the elementary basis
        sage: parent(s[3].itensor(e[3]))
        Symmetric Functions over Cyclotomic Field of order 12 and degree 4 in the Schur basis

    Note

    The currently existing implementation of this function is technically unsatisfactory. It distinguishes the case when the base ring is a \(\QQ\)-algebra (in which case the Kronecker product can be easily computed using the power sum basis) from the case where it isn’t. In the latter, it does a computation using universal coefficients, again distinguishing the case when it is able to compute the “corresponding” basis of the symmetric function algebra over \(\QQ\) (using the `corresponding_basis_over` hack) from the case when it isn’t (in which case it transforms everything into the Schur basis, which is slow).

  `internal_coproduct`()

    Return the inner coproduct of `self` in the basis of `self`.

    The inner coproduct (also known as the Kronecker coproduct, as the internal coproduct, or as the second comultiplication on the ring of symmetric functions) is a ring homomorphism \(\Delta^\times\) from the ring of symmetric functions to the tensor product (over the base ring) of this ring with itself. It is uniquely characterized by the formula

    \[\Delta^{\times}(h_n) = \sum_{\lambda \vdash n} s_{\lambda} \otimes s_{\lambda} = \sum_{\lambda \vdash n} h_{\lambda} \otimes m_{\lambda} = \sum_{\lambda \vdash n} m_{\lambda} \otimes h_{\lambda},\]

    where \(\lambda \vdash n\) means \(\lambda\) is a partition of \(n\), and \(n\) is any nonnegative integer. It also satisfies

    \[\Delta^\times (p_n) = p_n \otimes p_n\]

    for any positive integer \(n\). If the base ring is a \(\QQ\)-algebra, it also satisfies

    \[\Delta^{\times}(h_n) = \sum_{\lambda \vdash n} z_{\lambda}^{-1} p_{\lambda} \otimes p_{\lambda},\]

    where

    \[z_{\lambda} = \prod_{i=1}^\infty i^{m_i(\lambda)} m_i(\lambda)!\]

    with \(m_i(\lambda)\) meaning the number of appearances of \(i\) in \(\lambda\) (see `zee()`).

    The method `kronecker_coproduct()` is a synonym of `internal_coproduct()`.

    EXAMPLES:

        sage: s = SymmetricFunctions(ZZ).s()
        sage: a = s([2,1])
        sage: a.internal_coproduct()
        s[1, 1, 1] # s[2, 1] + s[2, 1] # s[1, 1, 1] + s[2, 1] # s[2, 1] + s[2, 1] # s[3] + s[3] # s[2, 1]

        sage: e = SymmetricFunctions(QQ).e()
        sage: b = e([2])
        sage: b.internal_coproduct()
        e[1, 1] # e[2] + e[2] # e[1, 1] - 2*e[2] # e[2]

    The internal coproduct is adjoint to the internal product with respect to the Hall inner product: Any three symmetric functions \(f\), \(g\) and \(h\) satisfy \(\langle f * g, h \rangle = \sum_i \langle f, h^{\prime}_i \rangle \langle g, h^{\prime\prime}_i \rangle\), where we write \(\Delta^{\times}(h)\) as \(\sum_i h^{\prime}_i \otimes h^{\prime\prime}_i\). Let us check this in degree \(4\):

        sage: e = SymmetricFunctions(FiniteField(29)).e()
        sage: s = SymmetricFunctions(FiniteField(29)).s()
        sage: m = SymmetricFunctions(FiniteField(29)).m()
        sage: def tensor_incopr(f, g, h):  # computes \sum_i \left< f, h'_i \right> \left< g, h''_i \right>
        ....:     result = h.base_ring().zero()
        ....:     for partition_pair, coeff in h.internal_coproduct():
        ....:         result += coeff * h.parent()(f).scalar(partition_pair[0]) * h.parent()(g).scalar(partition_pair[1])
        ....:     return result
        sage: all( all( all( tensor_incopr(e[u], s[v], m[w]) == (e[u].itensor(s[v])).scalar(m[w])  # long time (10s on sage.math, 2013)
        ....:                for w in Partitions(5) )
        ....:           for v in Partitions(2) )
        ....:      for u in Partitions(3) )
        True

    Let us check the formulas for \(\Delta^{\times}(h_n)\) and \(\Delta^{\times}(p_n)\) given in the description of this method:

        sage: e = SymmetricFunctions(QQ).e()
        sage: p = SymmetricFunctions(QQ).p()
        sage: h = SymmetricFunctions(QQ).h()
        sage: s = SymmetricFunctions(QQ).s()
        sage: all( s(h([n])).internal_coproduct() == sum([tensor([s(lam), s(lam)]) for lam in Partitions(n)])
        ....:      for n in range(6) )
        True
        sage: all( h([n]).internal_coproduct() == sum([tensor([h(lam), h(m(lam))]) for lam in Partitions(n)])
        ....:      for n in range(6) )
        True
        sage: all( factorial(n) * h([n]).internal_coproduct()
        ....:      == sum([lam.conjugacy_class_size() * tensor([h(p(lam)), h(p(lam))])
        ....:              for lam in Partitions(n)])
        ....:      for n in range(6) )
        True

    TESTS:

        sage: s = SymmetricFunctions(QQ).s()
        sage: s([]).internal_coproduct()
        s[] # s[]

  `internal_product`(x)

    Return the internal (tensor) product of `self` and `x` in the basis of `self`.

    The internal tensor product can be defined as the linear extension of the definition on power sums \(p_{\lambda} \ast p_{\mu} = \delta_{\lambda,\mu} z_{\lambda} p_{\lambda}\), where \(z_{\lambda} = (1^{r_1} r_1!) (2^{r_2} r_2!) \cdots\) for \(\lambda = (1^{r_1} 2^{r_2} \cdots )\) and where \(\ast\) denotes the internal tensor product. The internal tensor product is also known as the Kronecker product, or as the second multiplication on the ring of symmetric functions.

    Note that the internal product of any two homogeneous symmetric functions of equal degrees is a homogeneous symmetric function of the same degree. On the other hand, the internal product of two homogeneous symmetric functions of distinct degrees is \(0\).

    Note

    The internal product is sometimes referred to as “inner product” in the literature, but unfortunately this name is shared by a different operation, namely the Hall inner product (see `scalar()`).

    INPUT:

    - `x` – element of the ring of symmetric functions over the same base ring as `self`

    OUTPUT:

    - the internal product of `self` with `x` (an element of the ring of symmetric functions in the same basis as `self`)

    The methods `itensor()`, `internal_product()`, `kronecker_product()`, `inner_tensor()` are all synonyms.

    EXAMPLES:

        sage: s = SymmetricFunctions(QQ).s()
        sage: a = s([2,1])
        sage: b = s([3])
        sage: a.itensor(b)
        s[2, 1]
        sage: c = s([3,2,1])
        sage: c.itensor(c)
        s[1, 1, 1, 1, 1, 1] + 2*s[2, 1, 1, 1, 1] + 3*s[2, 2, 1, 1] + 2*s[2, 2, 2]
         + 4*s[3, 1, 1, 1] + 5*s[3, 2, 1] + 2*s[3, 3] + 4*s[4, 1, 1]
         + 3*s[4, 2] + 2*s[5, 1] + s[6]

    There are few quantitative results pertaining to Kronecker products in general, which makes their computation so difficult. Let us test a few of them in different bases.

    The Kronecker product of any homogeneous symmetric function \(f\) of degree \(n\) with the \(n\)-th complete homogeneous symmetric function `h[n]` (a.k.a. `s[n]`) is \(f\):

        sage: h = SymmetricFunctions(ZZ).h()
        sage: all( h([5]).itensor(h(p)) == h(p) for p in Partitions(5) )
        True

    The Kronecker product of a Schur function \(s_{\lambda}\) with the \(n\)-th elementary symmetric function `e[n]`, where \(n = \left| \lambda \right|\), is \(s_{\lambda'}\) (where \(\lambda'\) is the conjugate partition of \(\lambda\)):

        sage: F = CyclotomicField(12)
        sage: s = SymmetricFunctions(F).s()
        sage: e = SymmetricFunctions(F).e()
        sage: all( e([5]).itensor(s(p)) == s(p.conjugate()) for p in Partitions(5) )
        True

    The Kronecker product is commutative:

        sage: e = SymmetricFunctions(FiniteField(19)).e()
        sage: m = SymmetricFunctions(FiniteField(19)).m()
        sage: all( all( e(p).itensor(m(q)) == m(q).itensor(e(p)) for q in Partitions(4) )
        ....:      for p in Partitions(4) )
        True

        sage: F = FractionField(QQ['q','t'])
        sage: mq = SymmetricFunctions(F).macdonald().Q()
        sage: mh = SymmetricFunctions(F).macdonald().H()
        sage: all( all( mq(p).itensor(mh(r)) == mh(r).itensor(mq(p)) for r in Partitions(4) )
        ....:      for p in Partitions(3) )   # long time
        True

    Let us check (on examples) Proposition 5.2 of Gelfand, Krob, Lascoux, Leclerc, Retakh, Thibon, “Noncommutative symmetric functions”, Arxiv hep-th/9407124, for \(r = 2\):

        sage: e = SymmetricFunctions(FiniteField(29)).e()
        sage: s = SymmetricFunctions(FiniteField(29)).s()
        sage: m = SymmetricFunctions(FiniteField(29)).m()
        sage: def tensor_copr(u, v, w):  # computes \mu ((u \otimes v) * \Delta(w)) with
        ....:                            # * meaning Kronecker product and \mu meaning the
        ....:                            # usual multiplication.
        ....:     result = w.parent().zero()
        ....:     for partition_pair, coeff in w.coproduct():
        ....:         result += coeff * w.parent()(u).itensor(partition_pair[0]) * w.parent()(v).itensor(partition_pair[1])
        ....:     return result
        sage: all( all( all( tensor_copr(e[u], s[v], m[w]) == (e[u] * s[v]).itensor(m[w])
        ....:                for w in Partitions(5) )
        ....:           for v in Partitions(2) )
        ....:      for u in Partitions(3) )   # long time
        True

    Some examples from Briand, Orellana, Rosas, “The stability of the Kronecker products of Schur functions.” Arxiv 0907.4652:

        sage: s = SymmetricFunctions(ZZ).s()
        sage: s[2,2].itensor(s[2,2])
        s[1, 1, 1, 1] + s[2, 2] + s[4]
        sage: s[3,2].itensor(s[3,2])
        s[2, 1, 1, 1] + s[2, 2, 1] + s[3, 1, 1] + s[3, 2] + s[4, 1] + s[5]
        sage: s[4,2].itensor(s[4,2])
        s[2, 2, 2] + s[3, 1, 1, 1] + 2*s[3, 2, 1] + s[4, 1, 1] + 2*s[4, 2] + s[5, 1] + s[6]

    An example from p. 220 of Thibon, “Hopf algebras of symmetric functions and tensor products of symmetric group representations”, International Journal of Algebra and Computation, 1991:

        sage: s = SymmetricFunctions(QQbar).s()
        sage: s[2,1].itensor(s[2,1])
        s[1, 1, 1] + s[2, 1] + s[3]

    TESTS:

        sage: s = SymmetricFunctions(QQ).s()
        sage: a = s([8,8])
        sage: a.itensor(a)
        s[4, 4, 4, 4] + s[5, 5, 3, 3] + s[5, 5, 5, 1] + s[6, 4, 4, 2]
         + s[6, 6, 2, 2] + s[6, 6, 4] + s[7, 3, 3, 3] + s[7, 5, 3, 1]
         + s[7, 7, 1, 1] + s[8, 4, 2, 2] + s[8, 4, 4] + s[8, 6, 2]
         + s[8, 8] + s[9, 3, 3, 1] + s[9, 5, 1, 1] + s[10, 2, 2, 2]
         + s[10, 4, 2] + s[10, 6] + s[11, 3, 1, 1] + s[12, 2, 2]
         + s[12, 4] + s[13, 1, 1, 1] + s[14, 2] + s[16]
        sage: s[8].itensor(s[7])
        0
        sage: s(0).itensor(s(0))
        0
        sage: s(1).itensor(s(0))
        0
        sage: s(0).itensor(s(1))
        0
        sage: s(1).itensor(s(1))
        s[]

