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# Symmetric functions, with their multiple realizations

class `sage.combinat.sf.sf.``SymmetricFunctions`(R)

  Bases: `sage.structure.unique_representation.UniqueRepresentation`, `sage.structure.parent.Parent`

  The abstract algebra of commutative symmetric functions

  Symmetric Functions in Sage

  This document is an introduction to working with symmetric function theory in Sage. It is not intended to be an introduction to the theory of symmetric functions ([MAC] and [STA], Chapter 7, are two excellent references.) The reader is also expected to be familiar with Sage.

  The algebra of symmetric functions

  The algebra of symmetric functions is the unique free commutative graded connected algebra over the given ring, with one generator in each degree. It can also be thought of as the inverse limit (in the category of graded algebras) of the algebra of symmetric polynomials in \(n\) variables as \(n \rightarrow \infty\). Sage allows us to construct the algebra of symmetric functions over any ring. We will use a base ring of rational numbers in these first examples:

      sage: Sym = SymmetricFunctions(QQ)
      sage: Sym
      Symmetric Functions over Rational Field

  Sage knows certain categorical information about this algebra:

      sage: Sym.category()
      Join of Category of hopf algebras over Rational Field
          and Category of graded algebras over Rational Field
          and Category of monoids with realizations
          and Category of coalgebras over Rational Field with realizations

  Notice that `Sym` is an abstract algebra. This reflects the fact that there are multiple natural bases. To work with specific elements, we need a realization of this algebra. In practice, this means we need to specify a basis.

  An example basis - power sums

  Here is an example of how one might use the power sum realization:

      sage: p = Sym.powersum()
      sage: p
      Symmetric Functions over Rational Field in the powersum basis

  `p` now represents the realization of the symmetric function algebra on the power sum basis. The basis itself is accessible through:

      sage: p.basis()
      Lazy family (Term map from Partitions to Symmetric Functions over Rational Field in the powersum basis(i))_{i in Partitions}
      sage: p.basis().keys()
      Partitions

  This last line means that `p.basis()` is an association between the set of Partitions and the basis elements of the algebra `p`. To construct a specific element one can therefore do:

      sage: p.basis()[Partition([2,1,1])]
      p[2, 1, 1]

  As this is rather cumbersome, realizations of the symmetric function algebra allow for the following abuses of notation:

      sage: p[Partition([2, 1, 1])]
      p[2, 1, 1]
      sage: p[[2, 1, 1]]
      p[2, 1, 1]
      sage: p[2, 1, 1]
      p[2, 1, 1]

  or even:

      sage: p[(i for i in [2, 1, 1])]
      p[2, 1, 1]

  In the special case of the empty partition, due to a limitation in Python syntax, one cannot use:

      sage: p[]       # todo: not implemented

  Please use instead:

      sage: p[[]]
      p[]

  Note

  When elements are constructed using the `p[something ]` syntax , an error will be raised if the input cannot be interpreted as a partition. This is not the case when `p.basis()` is used:

      sage: p['something']
      Traceback (most recent call last):
      ...
      ValueError: ['s', 'o', 'm', 'e', 't', 'h', 'i', 'n', 'g'] is not an element of Partitions
      sage: p.basis()['something']
      p'something'

  Elements of `p` are linear combinations of such compositions:

      sage: p.an_element()
      2*p[] + 2*p[1] + 3*p[2]

  Algebra structure

  Algebraic combinations of basis elements can be entered in a natural way:

      sage: p[2,1,1] + 2 * p[1] * (p[4] + p[2,1])
      3*p[2, 1, 1] + 2*p[4, 1]

  Let us explore the other operations of `p`. We can ask for the mathematical properties of `p`:

      sage: p.categories()
      [Category of bases of Symmetric Functions over Rational Field, Category of graded hopf algebras with basis over Rational Field, ...]

  To start with, `p` is a graded algebra, the grading being induced by the size of the partitions. Due to this, the one is the basis element indexed by the empty partition:

      sage: p.one()
      p[]

  The `p` basis is multiplicative; that is, multiplication is induced by linearity from the (nonincreasingly sorted) concatenation of partitions:

      sage: p[3,1] * p[2,1]
      p[3, 2, 1, 1]

      sage: (p.one() + 2 * p[3,1]) * p[4, 2]
      p[4, 2] + 2*p[4, 3, 2, 1]

  The classical bases

  In addition to the power sum basis, the other classical bases of the symmetric function algebra are the elementary, complete homogeneous, monomial, and Schur bases. These can be defined as follows:

      sage: e = Sym.elementary()
      sage: h = Sym.homogeneous()
      sage: m = Sym.monomial()
      sage: s = Sym.schur()

  These can be defined all at once with the single command

      sage: Sym.inject_shorthands()
      doctest:...: RuntimeWarning: redefining global value `h`
      doctest:...: RuntimeWarning: redefining global value `s`
      doctest:...: RuntimeWarning: redefining global value `e`
      doctest:...: RuntimeWarning: redefining global value `m`
      doctest:...: RuntimeWarning: redefining global value `p`

  We can then do conversions from one basis to another:

      sage: s(p[2,1])
      -s[1, 1, 1] + s[3]

      sage: m(p[3])
      m[3]
      sage: m(p[3,2])
      m[3, 2] + m[5]

  For computations which mix bases, Sage will return a result with respect to a single (not necessarily predictable) basis:

      sage: p[2] * s[2] - m[4]
      1/2*p[2, 1, 1] + 1/2*p[2, 2] - p[4]

      sage: p( m[1] * ( e[3]*s[2] + 1 ))
      p[1] + 1/12*p[1, 1, 1, 1, 1, 1] - 1/6*p[2, 1, 1, 1, 1] - 1/4*p[2, 2, 1, 1] + 1/6*p[3, 1, 1, 1] + 1/6*p[3, 2, 1]

  The one for different bases such as the power sum and Schur function is the same:

      sage: s.one() == p.one()
      True

  Basic computations

  In this section, we explore some of the many methods that can be applied to an arbitrary symmetric function:

      sage: f = s[2]^2; f
      s[2, 2] + s[3, 1] + s[4]

  For more methods than discussed here, create a symmetric function as above, and use `f.<tab>`.