    Same over the ring of integers:

        sage: s = SymmetricFunctions(ZZ).s()
        sage: a = s([8,8])
        sage: a.itensor(a)
        s[4, 4, 4, 4] + s[5, 5, 3, 3] + s[5, 5, 5, 1] + s[6, 4, 4, 2]
         + s[6, 6, 2, 2] + s[6, 6, 4] + s[7, 3, 3, 3] + s[7, 5, 3, 1]
         + s[7, 7, 1, 1] + s[8, 4, 2, 2] + s[8, 4, 4] + s[8, 6, 2]
         + s[8, 8] + s[9, 3, 3, 1] + s[9, 5, 1, 1] + s[10, 2, 2, 2]
         + s[10, 4, 2] + s[10, 6] + s[11, 3, 1, 1] + s[12, 2, 2]
         + s[12, 4] + s[13, 1, 1, 1] + s[14, 2] + s[16]
        sage: s[8].itensor(s[7])
        0
        sage: s(0).itensor(s(0))
        0
        sage: s(1).itensor(s(0))
        0
        sage: s(0).itensor(s(1))
        0
        sage: s(1).itensor(s(1))
        s[]

    Theorem 2.1 in Bessenrodt, van Willigenburg, Arxiv 1105.3170v2:

        sage: s = SymmetricFunctions(ZZ).s()
        sage: all( all( max( r[0] for r in s(p).itensor(s(q)).monomial_coefficients().keys() )
        ....:           == sum( min(p[i], q.get_part(i)) for i in range(len(p)) )
        ....:           for p in Partitions(4) )
        ....:      for q in Partitions(4) )
        True
        sage: all( all( max( len(r) for r in s(p).itensor(s(q)).monomial_coefficients().keys() )
        ....:           == sum( min(p[i], q.conjugate().get_part(i)) for i in range(len(p)) )
        ....:           for p in Partitions(4) )
        ....:      for q in Partitions(4) )
        True

    Check that the basis and ground ring of `self` are preserved:

        sage: F = CyclotomicField(12)
        sage: s = SymmetricFunctions(F).s()
        sage: e = SymmetricFunctions(F).e()
        sage: e[3].itensor(s[3])
        e[3]
        sage: s[3].itensor(e[3])
        s[1, 1, 1]
        sage: parent(e[3].itensor(s[3]))
        Symmetric Functions over Cyclotomic Field of order 12 and degree 4 in the elementary basis
        sage: parent(s[3].itensor(e[3]))
        Symmetric Functions over Cyclotomic Field of order 12 and degree 4 in the Schur basis

    Note

    The currently existing implementation of this function is technically unsatisfactory. It distinguishes the case when the base ring is a \(\QQ\)-algebra (in which case the Kronecker product can be easily computed using the power sum basis) from the case where it isn’t. In the latter, it does a computation using universal coefficients, again distinguishing the case when it is able to compute the “corresponding” basis of the symmetric function algebra over \(\QQ\) (using the `corresponding_basis_over` hack) from the case when it isn’t (in which case it transforms everything into the Schur basis, which is slow).

  `is_schur_positive`()

    Return `True` if and only if `self` is Schur positive.

    If \(s\) is the space of Schur functions over `self`‘s base ring, then this is the same as `self._is_positive(s)`.

    EXAMPLES:

        sage: s = SymmetricFunctions(QQ).s()
        sage: a = s([2,1]) + s([3])
        sage: a.is_schur_positive()
        True
        sage: a = s([2,1]) - s([3])
        sage: a.is_schur_positive()
        False

        sage: QQx = QQ['x']
        sage: s = SymmetricFunctions(QQx).s()
        sage: x = QQx.gen()
        sage: a = (1+x)*s([2,1])
        sage: a.is_schur_positive()
        True
        sage: a = (1-x)*s([2,1])
        sage: a.is_schur_positive()
        False
        sage: s(0).is_schur_positive()
        True
        sage: s(1+x).is_schur_positive()
        True

  `itensor`(x)

    Return the internal (tensor) product of `self` and `x` in the basis of `self`.

    The internal tensor product can be defined as the linear extension of the definition on power sums \(p_{\lambda} \ast p_{\mu} = \delta_{\lambda,\mu} z_{\lambda} p_{\lambda}\), where \(z_{\lambda} = (1^{r_1} r_1!) (2^{r_2} r_2!) \cdots\) for \(\lambda = (1^{r_1} 2^{r_2} \cdots )\) and where \(\ast\) denotes the internal tensor product. The internal tensor product is also known as the Kronecker product, or as the second multiplication on the ring of symmetric functions.

    Note that the internal product of any two homogeneous symmetric functions of equal degrees is a homogeneous symmetric function of the same degree. On the other hand, the internal product of two homogeneous symmetric functions of distinct degrees is \(0\).

    Note

    The internal product is sometimes referred to as “inner product” in the literature, but unfortunately this name is shared by a different operation, namely the Hall inner product (see `scalar()`).

    INPUT:

    - `x` – element of the ring of symmetric functions over the same base ring as `self`

    OUTPUT:

    - the internal product of `self` with `x` (an element of the ring of symmetric functions in the same basis as `self`)

    The methods `itensor()`, `internal_product()`, `kronecker_product()`, `inner_tensor()` are all synonyms.

    EXAMPLES:

        sage: s = SymmetricFunctions(QQ).s()
        sage: a = s([2,1])
        sage: b = s([3])
        sage: a.itensor(b)
        s[2, 1]
        sage: c = s([3,2,1])
        sage: c.itensor(c)
        s[1, 1, 1, 1, 1, 1] + 2*s[2, 1, 1, 1, 1] + 3*s[2, 2, 1, 1] + 2*s[2, 2, 2]
         + 4*s[3, 1, 1, 1] + 5*s[3, 2, 1] + 2*s[3, 3] + 4*s[4, 1, 1]
         + 3*s[4, 2] + 2*s[5, 1] + s[6]

    There are few quantitative results pertaining to Kronecker products in general, which makes their computation so difficult. Let us test a few of them in different bases.

    The Kronecker product of any homogeneous symmetric function \(f\) of degree \(n\) with the \(n\)-th complete homogeneous symmetric function `h[n]` (a.k.a. `s[n]`) is \(f\):

        sage: h = SymmetricFunctions(ZZ).h()
        sage: all( h([5]).itensor(h(p)) == h(p) for p in Partitions(5) )
        True

    The Kronecker product of a Schur function \(s_{\lambda}\) with the \(n\)-th elementary symmetric function `e[n]`, where \(n = \left| \lambda \right|\), is \(s_{\lambda'}\) (where \(\lambda'\) is the conjugate partition of \(\lambda\)):

        sage: F = CyclotomicField(12)
        sage: s = SymmetricFunctions(F).s()
        sage: e = SymmetricFunctions(F).e()
        sage: all( e([5]).itensor(s(p)) == s(p.conjugate()) for p in Partitions(5) )
        True

    The Kronecker product is commutative:

        sage: e = SymmetricFunctions(FiniteField(19)).e()
        sage: m = SymmetricFunctions(FiniteField(19)).m()
        sage: all( all( e(p).itensor(m(q)) == m(q).itensor(e(p)) for q in Partitions(4) )
        ....:      for p in Partitions(4) )
        True

        sage: F = FractionField(QQ['q','t'])
        sage: mq = SymmetricFunctions(F).macdonald().Q()
        sage: mh = SymmetricFunctions(F).macdonald().H()
        sage: all( all( mq(p).itensor(mh(r)) == mh(r).itensor(mq(p)) for r in Partitions(4) )
        ....:      for p in Partitions(3) )   # long time
        True

    Let us check (on examples) Proposition 5.2 of Gelfand, Krob, Lascoux, Leclerc, Retakh, Thibon, “Noncommutative symmetric functions”, Arxiv hep-th/9407124, for \(r = 2\):

        sage: e = SymmetricFunctions(FiniteField(29)).e()
        sage: s = SymmetricFunctions(FiniteField(29)).s()
        sage: m = SymmetricFunctions(FiniteField(29)).m()
        sage: def tensor_copr(u, v, w):  # computes \mu ((u \otimes v) * \Delta(w)) with
        ....:                            # * meaning Kronecker product and \mu meaning the
        ....:                            # usual multiplication.
        ....:     result = w.parent().zero()
        ....:     for partition_pair, coeff in w.coproduct():
        ....:         result += coeff * w.parent()(u).itensor(partition_pair[0]) * w.parent()(v).itensor(partition_pair[1])
        ....:     return result
        sage: all( all( all( tensor_copr(e[u], s[v], m[w]) == (e[u] * s[v]).itensor(m[w])
        ....:                for w in Partitions(5) )
        ....:           for v in Partitions(2) )
        ....:      for u in Partitions(3) )   # long time
        True

    Some examples from Briand, Orellana, Rosas, “The stability of the Kronecker products of Schur functions.” Arxiv 0907.4652:

        sage: s = SymmetricFunctions(ZZ).s()
        sage: s[2,2].itensor(s[2,2])
        s[1, 1, 1, 1] + s[2, 2] + s[4]
        sage: s[3,2].itensor(s[3,2])
        s[2, 1, 1, 1] + s[2, 2, 1] + s[3, 1, 1] + s[3, 2] + s[4, 1] + s[5]
        sage: s[4,2].itensor(s[4,2])
        s[2, 2, 2] + s[3, 1, 1, 1] + 2*s[3, 2, 1] + s[4, 1, 1] + 2*s[4, 2] + s[5, 1] + s[6]

    An example from p. 220 of Thibon, “Hopf algebras of symmetric functions and tensor products of symmetric group representations”, International Journal of Algebra and Computation, 1991:

        sage: s = SymmetricFunctions(QQbar).s()
        sage: s[2,1].itensor(s[2,1])
        s[1, 1, 1] + s[2, 1] + s[3]

    TESTS:

        sage: s = SymmetricFunctions(QQ).s()
        sage: a = s([8,8])
        sage: a.itensor(a)
        s[4, 4, 4, 4] + s[5, 5, 3, 3] + s[5, 5, 5, 1] + s[6, 4, 4, 2]
         + s[6, 6, 2, 2] + s[6, 6, 4] + s[7, 3, 3, 3] + s[7, 5, 3, 1]
         + s[7, 7, 1, 1] + s[8, 4, 2, 2] + s[8, 4, 4] + s[8, 6, 2]
         + s[8, 8] + s[9, 3, 3, 1] + s[9, 5, 1, 1] + s[10, 2, 2, 2]
         + s[10, 4, 2] + s[10, 6] + s[11, 3, 1, 1] + s[12, 2, 2]
         + s[12, 4] + s[13, 1, 1, 1] + s[14, 2] + s[16]
        sage: s[8].itensor(s[7])
        0
        sage: s(0).itensor(s(0))
        0
        sage: s(1).itensor(s(0))
        0
        sage: s(0).itensor(s(1))
        0
        sage: s(1).itensor(s(1))
        s[]

    Same over the ring of integers:

        sage: s = SymmetricFunctions(ZZ).s()
        sage: a = s([8,8])
        sage: a.itensor(a)
        s[4, 4, 4, 4] + s[5, 5, 3, 3] + s[5, 5, 5, 1] + s[6, 4, 4, 2]
         + s[6, 6, 2, 2] + s[6, 6, 4] + s[7, 3, 3, 3] + s[7, 5, 3, 1]
         + s[7, 7, 1, 1] + s[8, 4, 2, 2] + s[8, 4, 4] + s[8, 6, 2]
         + s[8, 8] + s[9, 3, 3, 1] + s[9, 5, 1, 1] + s[10, 2, 2, 2]
         + s[10, 4, 2] + s[10, 6] + s[11, 3, 1, 1] + s[12, 2, 2]
         + s[12, 4] + s[13, 1, 1, 1] + s[14, 2] + s[16]
        sage: s[8].itensor(s[7])
        0
        sage: s(0).itensor(s(0))
        0
        sage: s(1).itensor(s(0))
        0
        sage: s(0).itensor(s(1))
        0
        sage: s(1).itensor(s(1))
        s[]

    Theorem 2.1 in Bessenrodt, van Willigenburg, Arxiv 1105.3170v2:

        sage: s = SymmetricFunctions(ZZ).s()
        sage: all( all( max( r[0] for r in s(p).itensor(s(q)).monomial_coefficients().keys() )
        ....:           == sum( min(p[i], q.get_part(i)) for i in range(len(p)) )
        ....:           for p in Partitions(4) )
        ....:      for q in Partitions(4) )
        True
        sage: all( all( max( len(r) for r in s(p).itensor(s(q)).monomial_coefficients().keys() )
        ....:           == sum( min(p[i], q.conjugate().get_part(i)) for i in range(len(p)) )
        ....:           for p in Partitions(4) )
        ....:      for q in Partitions(4) )
        True

    Check that the basis and ground ring of `self` are preserved:

        sage: F = CyclotomicField(12)
        sage: s = SymmetricFunctions(F).s()
        sage: e = SymmetricFunctions(F).e()
        sage: e[3].itensor(s[3])
        e[3]
        sage: s[3].itensor(e[3])
        s[1, 1, 1]
        sage: parent(e[3].itensor(s[3]))
        Symmetric Functions over Cyclotomic Field of order 12 and degree 4 in the elementary basis
        sage: parent(s[3].itensor(e[3]))
        Symmetric Functions over Cyclotomic Field of order 12 and degree 4 in the Schur basis

    Note

    The currently existing implementation of this function is technically unsatisfactory. It distinguishes the case when the base ring is a \(\QQ\)-algebra (in which case the Kronecker product can be easily computed using the power sum basis) from the case where it isn’t. In the latter, it does a computation using universal coefficients, again distinguishing the case when it is able to compute the “corresponding” basis of the symmetric function algebra over \(\QQ\) (using the `corresponding_basis_over` hack) from the case when it isn’t (in which case it transforms everything into the Schur basis, which is slow).