  Representation theory of the symmetric group

  The Schur functions \(s_\lambda\) can also be interpreted as irreducible characters of the symmetric group \(S_n\), where \(n\) is the size of the partition \(\lambda\). Since the Schur functions of degree \(n\) form a basis of the symmetric functions of degree \(n\), it follows that an arbitrary symmetric function (homogeneous of degree \(n\)) may be interpreted as a function on the symmetric group. In this interpretation the power sum symmetric function \(p_\lambda\) is the characteristic function of the conjugacy class with shape \(\lambda\), multiplied by the order of the centralizer of an element. Hence the irreducible characters can be computed as follows:

      sage: Sym = SymmetricFunctions(QQ)
      sage: s = Sym.schur()
      sage: p = Sym.power()
      sage: P = Partitions(5).list()
      sage: P = [P[i] for i in range(len(P)-1,-1,-1)]
      sage: M = matrix([[s[P[i]].scalar(p[P[j]]) for j in range(len(P))] for i in range(len(P))])
      sage: M
      [ 1 -1  1  1 -1 -1  1]
      [ 4 -2  0  1  1  0 -1]
      [ 5 -1  1 -1 -1  1  0]
      [ 6  0 -2  0  0  0  1]
      [ 5  1  1 -1  1 -1  0]
      [ 4  2  0  1 -1  0 -1]
      [ 1  1  1  1  1  1  1]

  We can indeed check that this agrees with the character table of \(S_5\):

      sage: SymmetricGroup(5).character_table() == M
      True

  In this interpretation of symmetric functions as characters on the symmetric group, the multiplication and comultiplication are interpreted as induction (from \(S_n\times S_m\) to \(S_{n+m}\)) and restriction, respectively. The Schur functions can also be interpreted as characters of \(GL_n\), see Partitions and Schur functions.

  The omega involution

  The \(\omega\) involution is the linear extension of the map which sends \(e_\lambda\) to \(h_{\lambda}\):

      sage: h(f)
      h[2, 2]
      sage: e(f.omega())
      e[2, 2]

  The Hall scalar product

  The Hall scalar product on the algebra of symmetric functions makes the Schur functions into an orthonormal basis:

      sage: f.scalar(f)
      3

  Skewing

  Skewing is the adjoint operation to multiplication with respect to this scalar product:

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

  In general, `s[la].skew_by(s[mu])` is the symmetric function typically denoted \(s_{\lambda \setminus \mu}\) or \(s_{\lambda / \mu}\).

  Expanding into variables

  We can expand a symmetric function into a symmetric polynomial in a specified number of variables:

      sage: f.expand(2)
      x0^4 + 2*x0^3*x1 + 3*x0^2*x1^2 + 2*x0*x1^3 + x1^4

  See the documentation for `expand` for more examples.

  The Kronecker product

  As in the section on the Representation theory of the symmetric group, a symmetric function may be considered as a class function on the symmetric group where the elements \(p_\mu/z_\mu\) are the indicators of a permutation having cycle structure \(\mu\). The Kronecker product of two symmetric functions corresponds to the pointwise product of these class functions.

  Since the Schur functions are the irreducible characters of the symmetric group under this identification, the Kronecker product of two Schur functions corresponds to the internal tensor product of two irreducible symmetric group representations.

  Under this identification, the Kronecker product of \(p_\mu/z_\mu\) and \(p_\nu/z_\nu\) is \(p_\mu/z_\mu\) if \(\mu=\nu\), and the result is equal to \(0\) otherwise.

  `internal_product`, `kronecker_product`, `inner_tensor` and `itensor` are different names for the same function.

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

  Plethysm

  The plethysm of symmetric functions is the operation corresponding to composition of representations of the general linear group. See [STA] Chapter 7, Appendix 2 for details.

      sage: s[2].plethysm(s[2])
      s[2, 2] + s[4]

  Plethysm can also be written as a composition of functions:

      sage: s[2]( s[2] )
      s[2, 2] + s[4]

  If the coefficient ring contains degree 1 elements, these are handled properly by plethysm:

      sage: R.<t> = QQ[]; s = SymmetricFunctions(R).schur()
      sage: s[2]( (1-t)*s[1] )
      (t^2-t)*s[1, 1] + (-t+1)*s[2]

  See the documentation for `plethysm` for more information.

  Inner plethysm

  The operation of inner plethysm `f.inner_plethysm(g)` models the composition of the \(S_n\) representation represented by \(g\) with the \(GL_m\) representation whose character is \(f\). See the documentation of `inner_plethysm`, [ST94] or [STA], exercise 7.74 solutions for more information:

      sage: s = SymmetricFunctions(QQ).schur()
      sage: f = s[2]^2
      sage: f.inner_plethysm(s[2])
      s[2]

  Hopf algebra structure

  The ring of symmetric functions is further endowed with a coalgebra structure. The coproduct is an algebra morphism, and therefore determined by its values on the generators; the power sum generators are primitive:

      sage: p[1].coproduct()
      p[] # p[1] + p[1] # p[]
      sage: p[2].coproduct()
      p[] # p[2] + p[2] # p[]

  The coproduct, being cocommutative on the generators, is cocommutative everywhere:

      sage: p[2, 1].coproduct()
      p[] # p[2, 1] + p[1] # p[2] + p[2] # p[1] + p[2, 1] # p[]

  This coproduct, along with the counit which sends every symmetric function to its \(0\)-th homogeneous component, makes the ring of symmetric functions into a graded connected bialgebra. It is known that every graded connected bialgebra has an antipode. For the ring of symmetric functions, the antipode can be characterized explicitly: The antipode is an anti-algebra morphism (thus an algebra morphism, since our algebra is commutative) which sends \(p_{\lambda}\) to \((-1)^{\mathrm{length}(\lambda)} p_{\lambda}\) for every partition \(\lambda\). Thus, in particular, it sends the generators on the `p` basis to their opposites:

      sage: p[3].antipode()
      -p[3]
      sage: p[3,2,1].antipode()
      -p[3, 2, 1]

  The graded connected bialgebra of symmetric functions over a \(\QQ\)-algebra has a rather simply-understood structure: It is (isomorphic to) the symmetric algebra of its space of primitives (which is spanned by the power-sum symmetric functions).