  `kronecker_coproduct`()

    Return the inner coproduct of `self` in the basis of `self`.

    The inner coproduct (also known as the Kronecker coproduct, as the internal coproduct, or as the second comultiplication on the ring of symmetric functions) is a ring homomorphism \(\Delta^\times\) from the ring of symmetric functions to the tensor product (over the base ring) of this ring with itself. It is uniquely characterized by the formula

    \[\Delta^{\times}(h_n) = \sum_{\lambda \vdash n} s_{\lambda} \otimes s_{\lambda} = \sum_{\lambda \vdash n} h_{\lambda} \otimes m_{\lambda} = \sum_{\lambda \vdash n} m_{\lambda} \otimes h_{\lambda},\]

    where \(\lambda \vdash n\) means \(\lambda\) is a partition of \(n\), and \(n\) is any nonnegative integer. It also satisfies

    \[\Delta^\times (p_n) = p_n \otimes p_n\]

    for any positive integer \(n\). If the base ring is a \(\QQ\)-algebra, it also satisfies

    \[\Delta^{\times}(h_n) = \sum_{\lambda \vdash n} z_{\lambda}^{-1} p_{\lambda} \otimes p_{\lambda},\]

    where

    \[z_{\lambda} = \prod_{i=1}^\infty i^{m_i(\lambda)} m_i(\lambda)!\]

    with \(m_i(\lambda)\) meaning the number of appearances of \(i\) in \(\lambda\) (see `zee()`).

    The method `kronecker_coproduct()` is a synonym of `internal_coproduct()`.

    EXAMPLES:

        sage: s = SymmetricFunctions(ZZ).s()
        sage: a = s([2,1])
        sage: a.internal_coproduct()
        s[1, 1, 1] # s[2, 1] + s[2, 1] # s[1, 1, 1] + s[2, 1] # s[2, 1] + s[2, 1] # s[3] + s[3] # s[2, 1]

        sage: e = SymmetricFunctions(QQ).e()
        sage: b = e([2])
        sage: b.internal_coproduct()
        e[1, 1] # e[2] + e[2] # e[1, 1] - 2*e[2] # e[2]

    The internal coproduct is adjoint to the internal product with respect to the Hall inner product: Any three symmetric functions \(f\), \(g\) and \(h\) satisfy \(\langle f * g, h \rangle = \sum_i \langle f, h^{\prime}_i \rangle \langle g, h^{\prime\prime}_i \rangle\), where we write \(\Delta^{\times}(h)\) as \(\sum_i h^{\prime}_i \otimes h^{\prime\prime}_i\). Let us check this in degree \(4\):

        sage: e = SymmetricFunctions(FiniteField(29)).e()
        sage: s = SymmetricFunctions(FiniteField(29)).s()
        sage: m = SymmetricFunctions(FiniteField(29)).m()
        sage: def tensor_incopr(f, g, h):  # computes \sum_i \left< f, h'_i \right> \left< g, h''_i \right>
        ....:     result = h.base_ring().zero()
        ....:     for partition_pair, coeff in h.internal_coproduct():
        ....:         result += coeff * h.parent()(f).scalar(partition_pair[0]) * h.parent()(g).scalar(partition_pair[1])
        ....:     return result
        sage: all( all( all( tensor_incopr(e[u], s[v], m[w]) == (e[u].itensor(s[v])).scalar(m[w])  # long time (10s on sage.math, 2013)
        ....:                for w in Partitions(5) )
        ....:           for v in Partitions(2) )
        ....:      for u in Partitions(3) )
        True

    Let us check the formulas for \(\Delta^{\times}(h_n)\) and \(\Delta^{\times}(p_n)\) given in the description of this method:

        sage: e = SymmetricFunctions(QQ).e()
        sage: p = SymmetricFunctions(QQ).p()
        sage: h = SymmetricFunctions(QQ).h()
        sage: s = SymmetricFunctions(QQ).s()
        sage: all( s(h([n])).internal_coproduct() == sum([tensor([s(lam), s(lam)]) for lam in Partitions(n)])
        ....:      for n in range(6) )
        True
        sage: all( h([n]).internal_coproduct() == sum([tensor([h(lam), h(m(lam))]) for lam in Partitions(n)])
        ....:      for n in range(6) )
        True
        sage: all( factorial(n) * h([n]).internal_coproduct()
        ....:      == sum([lam.conjugacy_class_size() * tensor([h(p(lam)), h(p(lam))])
        ....:              for lam in Partitions(n)])
        ....:      for n in range(6) )
        True

    TESTS:

        sage: s = SymmetricFunctions(QQ).s()
        sage: s([]).internal_coproduct()
        s[] # s[]

  `kronecker_product`(x)

    Return the internal (tensor) product of `self` and `x` in the basis of `self`.

    The internal tensor product can be defined as the linear extension of the definition on power sums \(p_{\lambda} \ast p_{\mu} = \delta_{\lambda,\mu} z_{\lambda} p_{\lambda}\), where \(z_{\lambda} = (1^{r_1} r_1!) (2^{r_2} r_2!) \cdots\) for \(\lambda = (1^{r_1} 2^{r_2} \cdots )\) and where \(\ast\) denotes the internal tensor product. The internal tensor product is also known as the Kronecker product, or as the second multiplication on the ring of symmetric functions.

    Note that the internal product of any two homogeneous symmetric functions of equal degrees is a homogeneous symmetric function of the same degree. On the other hand, the internal product of two homogeneous symmetric functions of distinct degrees is \(0\).

    Note

    The internal product is sometimes referred to as “inner product” in the literature, but unfortunately this name is shared by a different operation, namely the Hall inner product (see `scalar()`).

    INPUT:

    - `x` – element of the ring of symmetric functions over the same base ring as `self`

    OUTPUT:

    - the internal product of `self` with `x` (an element of the ring of symmetric functions in the same basis as `self`)

    The methods `itensor()`, `internal_product()`, `kronecker_product()`, `inner_tensor()` are all synonyms.

    EXAMPLES:

        sage: s = SymmetricFunctions(QQ).s()
        sage: a = s([2,1])
        sage: b = s([3])
        sage: a.itensor(b)
        s[2, 1]
        sage: c = s([3,2,1])
        sage: c.itensor(c)
        s[1, 1, 1, 1, 1, 1] + 2*s[2, 1, 1, 1, 1] + 3*s[2, 2, 1, 1] + 2*s[2, 2, 2]
         + 4*s[3, 1, 1, 1] + 5*s[3, 2, 1] + 2*s[3, 3] + 4*s[4, 1, 1]
         + 3*s[4, 2] + 2*s[5, 1] + s[6]

    There are few quantitative results pertaining to Kronecker products in general, which makes their computation so difficult. Let us test a few of them in different bases.

    The Kronecker product of any homogeneous symmetric function \(f\) of degree \(n\) with the \(n\)-th complete homogeneous symmetric function `h[n]` (a.k.a. `s[n]`) is \(f\):

        sage: h = SymmetricFunctions(ZZ).h()
        sage: all( h([5]).itensor(h(p)) == h(p) for p in Partitions(5) )
        True

    The Kronecker product of a Schur function \(s_{\lambda}\) with the \(n\)-th elementary symmetric function `e[n]`, where \(n = \left| \lambda \right|\), is \(s_{\lambda'}\) (where \(\lambda'\) is the conjugate partition of \(\lambda\)):

        sage: F = CyclotomicField(12)
        sage: s = SymmetricFunctions(F).s()
        sage: e = SymmetricFunctions(F).e()
        sage: all( e([5]).itensor(s(p)) == s(p.conjugate()) for p in Partitions(5) )
        True

    The Kronecker product is commutative:

        sage: e = SymmetricFunctions(FiniteField(19)).e()
        sage: m = SymmetricFunctions(FiniteField(19)).m()
        sage: all( all( e(p).itensor(m(q)) == m(q).itensor(e(p)) for q in Partitions(4) )
        ....:      for p in Partitions(4) )
        True

        sage: F = FractionField(QQ['q','t'])
        sage: mq = SymmetricFunctions(F).macdonald().Q()
        sage: mh = SymmetricFunctions(F).macdonald().H()
        sage: all( all( mq(p).itensor(mh(r)) == mh(r).itensor(mq(p)) for r in Partitions(4) )
        ....:      for p in Partitions(3) )   # long time
        True

    Let us check (on examples) Proposition 5.2 of Gelfand, Krob, Lascoux, Leclerc, Retakh, Thibon, “Noncommutative symmetric functions”, Arxiv hep-th/9407124, for \(r = 2\):

        sage: e = SymmetricFunctions(FiniteField(29)).e()
        sage: s = SymmetricFunctions(FiniteField(29)).s()
        sage: m = SymmetricFunctions(FiniteField(29)).m()
        sage: def tensor_copr(u, v, w):  # computes \mu ((u \otimes v) * \Delta(w)) with
        ....:                            # * meaning Kronecker product and \mu meaning the
        ....:                            # usual multiplication.
        ....:     result = w.parent().zero()
        ....:     for partition_pair, coeff in w.coproduct():
        ....:         result += coeff * w.parent()(u).itensor(partition_pair[0]) * w.parent()(v).itensor(partition_pair[1])
        ....:     return result
        sage: all( all( all( tensor_copr(e[u], s[v], m[w]) == (e[u] * s[v]).itensor(m[w])
        ....:                for w in Partitions(5) )
        ....:           for v in Partitions(2) )
        ....:      for u in Partitions(3) )   # long time
        True

    Some examples from Briand, Orellana, Rosas, “The stability of the Kronecker products of Schur functions.” Arxiv 0907.4652:

        sage: s = SymmetricFunctions(ZZ).s()
        sage: s[2,2].itensor(s[2,2])
        s[1, 1, 1, 1] + s[2, 2] + s[4]
        sage: s[3,2].itensor(s[3,2])
        s[2, 1, 1, 1] + s[2, 2, 1] + s[3, 1, 1] + s[3, 2] + s[4, 1] + s[5]
        sage: s[4,2].itensor(s[4,2])
        s[2, 2, 2] + s[3, 1, 1, 1] + 2*s[3, 2, 1] + s[4, 1, 1] + 2*s[4, 2] + s[5, 1] + s[6]

    An example from p. 220 of Thibon, “Hopf algebras of symmetric functions and tensor products of symmetric group representations”, International Journal of Algebra and Computation, 1991:

        sage: s = SymmetricFunctions(QQbar).s()
        sage: s[2,1].itensor(s[2,1])
        s[1, 1, 1] + s[2, 1] + s[3]

    TESTS:

        sage: s = SymmetricFunctions(QQ).s()
        sage: a = s([8,8])
        sage: a.itensor(a)
        s[4, 4, 4, 4] + s[5, 5, 3, 3] + s[5, 5, 5, 1] + s[6, 4, 4, 2]
         + s[6, 6, 2, 2] + s[6, 6, 4] + s[7, 3, 3, 3] + s[7, 5, 3, 1]
         + s[7, 7, 1, 1] + s[8, 4, 2, 2] + s[8, 4, 4] + s[8, 6, 2]
         + s[8, 8] + s[9, 3, 3, 1] + s[9, 5, 1, 1] + s[10, 2, 2, 2]
         + s[10, 4, 2] + s[10, 6] + s[11, 3, 1, 1] + s[12, 2, 2]
         + s[12, 4] + s[13, 1, 1, 1] + s[14, 2] + s[16]
        sage: s[8].itensor(s[7])
        0
        sage: s(0).itensor(s(0))
        0
        sage: s(1).itensor(s(0))
        0
        sage: s(0).itensor(s(1))
        0
        sage: s(1).itensor(s(1))
        s[]

    Same over the ring of integers:

        sage: s = SymmetricFunctions(ZZ).s()
        sage: a = s([8,8])
        sage: a.itensor(a)
        s[4, 4, 4, 4] + s[5, 5, 3, 3] + s[5, 5, 5, 1] + s[6, 4, 4, 2]
         + s[6, 6, 2, 2] + s[6, 6, 4] + s[7, 3, 3, 3] + s[7, 5, 3, 1]
         + s[7, 7, 1, 1] + s[8, 4, 2, 2] + s[8, 4, 4] + s[8, 6, 2]
         + s[8, 8] + s[9, 3, 3, 1] + s[9, 5, 1, 1] + s[10, 2, 2, 2]
         + s[10, 4, 2] + s[10, 6] + s[11, 3, 1, 1] + s[12, 2, 2]
         + s[12, 4] + s[13, 1, 1, 1] + s[14, 2] + s[16]
        sage: s[8].itensor(s[7])
        0
        sage: s(0).itensor(s(0))
        0
        sage: s(1).itensor(s(0))
        0
        sage: s(0).itensor(s(1))
        0
        sage: s(1).itensor(s(1))
        s[]

    Theorem 2.1 in Bessenrodt, van Willigenburg, Arxiv 1105.3170v2:

        sage: s = SymmetricFunctions(ZZ).s()
        sage: all( all( max( r[0] for r in s(p).itensor(s(q)).monomial_coefficients().keys() )
        ....:           == sum( min(p[i], q.get_part(i)) for i in range(len(p)) )
        ....:           for p in Partitions(4) )
        ....:      for q in Partitions(4) )
        True
        sage: all( all( max( len(r) for r in s(p).itensor(s(q)).monomial_coefficients().keys() )
        ....:           == sum( min(p[i], q.conjugate().get_part(i)) for i in range(len(p)) )
        ....:           for p in Partitions(4) )
        ....:      for q in Partitions(4) )
        True

    Check that the basis and ground ring of `self` are preserved:

        sage: F = CyclotomicField(12)
        sage: s = SymmetricFunctions(F).s()
        sage: e = SymmetricFunctions(F).e()
        sage: e[3].itensor(s[3])
        e[3]
        sage: s[3].itensor(e[3])
        s[1, 1, 1]
        sage: parent(e[3].itensor(s[3]))
        Symmetric Functions over Cyclotomic Field of order 12 and degree 4 in the elementary basis
        sage: parent(s[3].itensor(e[3]))
        Symmetric Functions over Cyclotomic Field of order 12 and degree 4 in the Schur basis

    Note

    The currently existing implementation of this function is technically unsatisfactory. It distinguishes the case when the base ring is a \(\QQ\)-algebra (in which case the Kronecker product can be easily computed using the power sum basis) from the case where it isn’t. In the latter, it does a computation using universal coefficients, again distinguishing the case when it is able to compute the “corresponding” basis of the symmetric function algebra over \(\QQ\) (using the `corresponding_basis_over` hack) from the case when it isn’t (in which case it transforms everything into the Schur basis, which is slow).

  `left_padded_kronecker_product`(x)

    Return the left-padded Kronecker product of `self` and `x` in the basis of `self`.

    The left-padded Kronecker product is a bilinear map mapping two symmetric functions to another, not necessarily preserving degree. It can be defined as follows: Let \(*\) denote the Kronecker product (`itensor()`) on the space of symmetric functions. For any partitions \(\alpha\), \(\beta\), \(\gamma\), let \(g^{\gamma}_{\alpha, \beta}\) denote the coefficient of the complete homogeneous symmetric function \(h_{\gamma}\) in the Kronecker product \(h_{\alpha} * h_{\beta}\). For every partition \(\lambda = (\lambda_1, \lambda_2, \lambda_3, \ldots)\) and every integer \(n > \left| \lambda \right| + \lambda_1\), let \(\lambda[n]\) denote the \(n\)-completion of \(\lambda\) (this is the partition \((n - \left| \lambda \right|, \lambda_1, \lambda_2, \lambda_3, \ldots)\); see `t_completion()`). Then, for any partitions \(\alpha\) and \(\beta\) and every integer \(n \geq \left|\alpha\right| + \left|\beta\right| + \alpha_1 + \beta_1\), we can write the Kronecker product \(h_{\alpha[n]} * h_{\beta[n]}\) in the form

    \[h_{\alpha[n]} * h_{\beta[n]} = \sum_{\gamma} g^{\gamma[n]}_{\alpha[n], \beta[n]} h_{\gamma[n]}\]

    with \(\gamma\) ranging over all partitions. The coefficients \(g^{\gamma[n]}_{\alpha[n], \beta[n]}\) are independent on \(n\). These coefficients \(g^{\gamma[n]}_{\alpha[n], \beta[n]}\) are denoted by \(\overline{g}^{\gamma}_{\alpha, \beta}\), and the symmetric function

    \[\sum_{\gamma} \overline{g}^{\gamma}_{\alpha, \beta} h_{\gamma}\]

    is said to be the left-padded Kronecker product of \(h_{\alpha}\) and \(h_{\beta}\). By bilinearity, this extends to a definition of a left-padded Kronecker product of any two symmetric functions.

    This notion of left-padded Kronecker product can be lifted to the non-commutative symmetric functions (`left_padded_kronecker_product()`).

    Warning

    Don’t mistake this product for the reduced Kronecker product (`reduced_kronecker_product()`), which uses the Schur functions instead of the complete homogeneous functions in its definition.

    INPUT:

    - `x` – element of the ring of symmetric functions over the same base ring as `self`

    OUTPUT:

    - the left-padded Kronecker product of `self` with `x` (an element of the ring of symmetric functions in the same basis as `self`)

    EXAMPLES:

        sage: Sym = SymmetricFunctions(QQ)
        sage: h = Sym.h()
        sage: h[2,1].left_padded_kronecker_product(h[3])
        h[1, 1, 1, 1] + h[2, 1] + h[2, 1, 1] + h[2, 1, 1, 1] + h[2, 2, 1] + h[3, 2, 1]
        sage: h[2,1].left_padded_kronecker_product(h[1])
        h[1, 1, 1] + h[2, 1] + h[2, 1, 1]
        sage: h[1].left_padded_kronecker_product(h[2,1])
        h[1, 1, 1] + h[2, 1] + h[2, 1, 1]
        sage: h[1,1].left_padded_kronecker_product(h[2])
        h[1, 1] + 2*h[1, 1, 1] + h[2, 1, 1]
        sage: h[1].left_padded_kronecker_product(h[2,1,1])
        h[1, 1, 1, 1] + 2*h[2, 1, 1] + h[2, 1, 1, 1]
        sage: h[2].left_padded_kronecker_product(h[3])
        h[2, 1] + h[2, 1, 1] + h[3, 2]

    Taking the left-padded Kronecker product with \(1 = h_{\empty}\) is the identity map on the ring of symmetric functions:

        sage: all( h[Partition([])].left_padded_kronecker_product(h[lam])
        ....:      == h[lam] for i in range(4)
        ....:      for lam in Partitions(i) )
        True

    Here is a rule for the left-padded Kronecker product of \(h_1\) (this is the same as \(h_{(1)}\)) with any complete homogeneous function: Let \(\lambda\) be a partition. Then, the left-padded Kronecker product of \(h_1\) and \(h_{\lambda}\) is \(\sum_{\mu} a_{\mu} h_{\mu}\), where the sum runs over all partitions \(\mu\), and the coefficient \(a_{\mu}\) is defined as the number of ways to obtain \(\mu\) from \(\lambda\) by one of the following two operations:

    - Insert a \(1\) into \(\lambda\).
    - Subtract \(1\) from one of the entries of \(\lambda\) (and remove the entry if it thus becomes \(0\)), and insert a \(1\) into \(\lambda\).

    We check this for partitions of size \(\leq 4\):

        sage: def mults1(I):
        ....:     # Left-padded Kronecker multiplication by h[1].
        ....:     res = h[I[:] + [1]]
        ....:     for k in range(len(I)):
        ....:         I2 = I[:]
        ....:         if I2[k] == 1:
        ....:             I2 = I2[:k] + I2[k+1:]
        ....:         else:
        ....:             I2[k] -= 1
        ....:         res += h[sorted(I2 + [1], reverse=True)]
        ....:     return res
        sage: all( mults1(I) == h[1].left_padded_kronecker_product(h[I])
        ....:                == h[I].left_padded_kronecker_product(h[1])
        ....:      for i in range(5) for I in Partitions(i) )
        True

    The left-padded Kronecker product is commutative:

        sage: all( h[lam].left_padded_kronecker_product(h[mu])
        ....:      == h[mu].left_padded_kronecker_product(h[lam])
        ....:      for lam in Partitions(3) for mu in Partitions(3) )
        True

    TESTS:

        sage: h = SymmetricFunctions(QQ).h()
        sage: (2*h([])).left_padded_kronecker_product(3*h([]))
        6*h[]

    Different bases and base rings:

        sage: h = SymmetricFunctions(ZZ).h()
        sage: e = SymmetricFunctions(ZZ).e()
        sage: h(e[2].left_padded_kronecker_product(h[2]))
        h[1, 1] + h[1, 1, 1] - h[2] + h[2, 1, 1] - h[2, 2]

        sage: F = CyclotomicField(12)
        sage: s = SymmetricFunctions(F).s()
        sage: e = SymmetricFunctions(F).e()
        sage: v = e[2].left_padded_kronecker_product(e[2]); v
        e[1, 1] + e[1, 1, 1] + (-1)*e[2] + e[2, 2]
        sage: parent(v)
        Symmetric Functions over Cyclotomic Field of order 12 and degree 4 in the elementary basis

        sage: s = SymmetricFunctions(ZZ).s()
        sage: v = s[1].left_padded_kronecker_product(s[1]); parent(v)
        Symmetric Functions over Integer Ring in the Schur basis

  `nabla`(q=None, t=None, power=1)

    Return the value of the nabla operator applied to `self`.

    The eigenvectors of the nabla operator are the Macdonald polynomials in the Ht basis.

    If the parameter `power` is an integer then it calculates nabla to that integer. The default value of `power` is 1.

    INPUT:

    - `q`, `t` – optional parameters (default: `None`, in which case `q` and `t` are used)
    - `power` – (default: `1`) an integer indicating how many times to apply the operator \(\nabla\). Negative values of `power` indicate powers of \(\nabla^{-1}\).

    EXAMPLES:

        sage: Sym = SymmetricFunctions(FractionField(QQ['q','t']))
        sage: p = Sym.power()
        sage: p([1,1]).nabla()
        (-1/2*q*t+1/2*q+1/2*t+1/2)*p[1, 1] + (1/2*q*t-1/2*q-1/2*t+1/2)*p[2]
        sage: p([2,1]).nabla(q=1)
        (-t-1)*p[1, 1, 1] + t*p[2, 1]
        sage: p([2]).nabla(q=1)*p([1]).nabla(q=1)
        (-t-1)*p[1, 1, 1] + t*p[2, 1]
        sage: s = Sym.schur()
        sage: s([2,1]).nabla()
        (-q^3*t-q^2*t^2-q*t^3)*s[1, 1, 1] + (-q^2*t-q*t^2)*s[2, 1]
        sage: s([1,1,1]).nabla()
        (q^3+q^2*t+q*t^2+t^3+q*t)*s[1, 1, 1] + (q^2+q*t+t^2+q+t)*s[2, 1] + s[3]
        sage: s([1,1,1]).nabla(t=1)
        (q^3+q^2+2*q+1)*s[1, 1, 1] + (q^2+2*q+2)*s[2, 1] + s[3]
        sage: s(0).nabla()
        0
        sage: s(1).nabla()
        s[]
        sage: s([2,1]).nabla(power=-1)
        ((-q-t)/(q^2*t^2))*s[2, 1] + ((-q^2-q*t-t^2)/(q^3*t^3))*s[3]
        sage: (s([2])+s([3])).nabla()
        (-q*t)*s[1, 1] + (q^3*t^2+q^2*t^3)*s[1, 1, 1] + q^2*t^2*s[2, 1]

  `omega`()

    Return the image of `self` under the omega automorphism.