  Here are further examples:

      sage: f = s[2]^2
      sage: f.antipode()
      s[1, 1, 1, 1] + s[2, 1, 1] + s[2, 2]
      sage: f.coproduct()
      s[] # s[2, 2] + s[] # s[3, 1] + s[] # s[4] + 2*s[1] # s[2, 1] + 2*s[1] # s[3] + s[1, 1] # s[1, 1]
      + s[1, 1] # s[2] + s[2] # s[1, 1] + 3*s[2] # s[2] + 2*s[2, 1] # s[1] + s[2, 2] # s[] + 2*s[3] # s[1]
      + s[3, 1] # s[] + s[4] # s[]
      sage: f.coproduct().apply_multilinear_morphism( lambda x,y: x*y.antipode() )
      0

  Transformations of symmetric functions

  There are many methods in Sage which make it easy to manipulate symmetric functions. For example, if we have some function which acts on partitions (say, conjugation), it is a simple matter to apply it to the support of a symmetric function. Here is an example:

      sage: conj = lambda mu: mu.conjugate()
      sage: f = h[4] + 2*h[3,1]
      sage: f.map_support(conj)
      h[1, 1, 1, 1] + 2*h[2, 1, 1]

  We can also easily modify the coefficients:

      sage: def foo(mu, coeff): return mu.conjugate(), -coeff
      sage: f.map_item(foo)
      -h[1, 1, 1, 1] - 2*h[2, 1, 1]

  See also `map_coefficients`.

  There are also methods for building functions directly:

      sage: s.sum_of_monomials(mu for mu in Partitions(3))
      s[1, 1, 1] + s[2, 1] + s[3]
      sage: s.sum_of_monomials(Partitions(3))
      s[1, 1, 1] + s[2, 1] + s[3]
      sage: s.sum_of_terms( (mu, mu[0]) for mu in Partitions(3))
      s[1, 1, 1] + 2*s[2, 1] + 3*s[3]

  These are the preferred way to build elements within a program; the result will usually be faster than using `sum()`. It also guarantees that empty sums yields the zero of `s` (see also `s.sum`).

  Note also that it is a good idea to use:

      sage: s.one()
      s[]
      sage: s.zero()
      0

  instead of `s(1)` and `s(0)` within programs where speed is important, in order to prevent unnecessary coercions.

  Different base rings

  Depending on the base ring, the different realizations of the symmetric function algebra may not span the same space:

      sage: SZ = SymmetricFunctions(ZZ)
      sage: p = SZ.power(); s = SZ.schur()
      sage: p(s[1,1,1])
      Traceback (most recent call last):
      ...
      TypeError: no conversion of this rational to integer

  Because of this, some functions may not behave as expected when working over the integers, even though they make mathematical sense:

      sage: s[1,1,1].plethysm(s[1,1,1])
      Traceback (most recent call last):
      ...
      TypeError: no conversion of this rational to integer

  It is possible to work over different base rings simultaneously:

      sage: s = SymmetricFunctions(QQ).schur()
      sage: p = SymmetricFunctions(QQ).power()
      sage: sz = SymmetricFunctions(ZZ).schur(); sz._prefix = 'sz'
      sage: pz = SymmetricFunctions(ZZ).power(); pz._prefix = 'pz'
      sage: p(sz[1,1,1])
      1/6*p[1, 1, 1] - 1/2*p[2, 1] + 1/3*p[3]
      sage: sz( 1/6*p[1, 1, 1] - 1/2*p[2, 1] + 1/3*p[3] )
      sz[1, 1, 1]

  As shown in this example, if you are working over multiple base rings simultaneously, it is a good idea to change the prefix in some cases, so that you can tell from the output which realization your result is in.

  Let us change the notation back for the remainder of this tutorial:

      sage: sz._prefix = 's'
      sage: pz._prefix = 'p'

  One can also use the Sage standard renaming idiom to get shorter outputs:

      sage: Sym = SymmetricFunctions(QQ)
      sage: Sym.rename("Sym")
      sage: Sym
      Sym
      sage: Sym.rename()

  And we name it back:

      sage: Sym.rename("Symmetric Functions over Rational Field"); Sym
      Symmetric Functions over Rational Field

  Other bases

  There are two additional basis of the symmetric functions which are not considered as classical bases:

  - forgotten basis
  - Witt basis

  The forgotten basis is the dual basis of the elementary symmetric functions basis with respect to the Hall scalar product. The Witt basis can be constructed by

  \[\prod_{d=1}^{\infty} (1 - w_d t^d)^{-1} = \sum_{n=0}^{\infty} h_n t^n\]

  where \(t\) is a formal variable.

  There are further bases of the ring of symmetric functions, in general over fields with parameters such as \(q\) and \(t\):

  - Hall-Littlewood bases
  - Jack bases
  - Macdonald bases
  - \(k\)-Schur functions

  We briefly demonstrate how to access these bases. For more information, see the documentation of the individual bases.

  The Jack polynomials can be obtained as:

      sage: Sym = SymmetricFunctions(FractionField(QQ['t']))
      sage: Jack = Sym.jack()
      sage: P = Jack.P(); J = Jack.J(); Q = Jack.Q()
      sage: J(P[2,1])
      (1/(t+2))*JackJ[2, 1]

  The parameter \(t\) can be specialized as follows:

      sage: Sym = SymmetricFunctions(QQ)
      sage: Jack = Sym.jack(t = 1)
      sage: P = Jack.P(); J = Jack.J(); Q = Jack.Q()
      sage: J(P[2,1])
      1/3*JackJ[2, 1]

  Similarly one can access the Hall-Littlewood and Macdonald polynomials, etc:

      sage: Sym = SymmetricFunctions(FractionField(QQ['q','t']))
      sage: Mcd = Sym.macdonald()
      sage: P = Mcd.P(); J = Mcd.J(); Q = Mcd.Q()
      sage: J(P[2,1])
      (1/(-q*t^4+2*q*t^3-q*t^2+t^2-2*t+1))*McdJ[2, 1]

  \(k\)-Schur functions

  The \(k\)-Schur functions live in the \(k\)-bounded subspace of the ring of symmetric functions. It is possible to compute in the \(k\)-bounded subspace directly:

      sage: Sym = SymmetricFunctions(QQ)
      sage: ks = Sym.kschur(3,1)
      sage: f = ks[2,1]*ks[2,1]; f
      ks3[2, 2, 1, 1] + ks3[2, 2, 2] + ks3[3, 1, 1, 1]

  or to lift to the ring of symmetric functions:

      sage: f.lift()
      s[2, 2, 1, 1] + s[2, 2, 2] + s[3, 1, 1, 1] + 2*s[3, 2, 1] + s[3, 3] + s[4, 1, 1] + s[4, 2]

  However, it is not always possible to convert a symmetric function to the \(k\)-bounded subspace:

      sage: s = Sym.schur()
      sage: ks(s[2,1,1])
      Traceback (most recent call last):
      ...
      ValueError: s[2, 1, 1] is not in the image

  The \(k\)-Schur functions are more generally defined with a parameter \(t\) and they are a basis of the subspace spanned by the Hall-Littlewood `Qp` symmetric functions indexed by partitions whose first part is less than or equal to \(k\):

      sage: Sym = SymmetricFunctions(QQ['t'].fraction_field())
      sage: SymS3 = Sym.kBoundedSubspace(3) # default t='t'
      sage: ks = SymS3.kschur()
      sage: Qp = Sym.hall_littlewood().Qp()
      sage: ks(Qp[2,1,1,1])
      ks3[2, 1, 1, 1] + (t^2+t)*ks3[2, 2, 1] + (t^3+t^2)*ks3[3, 1, 1] + t^4*ks3[3, 2]

  The subspace spanned by the \(k\)-Schur functions with a parameter \(t\) are not known to form a natural algebra. However it is known that the product of a \(k\)-Schur function and an \(\ell\)-Schur function is in the linear span of the \(k+\ell\)-Schur functions:

      sage: ks(ks[2,1]*ks[1,1])
      Traceback (most recent call last):
      ...
      ValueError: s[2, 1, 1, 1] + s[2, 2, 1] + s[3, 1, 1] + s[3, 2] is not in the image
      sage: ks[2,1]*ks[1,1]
      s[2, 1, 1, 1] + s[2, 2, 1] + s[3, 1, 1] + s[3, 2]
      sage: ks6 = Sym.kBoundedSubspace(6).kschur()
      sage: ks6(ks[3,1,1]*ks[3])
      ks6[3, 3, 1, 1] + ks6[4, 2, 1, 1] + (t+1)*ks6[4, 3, 1] + t*ks6[4, 4]
      + ks6[5, 1, 1, 1] + ks6[5, 2, 1] + t*ks6[5, 3] + ks6[6, 1, 1]

  dual \(k\)-Schur functions

  The dual space to the subspace spanned by the \(k\)-Schur functions is most naturally realized as a quotient of the ring of symmetric functions by an ideal. When \(t=1\) the ideal is generated by the monomial symmetric functions indexed by partitions whose first part is greater than \(k\).:

      sage: Sym = SymmetricFunctions(QQ)
      sage: SymQ3 = Sym.kBoundedQuotient(3,t=1)
      sage: km = SymQ3.kmonomial()
      sage: km[2,1]*km[2,1]
      4*m3[2, 2, 1, 1] + 6*m3[2, 2, 2] + 2*m3[3, 2, 1] + 2*m3[3, 3]
      sage: F = SymQ3.affineSchur()
      sage: F[2,1]*F[2,1]
      2*F3[1, 1, 1, 1, 1, 1] + 4*F3[2, 1, 1, 1, 1] + 4*F3[2, 2, 1, 1] + 4*F3[2, 2, 2]
      + 2*F3[3, 1, 1, 1] + 4*F3[3, 2, 1] + 2*F3[3, 3]

  When \(t\) is not equal to \(1\), the subspace spanned by the \(k\)-Schur functions is realized as a quotient of the ring of symmetric functions by the ideal generated by the Hall-Littlewood symmetric functions in the P basis indexed by partitions with first part greater than \(k\).:

      sage: Sym = SymmetricFunctions(FractionField(QQ['t']))
      sage: SymQ3 = Sym.kBoundedQuotient(3)
      sage: kHLP = SymQ3.kHallLittlewoodP()
      sage: kHLP[2,1]*kHLP[2,1]
      (t^2+2*t+1)*HLP3[2, 2, 1, 1] + (t^3+2*t^2+2*t+1)*HLP3[2, 2, 2]
      + (-t^4-t^3+t+1)*HLP3[3, 1, 1, 1] + (-t^2+t+2)*HLP3[3, 2, 1] + (t+1)*HLP3[3, 3]
      sage: HLP = Sym.hall_littlewood().P()
      sage: kHLP(HLP[3,1])
      HLP3[3, 1]
      sage: kHLP(HLP[4])
      0

  In this space, the basis which is dual to the \(k\)-Schur functions conjecturally expands positively in the \(k\)-bounded Hall-Littlewood functions and has positive structure coefficients.:

      sage: dks = SymQ3.dual_k_Schur()
      sage: kHLP(dks[2,2])
      (t^4+t^2)*HLP3[1, 1, 1, 1] + t*HLP3[2, 1, 1] + HLP3[2, 2]
      sage: dks[2,1]*dks[1,1]
      (t^2+t)*dks3[1, 1, 1, 1, 1] + (t+1)*dks3[2, 1, 1, 1] + (t+1)*dks3[2, 2, 1]
      + dks3[3, 1, 1] + dks3[3, 2]

  At \(t=1\) the \(k\)-bounded Hall-Littlewood basis is equal to the \(k\)-bounded monomial basis and the dual \(k\)-Schur elements are equal to the affine Schur basis. The \(k\)-bounded monomial basis and affine Schur functions are faster and should be used instead of the \(k\)-bounded Hall-Littlewood P basis and dual \(k\)-Schur functions when \(t=1\).:

      sage: SymQ3 = Sym.kBoundedQuotient(3,t=1)
      sage: dks = SymQ3.dual_k_Schur()
      sage: F = SymQ3.affineSchur()
      sage: F[3,1]==dks[3,1]
      True

  Implementing new bases

  Todo

  to be described

  Acknowledgements

  The design is heavily inspired from the implementation of symmetric functions in MuPAD-Combinat (see [HT04] and [FD06]).