    The omega automorphism is defined to be the unique algebra endomorphism \(\omega\) of the ring of symmetric functions that satisfies \(\omega(e_k) = h_k\) for all positive integers \(k\) (where \(e_k\) stands for the \(k\)-th elementary symmetric function, and \(h_k\) stands for the \(k\)-th complete homogeneous symmetric function). It furthermore is a Hopf algebra endomorphism and an involution, and it is also known as the omega involution. It sends the power-sum symmetric function \(p_k\) to \((-1)^{k-1} p_k\) for every positive integer \(k\).

    The images of some bases under the omega automorphism are given by

    \[\omega(e_{\lambda}) = h_{\lambda}, \qquad \omega(h_{\lambda}) = e_{\lambda}, \qquad \omega(p_{\lambda}) = (-1)^{|\lambda| - \ell(\lambda)} p_{\lambda}, \qquad \omega(s_{\lambda}) = s_{\lambda^{\prime}},\]

    where \(\lambda\) is any partition, where \(\ell(\lambda)\) denotes the length (`length()`) of the partition \(\lambda\), where \(\lambda^{\prime}\) denotes the conjugate partition (`conjugate()`) of \(\lambda\), and where the usual notations for bases are used (\(e\) = elementary, \(h\) = complete homogeneous, \(p\) = powersum, \(s\) = Schur).

    The default implementation converts to the Schur basis, then performs the automorphism and changes back.

    `omega_involution()` is a synonym for the `omega()` method.

    EXAMPLES:

        sage: J = SymmetricFunctions(QQ).jack(t=1).P()
        sage: a = J([2,1]) + J([1,1,1])
        sage: a.omega()
        JackP[2, 1] + JackP[3]
        sage: J(0).omega()
        0
        sage: J(1).omega()
        JackP[]

    The forgotten symmetric functions are the images of the monomial symmetric functions under omega:

        sage: Sym = SymmetricFunctions(ZZ)
        sage: m = Sym.m()
        sage: f = Sym.f()
        sage: all( f(lam) == m(lam).omega() for lam in Partitions(3) )
        True
        sage: all( m(lam) == f(lam).omega() for lam in Partitions(3) )
        True

  `omega_involution`()

    Return the image of `self` under the omega automorphism.

    The omega automorphism is defined to be the unique algebra endomorphism \(\omega\) of the ring of symmetric functions that satisfies \(\omega(e_k) = h_k\) for all positive integers \(k\) (where \(e_k\) stands for the \(k\)-th elementary symmetric function, and \(h_k\) stands for the \(k\)-th complete homogeneous symmetric function). It furthermore is a Hopf algebra endomorphism and an involution, and it is also known as the omega involution. It sends the power-sum symmetric function \(p_k\) to \((-1)^{k-1} p_k\) for every positive integer \(k\).

    The images of some bases under the omega automorphism are given by

    \[\omega(e_{\lambda}) = h_{\lambda}, \qquad \omega(h_{\lambda}) = e_{\lambda}, \qquad \omega(p_{\lambda}) = (-1)^{|\lambda| - \ell(\lambda)} p_{\lambda}, \qquad \omega(s_{\lambda}) = s_{\lambda^{\prime}},\]

    where \(\lambda\) is any partition, where \(\ell(\lambda)\) denotes the length (`length()`) of the partition \(\lambda\), where \(\lambda^{\prime}\) denotes the conjugate partition (`conjugate()`) of \(\lambda\), and where the usual notations for bases are used (\(e\) = elementary, \(h\) = complete homogeneous, \(p\) = powersum, \(s\) = Schur).

    The default implementation converts to the Schur basis, then performs the automorphism and changes back.

    `omega_involution()` is a synonym for the `omega()` method.

    EXAMPLES:

        sage: J = SymmetricFunctions(QQ).jack(t=1).P()
        sage: a = J([2,1]) + J([1,1,1])
        sage: a.omega()
        JackP[2, 1] + JackP[3]
        sage: J(0).omega()
        0
        sage: J(1).omega()
        JackP[]

    The forgotten symmetric functions are the images of the monomial symmetric functions under omega:

        sage: Sym = SymmetricFunctions(ZZ)
        sage: m = Sym.m()
        sage: f = Sym.f()
        sage: all( f(lam) == m(lam).omega() for lam in Partitions(3) )
        True
        sage: all( m(lam) == f(lam).omega() for lam in Partitions(3) )
        True

  `omega_qt`(q=None, t=None)

    Return the image of `self` under the \(q,t\)-deformed omega automorphism which sends \(p_k\) to \((-1)^{k-1} \cdot \frac{1-q^k}{1-t^k} \cdot p_k\) for all positive integers \(k\).

    In general, this is well-defined outside of the powersum basis only if the base ring is a \(\QQ\)-algebra.

    If \(q = t\), then this is the omega automorphism (`omega()`).

    INPUT:

    - `q`, `t` – parameters (default: `None`, in which case `'q'` and `'t'` are used)

    EXAMPLES:

        sage: QQqt = QQ['q,t'].fraction_field()
        sage: q,t = QQqt.gens()
        sage: p = SymmetricFunctions(QQqt).p()
        sage: p[5].omega_qt()
        ((-q^5+1)/(-t^5+1))*p[5]
        sage: p[5].omega_qt(q,t)
        ((-q^5+1)/(-t^5+1))*p[5]
        sage: p([2]).omega_qt(q,t)
        ((q^2-1)/(-t^2+1))*p[2]
        sage: p([2,1]).omega_qt(q,t)
        ((-q^3+q^2+q-1)/(t^3-t^2-t+1))*p[2, 1]
        sage: p([3,2]).omega_qt(5,q)
        -(2976/(q^5-q^3-q^2+1))*p[3, 2]
        sage: p(0).omega_qt()
        0
        sage: p(1).omega_qt()
        p[]
        sage: H = SymmetricFunctions(QQqt).macdonald().H()
        sage: H([1,1]).omega_qt()
        ((2*q^2-2*q*t-2*q+2*t)/(t^3-t^2-t+1))*McdH[1, 1] + ((q-1)/(t-1))*McdH[2]
        sage: H([1,1]).omega_qt(q,t)
        ((2*q^2-2*q*t-2*q+2*t)/(t^3-t^2-t+1))*McdH[1, 1] + ((q-1)/(t-1))*McdH[2]
        sage: H([1,1]).omega_qt(t,q)
        ((t^3-t^2-t+1)/(q^3-q^2-q+1))*McdH[2]
        sage: Sym = SymmetricFunctions(FractionField(QQ['q','t']))
        sage: S = Sym.macdonald().S()
        sage: S([1,1]).omega_qt()
        ((q^2-q*t-q+t)/(t^3-t^2-t+1))*McdS[1, 1] + ((-q^2*t+q*t+q-1)/(-t^3+t^2+t-1))*McdS[2]
        sage: s = Sym.schur()
        sage: s(S([1,1]).omega_qt())
        s[2]

  `plethysm`(x, include=None, exclude=None)

    Return the outer plethysm of `self` with `x`. This is implemented only over base rings which are \(\QQ\)-algebras. (To compute outer plethysms over general binomial rings, change bases to the fraction field.)

    By default, the degree one elements are taken to be the generators for the `self`‘s base ring. This setting can be modified by specifying the `include` and `exclude` keywords.

    INPUT:

    - `x` – a symmetric function over the same base ring as `self`
    - `include` – a list of variables to be treated as degree one elements instead of the default degree one elements
    - `exclude` – a list of variables to be excluded from the default degree one elements

    EXAMPLES:

        sage: Sym = SymmetricFunctions(QQ)
        sage: s = Sym.s()
        sage: h = Sym.h()
        sage: s ( h([3])( h([2]) ) )
        s[2, 2, 2] + s[4, 2] + s[6]
        sage: p = Sym.p()
        sage: p([3])( s([2,1]) )
        1/3*p[3, 3, 3] - 1/3*p[9]
        sage: e = Sym.e()
        sage: e([3])( e([2]) )
        e[3, 3] + e[4, 1, 1] - 2*e[4, 2] - e[5, 1] + e[6]

        sage: R.<t> = QQ[]
        sage: s = SymmetricFunctions(R).s()
        sage: a = s([3])
        sage: f = t*s([2])
        sage: a(f)
        t^3*s[2, 2, 2] + t^3*s[4, 2] + t^3*s[6]
        sage: f(a)
        t*s[4, 2] + t*s[6]
        sage: s(0).plethysm(s[1])
        0
        sage: s(1).plethysm(s[1])
        s[]
        sage: s(1).plethysm(s(0))
        s[]

    See also

    `frobenius()`

  `reduced_kronecker_product`(x)

    Return the reduced Kronecker product of `self` and `x` in the basis of `self`.

    The reduced Kronecker product is a bilinear map mapping two symmetric functions to another, not necessarily preserving degree. It can be defined as follows: Let \(*\) denote the Kronecker product (`itensor()`) on the space of symmetric functions. For any partitions \(\alpha\), \(\beta\), \(\gamma\), let \(g^{\gamma}_{\alpha, \beta}\) denote the coefficient of the Schur function \(s_{\gamma}\) in the Kronecker product \(s_{\alpha} * s_{\beta}\) (this is called a Kronecker coefficient). For every partition \(\lambda = (\lambda_1, \lambda_2, \lambda_3, \ldots)\) and every integer \(n > \left| \lambda \right| + \lambda_1\), let \(\lambda[n]\) denote the \(n\)-completion of \(\lambda\) (this is the partition \((n - \left| \lambda \right|, \lambda_1, \lambda_2, \lambda_3, \ldots)\); see `t_completion()`). Then, Theorem 1.2 of [BOR09] shows that for any partitions \(\alpha\) and \(\beta\) and every integer \(n \geq \left|\alpha\right| + \left|\beta\right| + \alpha_1 + \beta_1\), we can write the Kronecker product \(s_{\alpha[n]} * s_{\beta[n]}\) in the form

    \[s_{\alpha[n]} * s_{\beta[n]} = \sum_{\gamma} g^{\gamma[n]}_{\alpha[n], \beta[n]} s_{\gamma[n]}\]

    with \(\gamma\) ranging over all partitions. The coefficients \(g^{\gamma[n]}_{\alpha[n], \beta[n]}\) are independent on \(n\). These coefficients \(g^{\gamma[n]}_{\alpha[n], \beta[n]}\) are denoted by \(\overline{g}^{\gamma}_{\alpha, \beta}\), and the symmetric function

    \[\sum_{\gamma} \overline{g}^{\gamma}_{\alpha, \beta} s_{\gamma}\]

    is said to be the reduced Kronecker product of \(s_{\alpha}\) and \(s_{\beta}\). By bilinearity, this extends to a definition of a reduced Kronecker product of any two symmetric functions.

    The definition of the reduced Kronecker product goes back to Murnaghan, and has recently been studied in [BOR09], [BdVO12] and other places (our notation \(\overline{g}^{\gamma}_{\alpha, \beta}\) appears in these two sources).

    INPUT:

    - `x` – element of the ring of symmetric functions over the same base ring as `self`

    OUTPUT:

    - the reduced Kronecker product of `self` with `x` (an element of the ring of symmetric functions in the same basis as `self`)

    EXAMPLES:

    The example from page 2 of [BOR09]:

        sage: Sym = SymmetricFunctions(QQ)
        sage: s = Sym.schur()
        sage: s[2].reduced_kronecker_product(s[2])
        s[] + s[1] + s[1, 1] + s[1, 1, 1] + 2*s[2] + 2*s[2, 1] + s[2, 2] + s[3] + s[3, 1] + s[4]

    Taking the reduced Kronecker product with \(1 = s_{\empty}\) is the identity map on the ring of symmetric functions:

        sage: all( s[Partition([])].reduced_kronecker_product(s[lam])
        ....:      == s[lam] for i in range(4)
        ....:      for lam in Partitions(i) )
        True

    While reduced Kronecker products are hard to compute in general, there is a rule for taking reduced Kronecker products with \(s_1\). Namely, for every partition \(\lambda\), the reduced Kronecker product of \(s_{\lambda}\) with \(s_1\) is \(\sum_{\mu} a_{\mu} s_{\mu}\), where the sum runs over all partitions \(\mu\), and the coefficient \(a_{\mu}\) is defined as the number of ways to obtain \(\mu\) from \(\lambda\) by one of the following three operations:

    - Add an addable cell (`addable_cells()`) to \(\lambda\).
    - Remove a removable cell (`removable_cells()`) from \(\lambda\).
    - First remove a removable cell from \(\lambda\), then add an addable cell to the resulting Young diagram.