  REFERENCES:

  > [FD06] | Francois Descouens, Making research on symmetric functions using MuPAD-Combinat. In Andres Iglesias and Nobuki Takayama, editors, 2nd International Congress on Mathematical Software (ICMS‘06), volume 4151 of LNCS, pages 407-418, Castro Urdiales, Spain, September 2006. Springer-Verlag. Arxiv 0806.1873

  > [HT04] | Florent Hivert and Nicolas M. Thiery, MuPAD-Combinat, an open-source package for research in algebraic combinatorics. Sem. Lothar. Combin., 51 :Art. B51z, 70 pp. (electronic), 2004. http://mupad-combinat.sf.net/.

  > [MAC] | Ian Macdonald, Symmetric Functions and Orthogonal Polynomials, Second edition. With contributions by A. Zelevinsky. Oxford Mathematical Monographs. Oxford Science Publications. The Clarendon Press, Oxford University Press, New York, 1995. x+475 pp. ISBN: 0-19-853489-2

  > [STA] | (1, 2, 3) Richard Stanley, Enumerative combinatorics. Vol. 2. With a foreword by Gian-Carlo Rota and appendix 1 by Sergey Fomin. Cambridge Studies in Advanced Mathematics, 62. Cambridge University Press, Cambridge, 1999. xii+581 pp. ISBN: 0-521-56069-1; 0-521-78987-7

  > [ST94] | Scharf, Thomas, Thibon, Jean-Yves, A Hopf-algebra approach to inner plethysm. Adv. Math. 104 (1994), no. 1, 30-58. doi:10.1006/aima.1994.1019

  Further tests

  TESTS:

      sage: Sym = SymmetricFunctions(QQ)
      sage: Sym
      Symmetric Functions over Rational Field
      sage: h = Sym.h(); e = Sym.e(); s = Sym.s(); m = Sym.m(); p = Sym.p()
      sage: ( ( h[2,1] * ( 1 + 3 * h[2,1]) ) + s[2]. antipode()) . coproduct()
      h[] # h[1, 1] - h[] # h[2] + h[] # h[2, 1] + 3*h[] # h[2, 2, 1, 1] + h[1] # h[1] + h[1] # h[1, 1]
      + h[1] # h[2] + 6*h[1] # h[2, 1, 1, 1] + 6*h[1] # h[2, 2, 1] + h[1, 1] # h[] + h[1, 1] # h[1]
      + 3*h[1, 1] # h[1, 1, 1, 1] + 12*h[1, 1] # h[2, 1, 1] + 3*h[1, 1] # h[2, 2] + 6*h[1, 1, 1] # h[1, 1, 1]
      + 6*h[1, 1, 1] # h[2, 1] + 3*h[1, 1, 1, 1] # h[1, 1] - h[2] # h[] + h[2] # h[1] + 6*h[2] # h[2, 1, 1]
      + h[2, 1] # h[] + 6*h[2, 1] # h[1, 1, 1] + 12*h[2, 1] # h[2, 1] + 12*h[2, 1, 1] # h[1, 1]
      + 6*h[2, 1, 1] # h[2] + 6*h[2, 1, 1, 1] # h[1] + 3*h[2, 2] # h[1, 1] + 6*h[2, 2, 1] # h[1] + 3*h[2, 2, 1, 1] # h[]

  Todo

  - Introduce fields with degree 1 elements as in MuPAD-Combinat, to get proper plethysm.
  - Use UniqueRepresentation to get rid of all the manual cache handling for the bases
  - Devise a mechanism so that pickling bases of symmetric functions pickles the coercions which have a cache.
  `Schur`()

    The Schur basis of the Symmetric Functions

    EXAMPLES:

        sage: SymmetricFunctions(QQ).schur()
        Symmetric Functions over Rational Field in the Schur basis

  `Witt`(coerce_h=True, coerce_e=False, coerce_p=False)

    The Witt basis of the symmetric functions.

    EXAMPLES:

        sage: SymmetricFunctions(QQ).witt()
        Symmetric Functions over Rational Field in the Witt basis
        sage: SymmetricFunctions(QQ).witt(coerce_p=True)
        Symmetric Functions over Rational Field in the Witt basis
        sage: SymmetricFunctions(QQ).witt(coerce_h=False, coerce_e=True, coerce_p=True)
        Symmetric Functions over Rational Field in the Witt basis

  `a_realization`()

    Returns a particular realization of `self` (the Schur basis).

    EXAMPLES:

        sage: Sym = SymmetricFunctions(QQ)
        sage: Sym.a_realization()
        Symmetric Functions over Rational Field in the Schur basis

  `complete`()

    The complete basis of the Symmetric Functions

    EXAMPLES:

        sage: SymmetricFunctions(QQ).complete()
        Symmetric Functions over Rational Field in the homogeneous basis

  `e`()

    The elementary basis of the Symmetric Functions

    EXAMPLES:

        sage: SymmetricFunctions(QQ).elementary()
        Symmetric Functions over Rational Field in the elementary basis

  `elementary`()

    The elementary basis of the Symmetric Functions

    EXAMPLES:

        sage: SymmetricFunctions(QQ).elementary()
        Symmetric Functions over Rational Field in the elementary basis

  `f`()

    The forgotten basis of the Symmetric Functions (or the basis dual to the elementary basis with respect to the Hall scalar product).

    EXAMPLES:

        sage: SymmetricFunctions(QQ).forgotten()
        Symmetric Functions over Rational Field in the forgotten basis

    TESTS:

    Over the rationals:

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

    Over the integers:

        sage: Sym = SymmetricFunctions(ZZ)
        sage: e = Sym.e()
        sage: f = Sym.f()
        sage: h = Sym.h()
        sage: p = Sym.p()
        sage: s = Sym.s()
        sage: m = Sym.m()
        sage: e(f([2,1]))
        -2*e[1, 1, 1] + 5*e[2, 1] - 3*e[3]
        sage: f(e([2,1]))
        3*f[1, 1, 1] + 2*f[2, 1] + f[3]
        sage: h(f([2,1]))
        h[2, 1] - 3*h[3]
        sage: f(h([2,1]))
        3*f[1, 1, 1] + f[2, 1]
        sage: f(p([2,1]))
        -f[2, 1] - f[3]
        sage: s(f([2,1]))
        s[2, 1] - 2*s[3]
        sage: f(s([2,1]))
        2*f[1, 1, 1] + f[2, 1]
        sage: m(f([2,1]))
        -m[2, 1] - 2*m[3]
        sage: f(m([2,1]))
        -f[2, 1] - 2*f[3]

    Conversion from the forgotten basis to the power-sum basis over the integers is not well-defined in general, even if the result happens to have integral coefficients:

        sage: p(f([2,1]))
        Traceback (most recent call last):
        ...
        TypeError: no conversion of this rational to integer

    Fun exercise: prove that \(p(f_{\lambda})\) and \(p(m_{\lambda})\) have integral coefficients whenever \(\lambda\) is a strict partition.

  `forgotten`()

    The forgotten basis of the Symmetric Functions (or the basis dual to the elementary basis with respect to the Hall scalar product).

    EXAMPLES:

        sage: SymmetricFunctions(QQ).forgotten()
        Symmetric Functions over Rational Field in the forgotten basis

    TESTS:

    Over the rationals:

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

    Over the integers:

        sage: Sym = SymmetricFunctions(ZZ)
        sage: e = Sym.e()
        sage: f = Sym.f()
        sage: h = Sym.h()
        sage: p = Sym.p()
        sage: s = Sym.s()
        sage: m = Sym.m()
        sage: e(f([2,1]))
        -2*e[1, 1, 1] + 5*e[2, 1] - 3*e[3]
        sage: f(e([2,1]))
        3*f[1, 1, 1] + 2*f[2, 1] + f[3]
        sage: h(f([2,1]))
        h[2, 1] - 3*h[3]
        sage: f(h([2,1]))
        3*f[1, 1, 1] + f[2, 1]
        sage: f(p([2,1]))
        -f[2, 1] - f[3]
        sage: s(f([2,1]))
        s[2, 1] - 2*s[3]
        sage: f(s([2,1]))
        2*f[1, 1, 1] + f[2, 1]
        sage: m(f([2,1]))
        -m[2, 1] - 2*m[3]
        sage: f(m([2,1]))
        -f[2, 1] - 2*f[3]

    Conversion from the forgotten basis to the power-sum basis over the integers is not well-defined in general, even if the result happens to have integral coefficients:

        sage: p(f([2,1]))
        Traceback (most recent call last):
        ...
        TypeError: no conversion of this rational to integer

    Fun exercise: prove that \(p(f_{\lambda})\) and \(p(m_{\lambda})\) have integral coefficients whenever \(\lambda\) is a strict partition.

  `from_polynomial`(f)

    Converts a symmetric polynomial `f` to a symmetric function.

    INPUT:

    - `f` – a symmetric polynomial

    This function converts a symmetric polynomial \(f\) in a polynomial ring in finitely many variables to a symmetric function in the monomial basis of the ring of symmetric functions over the same base ring.

    EXAMPLES:

        sage: P = PolynomialRing(QQ, 'x', 3)
        sage: x = P.gens()
        sage: f = x[0] + x[1] + x[2]
        sage: S = SymmetricFunctions(QQ)
        sage: S.from_polynomial(f)
        m[1]

        sage: f = x[0] + 2*x[1] + x[2]
        sage: S.from_polynomial(f)
        Traceback (most recent call last):
        ...
        ValueError: x0 + 2*x1 + x2 is not a symmetric polynomial

  `h`()

    The complete basis of the Symmetric Functions

    EXAMPLES:

        sage: SymmetricFunctions(QQ).complete()
        Symmetric Functions over Rational Field in the homogeneous basis

  `hall_littlewood`(t='t')

    Returns the entry point for the various Hall-Littlewood bases.

    INPUT:

    - `t` – parameter

    Hall-Littlewood symmetric functions including bases \(P\), \(Q\), \(Qp\). The Hall-Littlewood \(P\) and \(Q\) functions at \(t=-1\) are the Schur-P and Schur-Q functions when indexed by strict partitions.

    The parameter \(t\) must be in the base ring of parent.

    EXAMPLES:

        sage: Sym = SymmetricFunctions(FractionField(QQ['t']))
        sage: P = Sym.hall_littlewood().P(); P
        Symmetric Functions over Fraction Field of Univariate Polynomial Ring in t over Rational Field in the Hall-Littlewood P basis
        sage: P[2]
        HLP[2]
        sage: Q = Sym.hall_littlewood().Q(); Q
        Symmetric Functions over Fraction Field of Univariate Polynomial Ring in t over Rational Field in the Hall-Littlewood Q basis
        sage: Q[2]
        HLQ[2]
        sage: Qp = Sym.hall_littlewood().Qp(); Qp
        Symmetric Functions over Fraction Field of Univariate Polynomial Ring in t over Rational Field in the Hall-Littlewood Qp basis
        sage: Qp[2]
        HLQp[2]

  `homogeneous`()

    The complete basis of the Symmetric Functions

    EXAMPLES:

        sage: SymmetricFunctions(QQ).complete()
        Symmetric Functions over Rational Field in the homogeneous basis

  `inject_shorthands`(shorthands=set(['h', 's', 'e', 'm', 'p']))

    Imports standard shorthands into the global namespace

    INPUT:

    - `shorthands` – a list (or iterable) of strings (default: [‘e’, ‘h’, ‘m’, ‘p’, ‘s’])

    EXAMPLES:

        sage: S = SymmetricFunctions(ZZ)
        sage: S.inject_shorthands()
        sage: s[1] + e[2] * p[1,1] + 2*h[3] + m[2,1]
        s[1] - 2*s[1, 1, 1] + s[1, 1, 1, 1] + s[2, 1] + 2*s[2, 1, 1] + s[2, 2] + 2*s[3] + s[3, 1]
        sage: e
        Symmetric Functions over Integer Ring in the elementary basis
        sage: p
        Symmetric Functions over Integer Ring in the powersum basis
        sage: s
        Symmetric Functions over Integer Ring in the Schur basis

        sage: e == S.e(), h == S.h(), m == S.m(), p == S.p(), s == S.s()
        (True, True, True, True, True)

    One can also just import a subset of the shorthands:

        sage: S = SymmetricFunctions(QQ)
        sage: S.inject_shorthands(['p', 's'])
        sage: p
        Symmetric Functions over Rational Field in the powersum basis
        sage: s
        Symmetric Functions over Rational Field in the Schur basis

    Note that `e` is left unchanged:

        sage: e
        Symmetric Functions over Integer Ring in the elementary basis

  `jack`(t='t')

    Returns the entry point for the various Jack bases.