    This is, in fact, Proposition 5.15 of [CO10] in an elementary wording. We check this for partitions of size \(\leq 4\):

        sage: def mults1(lam):
        ....:     # Reduced Kronecker multiplication by s[1], according
        ....:     # to [CO10]_.
        ....:     res = s.zero()
        ....:     for mu in lam.up_list():
        ....:         res += s(mu)
        ....:     for mu in lam.down_list():
        ....:         res += s(mu)
        ....:         for nu in mu.up_list():
        ....:             res += s(nu)
        ....:     return res
        sage: all( mults1(lam) == s[1].reduced_kronecker_product(s[lam])
        ....:      for i in range(5) for lam in Partitions(i) )
        True

    Here is the example on page 3 of Christian Gutschwager’s Arxiv 0912.4411v3:

        sage: s[1,1].reduced_kronecker_product(s[2])
        s[1] + 2*s[1, 1] + s[1, 1, 1] + s[2] + 2*s[2, 1] + s[2, 1, 1] + s[3] + s[3, 1]

    Example 39 from F. D. Murnaghan, “The analysis of the Kronecker product of irreducible representations of the symmetric group”, American Journal of Mathematics, Vol. 60, No. 3, Jul. 1938:

        sage: s[3].reduced_kronecker_product(s[2,1])
        s[1] + 2*s[1, 1] + 2*s[1, 1, 1] + s[1, 1, 1, 1] + 2*s[2] + 5*s[2, 1] + 4*s[2, 1, 1]
        + s[2, 1, 1, 1] + 3*s[2, 2] + 2*s[2, 2, 1] + 2*s[3] + 5*s[3, 1] + 3*s[3, 1, 1]
        + 3*s[3, 2] + s[3, 2, 1] + 2*s[4] + 3*s[4, 1] + s[4, 1, 1] + s[4, 2] + s[5]
        + s[5, 1]

    TESTS:

        sage: h = SymmetricFunctions(QQ).h()
        sage: (2*h([])).reduced_kronecker_product(3*h([]))
        6*h[]

    Different bases and base rings:

        sage: h = SymmetricFunctions(ZZ).h()
        sage: e = SymmetricFunctions(ZZ).e()
        sage: h(e[2].reduced_kronecker_product(h[2]))
        h[1] + 2*h[1, 1] + h[1, 1, 1] - h[2] + h[2, 1, 1] - h[2, 2]

        sage: F = CyclotomicField(12)
        sage: s = SymmetricFunctions(F).s()
        sage: e = SymmetricFunctions(F).e()
        sage: v = e[2].reduced_kronecker_product(e[2]); v
        e[] + e[1] + 2*e[1, 1] + e[1, 1, 1] + (-1)*e[2] + e[2, 2]
        sage: parent(v)
        Symmetric Functions over Cyclotomic Field of order 12 and degree 4 in the elementary basis

        sage: s = SymmetricFunctions(ZZ).s()
        sage: v = s[1].reduced_kronecker_product(s[1]); parent(v)
        Symmetric Functions over Integer Ring in the Schur basis

    Todo

    This implementation of the reduced Kronecker product is painfully slow.

  `restrict_degree`(d, exact=True)

    Return the degree `d` component of `self`.

    INPUT:

    - `d` – positive integer, degree of the terms to be returned
    - `exact` – boolean, if `True`, returns the terms of degree exactly `d`, otherwise returns all terms of degree less than or equal to `d`

    OUTPUT:

    - the homogeneous component of `self` of degree `d`

    EXAMPLES:

        sage: s = SymmetricFunctions(QQ).s()
        sage: z = s([4]) + s([2,1]) + s([1,1,1]) + s([1])
        sage: z.restrict_degree(2)
        0
        sage: z.restrict_degree(1)
        s[1]
        sage: z.restrict_degree(3)
        s[1, 1, 1] + s[2, 1]
        sage: z.restrict_degree(3, exact=False)
        s[1] + s[1, 1, 1] + s[2, 1]
        sage: z.restrict_degree(0)
        0

  `restrict_partition_lengths`(l, exact=True)

    Return the terms of `self` labelled by partitions of length `l`.

    INPUT:

    - `l` – nonnegative integer
    - `exact` – boolean, defaulting to `True`

    OUTPUT:

    - if `True`, returns the terms labelled by partitions of length precisely `l`; otherwise returns all terms labelled by partitions of length less than or equal to `l`

    EXAMPLES:

        sage: s = SymmetricFunctions(QQ).s()
        sage: z = s([4]) + s([2,1]) + s([1,1,1]) + s([1])
        sage: z.restrict_partition_lengths(2)
        s[2, 1]
        sage: z.restrict_partition_lengths(0)
        0
        sage: z.restrict_partition_lengths(2, exact = False)
        s[1] + s[2, 1] + s[4]

  `restrict_parts`(n)

    Return the terms of `self` labelled by partitions \(\lambda\) with \(\lambda_1 \leq n\).

    INPUT:

    - `n` – positive integer, to restrict the parts of the partitions of the terms to be returned

    EXAMPLES:

        sage: s = SymmetricFunctions(QQ).s()
        sage: z = s([4]) + s([2,1]) + s([1,1,1]) + s([1])
        sage: z.restrict_parts(2)
        s[1] + s[1, 1, 1] + s[2, 1]
        sage: z.restrict_parts(1)
        s[1] + s[1, 1, 1]

  `scalar`(x, zee=None)

    Return standard scalar product between `self` and `x`.

    INPUT:

    - `x` – element of the ring of symmetric functions over the same base ring as `self`
    - `zee` – an optional function on partitions giving the value for the scalar product between \(p_{\mu}\) and \(p_{\mu}\) (default is to use the standard `zee()` function)

    This is the default implementation that converts both `self` and `x` into either Schur functions (if `zee` is not specified) or power-sum functions (if `zee` is specified) and performs the scalar product in that basis.

    EXAMPLES:

        sage: e = SymmetricFunctions(QQ).e()
        sage: h = SymmetricFunctions(QQ).h()
        sage: m = SymmetricFunctions(QQ).m()
        sage: p4 = Partitions(4)
        sage: matrix([ [e(a).scalar(h(b)) for a in p4] for b in p4])
        [ 0  0  0  0  1]
        [ 0  0  0  1  4]
        [ 0  0  1  2  6]
        [ 0  1  2  5 12]
        [ 1  4  6 12 24]
        sage: matrix([ [h(a).scalar(e(b)) for a in p4] for b in p4])
        [ 0  0  0  0  1]
        [ 0  0  0  1  4]
        [ 0  0  1  2  6]
        [ 0  1  2  5 12]
        [ 1  4  6 12 24]
        sage: matrix([ [m(a).scalar(e(b)) for a in p4] for b in p4])
        [-1  2  1 -3  1]
        [ 0  1  0 -2  1]
        [ 0  0  1 -2  1]
        [ 0  0  0 -1  1]
        [ 0  0  0  0  1]
        sage: matrix([ [m(a).scalar(h(b)) for a in p4] for b in p4])
        [1 0 0 0 0]
        [0 1 0 0 0]
        [0 0 1 0 0]
        [0 0 0 1 0]
        [0 0 0 0 1]

        sage: p = SymmetricFunctions(QQ).p()
        sage: m(p[3,2]).scalar(p[3,2], zee=lambda mu: 2**mu.length())
        4
        sage: m(p[3,2]).scalar(p[2,2,1], lambda mu: 1)
        0
        sage: m[3,2].scalar(h[3,2], zee=lambda mu: 2**mu.length())
        2/3

    TESTS:

        sage: m(1).scalar(h(1))
        1
        sage: m(0).scalar(h(1))
        0
        sage: m(1).scalar(h(0))
        0
        sage: m(0).scalar(h(0))
        0

    Over the integers, too (as long as `zee` is not set):

        sage: Sym = SymmetricFunctions(ZZ)
        sage: m = Sym.m()
        sage: m([2]).scalar(m([2]))
        2

  `scalar_hl`(x, t=None)

    Return the \(t\)-deformed standard Hall-Littlewood scalar product of `self` and `x`.

    INPUT:

    - `x` – element of the ring of symmetric functions over the same base ring as `self`
    - `t` – parameter (default: `None`, in which case `t` is used)

    EXAMPLES:

        sage: s = SymmetricFunctions(QQ).s()
        sage: a = s([2,1])
        sage: sp = a.scalar_t(a); sp
        (-t^2 - 1)/(t^5 - 2*t^4 + t^3 - t^2 + 2*t - 1)
        sage: sp.parent()
        Fraction Field of Univariate Polynomial Ring in t over Rational Field

  `scalar_jack`(x, t=None)

    Return the Jack-scalar product beween `self` and `x`.

    This scalar product is defined so that the power sum elements \(p_{\mu}\) are orthogonal and \(\langle p_{\mu}, p_{\mu} \rangle = z_{\mu} t^{\ell(\mu)}\), where \(\ell(\mu)\) denotes the length of \(\mu\).

    INPUT:

    - `x` – element of the ring of symmetric functions over the same base ring as `self`
    - `t` – an optional parameter (default: `None` in which case `t` is used)

    EXAMPLES:

        sage: p = SymmetricFunctions(QQ['t']).power()
        sage: matrix([[p(mu).scalar_jack(p(nu)) for nu in Partitions(4)] for mu in Partitions(4)])
        [   4*t      0      0      0      0]
        [     0  3*t^2      0      0      0]
        [     0      0  8*t^2      0      0]
        [     0      0      0  4*t^3      0]
        [     0      0      0      0 24*t^4]
        sage: matrix([[p(mu).scalar_jack(p(nu),2) for nu in Partitions(4)] for mu in Partitions(4)])
        [  8   0   0   0   0]
        [  0  12   0   0   0]
        [  0   0  32   0   0]
        [  0   0   0  32   0]
        [  0   0   0   0 384]
        sage: JQ = SymmetricFunctions(QQ['t'].fraction_field()).jack().Q()
        sage: matrix([[JQ(mu).scalar_jack(JQ(nu)) for nu in Partitions(3)] for mu in Partitions(3)])
        [(2*t^2 + 3*t + 1)/(6*t^3)                         0                         0]
        [                        0     (t + 2)/(2*t^3 + t^2)                         0]
        [                        0                         0     6/(t^3 + 3*t^2 + 2*t)]

  `scalar_qt`(x, q=None, t=None)

    Returns the \(q,t\)-deformed standard Hall-Littlewood scalar product of `self` and `x`.

    INPUT:

    - `x` – element of the ring of symmetric functions over the same base ring as `self`
    - `q`, `t` – parameters (default: `None` in which case `q` and `t` are used)

    EXAMPLES:

        sage: s = SymmetricFunctions(QQ).s()
        sage: a = s([2,1])
        sage: sp = a.scalar_qt(a); factor(sp)
        (t - 1)^-3 * (q - 1) * (t^2 + t + 1)^-1 * (q^2*t^2 - q*t^2 + q^2 - 2*q*t + t^2 - q + 1)
        sage: sp.parent()
        Fraction Field of Multivariate Polynomial Ring in q, t over Rational Field
        sage: a.scalar_qt(a,q=0)
        (-t^2 - 1)/(t^5 - 2*t^4 + t^3 - t^2 + 2*t - 1)
        sage: a.scalar_qt(a,t=0)
        -q^3 + 2*q^2 - 2*q + 1
        sage: a.scalar_qt(a,5,7) # q=5 and t=7
        490/1539
        sage: (x,y) = var('x,y')
        sage: a.scalar_qt(a,q=x,t=y)
        1/3*(x^3 - 1)/(y^3 - 1) + 2/3*(x - 1)^3/(y - 1)^3
        sage: Rn = QQ['q','t','y','z'].fraction_field()
        sage: (q,t,y,z) = Rn.gens()
        sage: Mac = SymmetricFunctions(Rn).macdonald(q=y,t=z)
        sage: a = Mac._sym.schur()([2,1])
        sage: factor(Mac.P()(a).scalar_qt(Mac.Q()(a),q,t))
        (t - 1)^-3 * (q - 1) * (t^2 + t + 1)^-1 * (q^2*t^2 - q*t^2 + q^2 - 2*q*t + t^2 - q + 1)
        sage: factor(Mac.P()(a).scalar_qt(Mac.Q()(a)))
        (z - 1)^-3 * (y - 1) * (z^2 + z + 1)^-1 * (y^2*z^2 - y*z^2 + y^2 - 2*y*z + z^2 - y + 1)

  `scalar_t`(x, t=None)

    Return the \(t\)-deformed standard Hall-Littlewood scalar product of `self` and `x`.