    INPUT:

    - `t` – parameter

    Jack symmetric functions including bases \(P\), \(Q\), \(Qp\).

    The parameter \(t\) must be in the base ring of parent.

    EXAMPLES:

        sage: Sym = SymmetricFunctions(FractionField(QQ['t']))
        sage: JP = Sym.jack().P(); JP
        Symmetric Functions over Fraction Field of Univariate Polynomial Ring in t over Rational Field in the Jack P basis
        sage: JQ = Sym.jack().Q(); JQ
        Symmetric Functions over Fraction Field of Univariate Polynomial Ring in t over Rational Field in the Jack Q basis
        sage: JJ = Sym.jack().J(); JJ
        Symmetric Functions over Fraction Field of Univariate Polynomial Ring in t over Rational Field in the Jack J basis
        sage: JQp = Sym.jack().Qp(); JQp
        Symmetric Functions over Fraction Field of Univariate Polynomial Ring in t over Rational Field in the Jack Qp basis

  `kBoundedQuotient`(k, t='t')

    Returns the \(k\)-bounded quotient space of the ring of symmetric functions.

    INPUT:

    - `k` - a positive integer

    The quotient of the ring of symmetric functions ...

    See also

    `sage.combinat.sf.k_dual.KBoundedQuotient()`

    EXAMPLES:

        sage: Sym = SymmetricFunctions(QQ)
        sage: KQ = Sym.kBoundedQuotient(3); KQ
        Traceback (most recent call last):
        ...
        TypeError: unable to convert t to a rational
        sage: KQ = Sym.kBoundedQuotient(3,t=1); KQ
        3-Bounded Quotient of Symmetric Functions over Rational Field with t=1
        sage: Sym = SymmetricFunctions(QQ['t'].fraction_field())
        sage: KQ = Sym.kBoundedQuotient(3); KQ
        3-Bounded Quotient of Symmetric Functions over Fraction Field of Univariate Polynomial Ring in t over Rational Field

  `kBoundedSubspace`(k, t='t')

    Return the \(k\)-bounded subspace of the ring of symmetric functions.

    INPUT:

    - `k` - a positive integer
    - `t` a formal parameter; \(t=1\) yields a subring

    The subspace of the ring of symmetric functions spanned by \(\{ s_{\lambda}[X/(1-t)] \}_{\lambda_1\le k} = \{ s_{\lambda}^{(k)}[X,t]\}_{\lambda_1 \le k}\) over the base ring \(\mathbb{Q}[t]\). When \(t=1\), this space is in fact a subalgebra of the ring of symmetric functions generated by the complete homogeneous symmetric functions \(h_i\) for \(1\le i \le k\).

    See also

    `sage.combinat.sf.new_kschur.KBoundedSubspace()`

    EXAMPLES:

        sage: Sym = SymmetricFunctions(QQ)
        sage: KB = Sym.kBoundedSubspace(3,1); KB
        3-bounded Symmetric Functions over Rational Field with t=1

        sage: Sym = SymmetricFunctions(QQ['t'])
        sage: Sym.kBoundedSubspace(3)
        3-bounded Symmetric Functions over Univariate Polynomial Ring in t over Rational Field

        sage: Sym = SymmetricFunctions(QQ['z'])
        sage: z = Sym.base_ring().gens()[0]
        sage: Sym.kBoundedSubspace(3,t=z)
        3-bounded Symmetric Functions over Univariate Polynomial Ring in z over Rational Field with t=z

  `khomogeneous`(k)

    Returns the homogeneous symmetric functions in the \(k\)-bounded subspace.

    EXAMPLES:

        sage: Sym = SymmetricFunctions(QQ)
        sage: kh = Sym.khomogeneous(4)
        sage: kh[3]*kh[4]
        h4[4, 3]
        sage: kh[4].lift()
        h[4]

  `kschur`(k, t='t')

    Returns the \(k\)-Schur functions.

    EXAMPLES:

        sage: Sym = SymmetricFunctions(QQ)
        sage: ks = Sym.kschur(3,1)
        sage: ks[2]*ks[2]
        ks3[2, 2] + ks3[3, 1]
        sage: ks[2,1,1].lift()
        s[2, 1, 1] + s[3, 1]

        sage: Sym = SymmetricFunctions(QQ['t'])
        sage: ks = Sym.kschur(3)
        sage: ks[2,2,1].lift()
        s[2, 2, 1] + t*s[3, 2]

  `llt`(k, t='t')

    The LLT symmetric functions.

    INPUT:

    - `k` – a positive integer indicating the level
    - `t` – a parameter (default: \(t\))

    LLT polynomials in \(hspin\) and \(hcospin\) bases.

    EXAMPLES:

        sage: llt3 = SymmetricFunctions(QQ['t'].fraction_field()).llt(3); llt3
        level 3 LLT polynomials over Fraction Field of Univariate Polynomial Ring in t over Rational Field
        sage: llt3.hspin()
        Symmetric Functions over Fraction Field of Univariate Polynomial Ring in t over Rational Field in the level 3 LLT spin basis
        sage: llt3.hcospin()
        Symmetric Functions over Fraction Field of Univariate Polynomial Ring in t over Rational Field in the level 3 LLT cospin basis
        sage: llt3.hcospin()
        Symmetric Functions over Fraction Field of Univariate Polynomial Ring in t over Rational Field in the level 3 LLT cospin basis

  `m`()

    The monomial basis of the Symmetric Functions

    EXAMPLES:

        sage: SymmetricFunctions(QQ).monomial()
        Symmetric Functions over Rational Field in the monomial basis

  `macdonald`(q='q', t='t')

    Returns the entry point for the various Macdonald bases.

    INPUT:

    - `q`, `t` – parameters

    Macdonald symmetric functions including bases \(P\), \(Q\), \(J\), \(H\), \(Ht\). This also contains the \(S\) basis which is dual to the Schur basis with respect to the \(q,t\) scalar product.