    INPUT:

    - `x` – element of the ring of symmetric functions over the same base ring as `self`
    - `t` – parameter (default: `None`, in which case `t` is used)

    EXAMPLES:

        sage: s = SymmetricFunctions(QQ).s()
        sage: a = s([2,1])
        sage: sp = a.scalar_t(a); sp
        (-t^2 - 1)/(t^5 - 2*t^4 + t^3 - t^2 + 2*t - 1)
        sage: sp.parent()
        Fraction Field of Univariate Polynomial Ring in t over Rational Field

  `skew_by`(x)

    Return the result of skewing `self` by `x`. (Skewing by `x` is the endomorphism (as additive group) of the ring of symmetric functions adjoint to multiplication by `x` with respect to the Hall inner product.)

    INPUT:

    - `x` – element of the ring of symmetric functions over the same base ring as `self`

    EXAMPLES:

        sage: s = SymmetricFunctions(QQ).s()
        sage: s([3,2]).skew_by(s([2]))
        s[2, 1] + s[3]
        sage: s([3,2]).skew_by(s([1,1,1]))
        0
        sage: s([3,2,1]).skew_by(s([2,1]))
        s[1, 1, 1] + 2*s[2, 1] + s[3]

        sage: p = SymmetricFunctions(QQ).powersum()
        sage: p([4,3,3,2,2,1]).skew_by(p([2,1]))
        4*p[4, 3, 3, 2]
        sage: zee = sage.combinat.sf.sfa.zee
        sage: zee([4,3,3,2,2,1])/zee([4,3,3,2])
        4
        sage: s(0).skew_by(s([1]))
        0
        sage: s(1).skew_by(s([1]))
        0
        sage: s([]).skew_by(s([]))
        s[]
        sage: s([]).skew_by(s[1])
        0

    TESTS:

        sage: f=s[3,2]
        sage: f.skew_by([1])
        Traceback (most recent call last):
        ...
        ValueError: x needs to be a symmetric function

  `theta`(a)

    Return the image of `self` under the theta endomorphism which sends \(p_k\) to \(a \cdot p_k\) for every positive integer \(k\).

    In general, this is well-defined outside of the powersum basis only if the base ring is a \(\QQ\)-algebra.

    INPUT:

    - `a` – an element of the base ring

    EXAMPLES:

        sage: s = SymmetricFunctions(QQ).s()
        sage: s([2,1]).theta(2)
        2*s[1, 1, 1] + 6*s[2, 1] + 2*s[3]
        sage: p = SymmetricFunctions(QQ).p()
        sage: p([2]).theta(2)
        2*p[2]
        sage: p(0).theta(2)
        0
        sage: p(1).theta(2)
        p[]

  `theta_qt`(q=None, t=None)

    Return the image of `self` under the \(q,t\)-deformed theta endomorphism which sends \(p_k\) to \(\frac{1-q^k}{1-t^k} \cdot p_k\) for all positive integers \(k\).

    In general, this is well-defined outside of the powersum basis only if the base ring is a \(\QQ\)-algebra.

    INPUT:

    - `q`, `t` – parameters (default: `None`, in which case ‘q’ and ‘t’ are used)

    EXAMPLES:

        sage: QQqt = QQ['q,t'].fraction_field()
        sage: q,t = QQqt.gens()
        sage: p = SymmetricFunctions(QQqt).p()
        sage: p([2]).theta_qt(q,t)
        ((-q^2+1)/(-t^2+1))*p[2]
        sage: p([2,1]).theta_qt(q,t)
        ((q^3-q^2-q+1)/(t^3-t^2-t+1))*p[2, 1]
        sage: p(0).theta_qt(q=1,t=3)
        0
        sage: p([2,1]).theta_qt(q=2,t=3)
        3/16*p[2, 1]
        sage: s = p.realization_of().schur()
        sage: s([3]).theta_qt(q=0)*(1-t)*(1-t^2)*(1-t^3)
        t^3*s[1, 1, 1] + (t^2+t)*s[2, 1] + s[3]
        sage: p(1).theta_qt()
        p[]

  `verschiebung`(n)

    Return the image of the symmetric function `self` under the \(n\)-th Verschiebung operator.

    The \(n\)-th Verschiebung operator \(\mathbf{V}_n\) is defined to be the unique algebra endomorphism \(V\) of the ring of symmetric functions that satisfies \(V(h_r) = h_{r/n}\) for every positive integer \(r\) divisible by \(n\), and satisfies \(V(h_r) = 0\) for every positive integer \(r\) not divisible by \(n\). This operator \(\mathbf{V}_n\) is a Hopf algebra endomorphism. For every nonnegative integer \(r\) with \(n \mid r\), it satisfies

    \[\mathbf{V}_n(h_r) = h_{r/n}, \quad \mathbf{V}_n(p_r) = n p_{r/n}, \quad \mathbf{V}_n(e_r) = (-1)^{r - r/n} e_{r/n}\]

    (where \(h\) is the complete homogeneous basis, \(p\) is the powersum basis, and \(e\) is the elementary basis). For every nonnegative integer \(r\) with \(n \nmid r\), it satisfes

    \[\mathbf{V}_n(h_r) = \mathbf{V}_n(p_r) = \mathbf{V}_n(e_r) = 0.\]

    The \(n\)-th Verschiebung operator is also called the \(n\)-th Verschiebung endomorphism. Its name derives from the Verschiebung (German for “shift”) endomorphism of the Witt vectors.

    The \(n\)-th Verschiebung operator is adjoint to the \(n\)-th Frobenius operator (see `frobenius()` for its definition) with respect to the Hall scalar product (`scalar()`).

    The action of the \(n\)-th Verschiebung operator on the Schur basis can also be computed explicitly. The following (probably clumsier than necessary) description can be obtained by solving exercise 7.61 in Stanley’s [STA].

    Let \(\lambda\) be a partition. Let \(n\) be a positive integer. If the \(n\)-core of \(\lambda\) is nonempty, then \(\mathbf{V}_n(s_\lambda) = 0\). Otherwise, the following method computes \(\mathbf{V}_n(s_\lambda)\): Write the partition \(\lambda\) in the form \((\lambda_1, \lambda_2, \ldots, \lambda_{ns})\) for some nonnegative integer \(s\). (If \(n\) does not divide the length of \(\lambda\), then this is achieved by adding trailing zeroes to \(\lambda\).) Set \(\beta_i = \lambda_i + ns - i\) for every \(s \in \{ 1, 2, \ldots, ns \}\). Then, \((\beta_1, \beta_2, \ldots, \beta_{ns})\) is a strictly decreasing sequence of nonnegative integers. Stably sort the list \((1, 2, \ldots, ns)\) in order of (weakly) increasing remainder of \(-1 - \beta_i\) modulo \(n\). Let \(\xi\) be the sign of the permutation that is used for this sorting. Let \(\psi\) be the sign of the permutation that is used to stably sort the list \((1, 2, \ldots, ns)\) in order of (weakly) increasing remainder of \(i - 1\) modulo \(n\). (Notice that \(\psi = (-1)^{n(n-1)s(s-1)/4}\).) Then, \(\mathbf{V}_n(s_\lambda) = \xi \psi \prod_{i = 0}^{n - 1} s_{\lambda^{(i)}}\), where \((\lambda^{(0)}, \lambda^{(1)}, \ldots, \lambda^{(n - 1)})\) is the \(n\)-quotient of \(\lambda\).

    INPUT:

    - `n` – a positive integer

    OUTPUT:

    The result of applying the \(n\)-th Verschiebung operator (on the ring of symmetric functions) to `self`.

    EXAMPLES:

        sage: Sym = SymmetricFunctions(ZZ)
        sage: p = Sym.p()
        sage: h = Sym.h()
        sage: s = Sym.s()
        sage: m = Sym.m()
        sage: s[3].verschiebung(2)
        0
        sage: s[3].verschiebung(3)
        s[1]
        sage: p[3].verschiebung(3)
        3*p[1]
        sage: m[3,2,1].verschiebung(3)
        -18*m[1, 1] - 3*m[2]
        sage: p[3,2,1].verschiebung(3)
        0
        sage: h[4].verschiebung(2)
        h[2]
        sage: p[2].verschiebung(2)
        2*p[1]
        sage: m[3,2,1].verschiebung(6)
        12*m[1]

    The Verschiebung endomorphisms are multiplicative:

        sage: all( all( s(lam).verschiebung(2) * s(mu).verschiebung(2)
        ....:           == (s(lam) * s(mu)).verschiebung(2)
        ....:           for mu in Partitions(4) )
        ....:      for lam in Partitions(4) )
        True

    Being Hopf algebra endomorphisms, the Verschiebung operators commute with the antipode:

        sage: all( p(lam).verschiebung(3).antipode()
        ....:      == p(lam).antipode().verschiebung(3)
        ....:      for lam in Partitions(6) )
        True

    Testing the adjointness between the Frobenius operators \(\mathbf{f}_n\) and the Verschiebung operators \(\mathbf{V}_n\):

        sage: Sym = SymmetricFunctions(QQ)
        sage: s = Sym.s()
        sage: p = Sym.p()
        sage: all( all( s(lam).verschiebung(2).scalar(p(mu))
        ....:           == s(lam).scalar(p(mu).frobenius(2))
        ....:           for mu in Partitions(3) )
        ....:      for lam in Partitions(6) )
        True

class `sage.combinat.sf.sfa.``SymmetricFunctionsBases`(base)

  Bases: `sage.categories.realizations.Category_realization_of_parent`

  The category of bases of the ring of symmetric functions.

  class `ElementMethods`
    `degree_negation`()

      Return the image of `self` under the degree negation automorphism of the ring of symmetric functions.

      The degree negation is the automorphism which scales every homogeneous element of degree \(k\) by \((-1)^k\) (for all \(k\)).

      Calling `degree_negation(self)` is equivalent to calling `self.parent().degree_negation(self)`.

      EXAMPLES:

          sage: Sym = SymmetricFunctions(ZZ)
          sage: m = Sym.monomial()
          sage: f = 2*m[2,1] + 4*m[1,1] - 5*m[1] - 3*m[[]]
          sage: f.degree_negation()
          -3*m[] + 5*m[1] + 4*m[1, 1] - 2*m[2, 1]
          sage: x = m.zero().degree_negation(); x
          0
          sage: parent(x) is m
          True

    `degree_zero_coefficient`()

      Returns the degree zero coefficient of `self`.

      INPUT:

      - `self` – an element of the symmetric functions

      EXAMPLES:

          sage: Sym = SymmetricFunctions(QQ)
          sage: m = Sym.monomial()
          sage: f = 2*m[2,1] + 3*m[[]]
          sage: f.degree_zero_coefficient()
          3

  class `SymmetricFunctionsBases.``ParentMethods`
    `Eulerian`(n, j, k=None)

      Return the Eulerian symmetric function \(Q_{n,j}\) (with \(n\) either an integer or a partition) or \(Q_{n,j,k}\) (if the optional argument `k` is specified) in terms of the basis `self`.

      It is known that the Eulerian quasisymmetric functions are in fact symmetric functions [SW2010]. For more information, see `QuasiSymmetricFunctions.Fundamental.Eulerian()`, which accepts the same syntax as this method.

      INPUT:

      - `n` – the nonnegative integer \(n\) or a partition
      - `j` – the number of excedances
      - `k` – (optional) if specified, determines the number of fixed points of the permutations which are being summed over

      EXAMPLES:

          sage: Sym = SymmetricFunctions(QQ)
          sage: m = Sym.m()
          sage: m.Eulerian(3, 1)
          4*m[1, 1, 1] + 3*m[2, 1] + 2*m[3]
          sage: h = Sym.h()
          sage: h.Eulerian(4, 2)
          h[2, 2] + h[3, 1] + h[4]
          sage: s = Sym.s()
          sage: s.Eulerian(5, 2)
          s[2, 2, 1] + s[3, 1, 1] + 5*s[3, 2] + 6*s[4, 1] + 6*s[5]
          sage: s.Eulerian([2,2,1], 2)
          s[2, 2, 1] + s[3, 2] + s[4, 1] + s[5]
          sage: s.Eulerian(5, 2, 2)
          s[3, 2] + s[4, 1] + s[5]

      We check Equation (5.4) in [SW2010]:

          sage: h.Eulerian([6], 3)
          h[3, 2, 1] - h[4, 1, 1] + 2*h[4, 2] + h[5, 1]
          sage: s.Eulerian([6], 3)
          s[3, 2, 1] + s[3, 3] + 3*s[4, 2] + 3*s[5, 1] + 3*s[6]

    `antipode_by_coercion`(element)

      The antipode of `element`.

      INPUT:

      - `element` – element in a basis of the ring of symmetric functions

      EXAMPLES:

          sage: Sym = SymmetricFunctions(QQ)
          sage: p = Sym.p()
          sage: s = Sym.s()
          sage: e = Sym.e()
          sage: h = Sym.h()
          sage: (h([]) + h([1])).antipode() # indirect doctest
          h[] - h[1]
          sage: (s([]) + s([1]) + s[2]).antipode()
          s[] - s[1] + s[1, 1]
          sage: (p([2]) + p([3])).antipode()
          -p[2] - p[3]
          sage: (e([2]) + e([3])).antipode()
          e[1, 1] - e[1, 1, 1] - e[2] + 2*e[2, 1] - e[3]
          sage: f = Sym.f()
          sage: f([3,2,1]).antipode()
          -f[3, 2, 1] - 4*f[3, 3] - 2*f[4, 2] - 2*f[5, 1] - 6*f[6]

      The antipode is an involution:

          sage: Sym = SymmetricFunctions(ZZ)
          sage: s = Sym.s()
          sage: all( s[u].antipode().antipode() == s[u] for u in Partitions(4) )
          True

      The antipode is an algebra homomorphism:

          sage: Sym = SymmetricFunctions(FiniteField(23))
          sage: h = Sym.h()
          sage: all( all( (s[u] * s[v]).antipode() == s[u].antipode() * s[v].antipode()
          ....:           for u in Partitions(3) )
          ....:      for v in Partitions(3) )
          True

      TESTS:

      Everything works over \(\ZZ\):

          sage: Sym = SymmetricFunctions(ZZ)
          sage: p = Sym.p()
          sage: s = Sym.s()
          sage: e = Sym.e()
          sage: h = Sym.h()
          sage: (h([]) + h([1])).antipode() # indirect doctest
          h[] - h[1]
          sage: (s([]) + s([1]) + s[2]).antipode()
          s[] - s[1] + s[1, 1]
          sage: (p([2]) + p([3])).antipode()
          -p[2] - p[3]
          sage: (e([2]) + e([3])).antipode()
          e[1, 1] - e[1, 1, 1] - e[2] + 2*e[2, 1] - e[3]

    `corresponding_basis_over`(R)

      Return the realization of symmetric functions corresponding to `self` but over the base ring `R`. Only works when `self` is one of the classical bases, not one of the \(q,t\)-dependent ones. In the latter case, `None` is returned instead.

      INPUT:

      - `R` – a commutative ring

      EXAMPLES:

          sage: Sym = SymmetricFunctions(QQ)
          sage: m = Sym.monomial()
          sage: m.corresponding_basis_over(ZZ)
          Symmetric Functions over Integer Ring in the monomial basis

          sage: Sym = SymmetricFunctions(CyclotomicField())
          sage: s = Sym.schur()
          sage: s.corresponding_basis_over(Integers(13))
          Symmetric Functions over Ring of integers modulo 13 in the Schur basis

          sage: P = ZZ['q','t']
          sage: Sym = SymmetricFunctions(P)
          sage: mj = Sym.macdonald().J()
          sage: mj.corresponding_basis_over(Integers(13))

      TESTS:

      Let’s check that this handles each of the bases properly:

          sage: P = QQ['q','t']
          sage: Sym = SymmetricFunctions(P)
          sage: Q = CyclotomicField()['q','t']
          sage: Sym.s().corresponding_basis_over(CyclotomicField())
          Symmetric Functions over Universal Cyclotomic Field in the Schur basis
          sage: Sym.p().corresponding_basis_over(CyclotomicField())
          Symmetric Functions over Universal Cyclotomic Field in the powersum basis
          sage: Sym.m().corresponding_basis_over(CyclotomicField())
          Symmetric Functions over Universal Cyclotomic Field in the monomial basis
          sage: Sym.e().corresponding_basis_over(CyclotomicField())
          Symmetric Functions over Universal Cyclotomic Field in the elementary basis
          sage: Sym.h().corresponding_basis_over(CyclotomicField())
          Symmetric Functions over Universal Cyclotomic Field in the homogeneous basis
          sage: Sym.f().corresponding_basis_over(CyclotomicField())
          Symmetric Functions over Universal Cyclotomic Field in the forgotten basis
          sage: Sym.w().corresponding_basis_over(CyclotomicField())
          Symmetric Functions over Universal Cyclotomic Field in the Witt basis
          sage: Sym.macdonald().P().corresponding_basis_over(CyclotomicField())
          sage: Sym.macdonald().Q().corresponding_basis_over(CyclotomicField())
          sage: Sym.macdonald().J().corresponding_basis_over(CyclotomicField())
          sage: Sym.macdonald().H().corresponding_basis_over(CyclotomicField())
          sage: Sym.macdonald().Ht().corresponding_basis_over(CyclotomicField())
          sage: Sym.macdonald().S().corresponding_basis_over(CyclotomicField())
          sage: Sym.macdonald(q=1).S().corresponding_basis_over(CyclotomicField())
          sage: Sym.macdonald(q=1,t=3).P().corresponding_basis_over(CyclotomicField())
          sage: Sym.hall_littlewood().P().corresponding_basis_over(CyclotomicField())
          sage: Sym.hall_littlewood().Q().corresponding_basis_over(CyclotomicField())
          sage: Sym.hall_littlewood().Qp().corresponding_basis_over(CyclotomicField())
          sage: Sym.hall_littlewood(t=1).P().corresponding_basis_over(CyclotomicField())
          sage: Sym.jack().J().corresponding_basis_over(CyclotomicField())
          sage: Sym.jack().P().corresponding_basis_over(CyclotomicField())
          sage: Sym.jack().Q().corresponding_basis_over(CyclotomicField())
          sage: Sym.jack().Qp().corresponding_basis_over(CyclotomicField())
          sage: Sym.jack(t=1).J().corresponding_basis_over(CyclotomicField())
          sage: Sym.zonal().corresponding_basis_over(CyclotomicField())
          Symmetric Functions over Universal Cyclotomic Field in the zonal basis
          sage: Sym.llt(3).hspin().corresponding_basis_over(CyclotomicField())
          sage: Sym.llt(3).hcospin().corresponding_basis_over(CyclotomicField())
          sage: Sym.llt(3, t=1).hspin().corresponding_basis_over(CyclotomicField())
          sage: Sym.llt(3, t=1).hcospin().corresponding_basis_over(CyclotomicField())

      Todo

      This function is an ugly hack using strings. It should be rewritten as soon as the bases of `SymmetricFunctions` are put on a more robust and systematic footing.

    `counit`(element)

      Return the counit of `element`.

      The counit is the constant term of `element`.

      INPUT:

      - `element` – element in a basis of the ring of symmetric functions

      EXAMPLES:

          sage: Sym = SymmetricFunctions(QQ)
          sage: m = Sym.monomial()
          sage: f = 2*m[2,1] + 3*m[[]]
          sage: f.counit()
          3

    `degree_negation`(element)

      Return the image of `element` under the degree negation automorphism of the ring of symmetric functions.

      The degree negation is the automorphism which scales every homogeneous element of degree \(k\) by \((-1)^k\) (for all \(k\)).

      INPUT:

      - `element` – symmetric function written in `self`

      EXAMPLES:

          sage: Sym = SymmetricFunctions(ZZ)
          sage: m = Sym.monomial()
          sage: f = 2*m[2,1] + 4*m[1,1] - 5*m[1] - 3*m[[]]
          sage: m.degree_negation(f)
          -3*m[] + 5*m[1] + 4*m[1, 1] - 2*m[2, 1]

      TESTS:

      Using `degree_negation()` on an element of a different basis works correctly:

          sage: e = Sym.elementary()
          sage: m.degree_negation(e[3])
          -m[1, 1, 1]
          sage: m.degree_negation(m(e[3]))
          -m[1, 1, 1]

    `degree_on_basis`(b)

      Return the degree of the basis element indexed by `b`.

      INPUT:

      - `self` – a basis of the symmetric functions
      - `b` – a partition

      EXAMPLES:

          sage: Sym = SymmetricFunctions(QQ['q,t'].fraction_field())
          sage: m = Sym.monomial()
          sage: m.degree_on_basis(Partition([3,2]))
          5
          sage: P = Sym.macdonald().P()
          sage: P.degree_on_basis(Partition([]))
          0

    `is_commutative`()

      Returns whether this symmetric function algebra is commutative.

      INPUT:

      - `self` – a basis of the symmetric functions

      EXAMPLES:

          sage: s = SymmetricFunctions(QQ).s()
          sage: s.is_commutative()
          True

    `is_field`(proof=True)

      Return whether `self` is a field. (It is not.)

      INPUT:

      - `self` – a basis of the symmetric functions
      - `proof` – an optional argument (default value: `True`)

      EXAMPLES:

          sage: s = SymmetricFunctions(QQ).s()
          sage: s.is_field()
          False

    `is_integral_domain`(proof=True)

      Return whether `self` is an integral domain. (It is if and only if the base ring is an integral domain.)

      INPUT:

      - `self` – a basis of the symmetric functions
      - `proof` – an optional argument (default value: `True`)

      EXAMPLES:

          sage: s = SymmetricFunctions(QQ).s()
          sage: s.is_integral_domain()
          True

      The following doctest is disabled pending trac ticket #15475:

          sage: s = SymmetricFunctions(Zmod(14)).s() # not tested
          sage: s.is_integral_domain() # not tested
          False

    `one_basis`()

      Returns the empty partition, as per `AlgebrasWithBasis.ParentMethods.one_basis`

      INPUT:

      - `self` – a basis of the ring of symmetric functions

      EXAMPLES:

          sage: Sym = SymmetricFunctions(QQ['t'].fraction_field())
          sage: s = Sym.s()
          sage: s.one_basis()
          []
          sage: Q = Sym.hall_littlewood().Q()
          sage: Q.one_basis()
          []

      Todo

      generalize to Modules.Graded.Connected.ParentMethods

  `SymmetricFunctionsBases.``super_categories`()

    The super categories of `self`.

    INPUT:

    - `self` – a category of bases for the symmetric functions

    EXAMPLES:

        sage: from sage.combinat.sf.sfa import SymmetricFunctionsBases
        sage: Sym = SymmetricFunctions(QQ)
        sage: bases = SymmetricFunctionsBases(Sym)
        sage: bases.super_categories()
        [Category of commutative graded hopf algebras with basis over Rational Field,
         Category of realizations of Symmetric Functions over Rational Field]

`sage.combinat.sf.sfa.``is_SymmetricFunction`(x)

  Checks whether `x` is a symmetric function.

  EXAMPLES:

      sage: from sage.combinat.sf.sfa import is_SymmetricFunction
      sage: s = SymmetricFunctions(QQ).s()
      sage: is_SymmetricFunction(2)
      False
      sage: is_SymmetricFunction(s(2))
      True
      sage: is_SymmetricFunction(s([2,1]))
      True

`sage.combinat.sf.sfa.``is_SymmetricFunctionAlgebra`(x)

  Checks whether `x` is a symmetric function algebra.

  EXAMPLES:

      sage: from sage.combinat.sf.sfa import is_SymmetricFunctionAlgebra
      sage: is_SymmetricFunctionAlgebra(5)
      False
      sage: is_SymmetricFunctionAlgebra(ZZ)
      False
      sage: is_SymmetricFunctionAlgebra(SymmetricFunctions(ZZ).schur())
      True
      sage: is_SymmetricFunctionAlgebra(SymmetricFunctions(QQ).e())
      True
      sage: is_SymmetricFunctionAlgebra(SymmetricFunctions(QQ).macdonald(q=1,t=1).P())
      True
      sage: is_SymmetricFunctionAlgebra(SymmetricFunctions(FractionField(QQ['q','t'])).macdonald().P())
      True

`sage.combinat.sf.sfa.``zee`(part)

  Return the size of the centralizer of any permutation of cycle type `part`.

  Note that the size of the centralizer is the inner product between `p(part)` and itself, where \(p\) is the power-sum symmetric functions.

  INPUT:

  - `part` – an integer partition (for example, `[2,1,1]`)

  OUTPUT:

  - the integer \(\prod_{i} i^{m_i(part)} m_i(part)!\) where \(m_i(part)\) is the number of parts in the partition `part` equal to \(i\)

  EXAMPLES:

      sage: from sage.combinat.sf.sfa import zee
      sage: zee([2,1,1])
      4

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