    The parameters \(q\) and \(t\) must be in the base_ring of parent.

    EXAMPLES:

        sage: Sym = SymmetricFunctions(FractionField(QQ['q','t']))
        sage: P = Sym.macdonald().P(); P
        Symmetric Functions over Fraction Field of Multivariate Polynomial Ring in q, t over Rational Field in the Macdonald P basis
        sage: P[2]
        McdP[2]
        sage: Q = Sym.macdonald().Q(); Q
        Symmetric Functions over Fraction Field of Multivariate Polynomial Ring in q, t over Rational Field in the Macdonald Q basis
        sage: S = Sym.macdonald().S()
        sage: s = Sym.schur()
        sage: matrix([[S(la).scalar_qt(s(mu)) for la in Partitions(3)] for mu in Partitions(3)])
        [1 0 0]
        [0 1 0]
        [0 0 1]
        sage: H = Sym.macdonald().H()
        sage: s(H[2,2])
        q^2*s[1, 1, 1, 1] + (q^2*t+q*t+q)*s[2, 1, 1] + (q^2*t^2+1)*s[2, 2] + (q*t^2+q*t+t)*s[3, 1] + t^2*s[4]

        sage: Sym = SymmetricFunctions(QQ['z','q'].fraction_field())
        sage: (z,q) = Sym.base_ring().gens()
        sage: Hzq = Sym.macdonald(q=z,t=q).H()
        sage: H1z = Sym.macdonald(q=1,t=z).H()
        sage: s = Sym.schur()
        sage: s(H1z([2,2]))
        s[1, 1, 1, 1] + (2*z+1)*s[2, 1, 1] + (z^2+1)*s[2, 2] + (z^2+2*z)*s[3, 1] + z^2*s[4]
        sage: s(Hzq[2,2])
        z^2*s[1, 1, 1, 1] + (z^2*q+z*q+z)*s[2, 1, 1] + (z^2*q^2+1)*s[2, 2] + (z*q^2+z*q+q)*s[3, 1] + q^2*s[4]
        sage: s(H1z(Hzq[2,2]))
        z^2*s[1, 1, 1, 1] + (z^2*q+z*q+z)*s[2, 1, 1] + (z^2*q^2+1)*s[2, 2] + (z*q^2+z*q+q)*s[3, 1] + q^2*s[4]

  `monomial`()

    The monomial basis of the Symmetric Functions

    EXAMPLES:

        sage: SymmetricFunctions(QQ).monomial()
        Symmetric Functions over Rational Field in the monomial basis

  `p`()

    The power sum basis of the Symmetric Functions

    EXAMPLES:

        sage: SymmetricFunctions(QQ).powersum()
        Symmetric Functions over Rational Field in the powersum basis

  `power`()

    The power sum basis of the Symmetric Functions

    EXAMPLES:

        sage: SymmetricFunctions(QQ).powersum()
        Symmetric Functions over Rational Field in the powersum basis

  `powersum`()

    The power sum basis of the Symmetric Functions

    EXAMPLES:

        sage: SymmetricFunctions(QQ).powersum()
        Symmetric Functions over Rational Field in the powersum basis

  `register_isomorphism`(morphism, only_conversion=False)

    Register an isomorphism between two bases of `self`, as a canonical coercion (unless the optional keyword `only_conversion` is set to `True`, in which case the isomorphism is registered as conversion only).

    EXAMPLES:

    We override the canonical coercion from the Schur basis to the powersum basis by a (stupid!) map \(s_\lambda\mapsto 2p_\lambda\).

        sage: Sym = SymmetricFunctions(QQ['zorglub']) # make sure we are not going to screw up later tests
        sage: s = Sym.s(); p = Sym.p().dual_basis()
        sage: phi = s.module_morphism(diagonal = lambda t: 2, codomain = p)
        sage: phi(s[2, 1])
        2*d_p[2, 1]
        sage: Sym.register_isomorphism(phi)
        sage: p(s[2,1])
        2*d_p[2, 1]

    The map is supposed to implement the canonical isomorphism between the two bases. Otherwise, the results will be mathematically wrong, as above. Use with care!

  `s`()

    The Schur basis of the Symmetric Functions

    EXAMPLES:

        sage: SymmetricFunctions(QQ).schur()
        Symmetric Functions over Rational Field in the Schur basis

  `schur`()

    The Schur basis of the Symmetric Functions

    EXAMPLES:

        sage: SymmetricFunctions(QQ).schur()
        Symmetric Functions over Rational Field in the Schur basis

  `w`(coerce_h=True, coerce_e=False, coerce_p=False)

    The Witt basis of the symmetric functions.

    EXAMPLES:

        sage: SymmetricFunctions(QQ).witt()
        Symmetric Functions over Rational Field in the Witt basis
        sage: SymmetricFunctions(QQ).witt(coerce_p=True)
        Symmetric Functions over Rational Field in the Witt basis
        sage: SymmetricFunctions(QQ).witt(coerce_h=False, coerce_e=True, coerce_p=True)
        Symmetric Functions over Rational Field in the Witt basis

  `witt`(coerce_h=True, coerce_e=False, coerce_p=False)

    The Witt basis of the symmetric functions.

    EXAMPLES:

        sage: SymmetricFunctions(QQ).witt()
        Symmetric Functions over Rational Field in the Witt basis
        sage: SymmetricFunctions(QQ).witt(coerce_p=True)
        Symmetric Functions over Rational Field in the Witt basis
        sage: SymmetricFunctions(QQ).witt(coerce_h=False, coerce_e=True, coerce_p=True)
        Symmetric Functions over Rational Field in the Witt basis

  `zonal`()

    The zonal basis of the Symmetric Functions

    EXAMPLES:

        sage: SymmetricFunctions(QQ).zonal()
        Symmetric Functions over Rational Field in the zonal basis

class `sage.combinat.sf.sf.``SymmetricaConversionOnBasis`(t, domain, codomain)

  Initialization of `self`.

  INPUT:

  - `t` – a function taking a monomial in CombinatorialFreeModule(QQ, Partitions()),

      and returning a (partition, coefficient) list.

  - `domain`, `codomain` – parents

  Construct a function mapping a partition to an element of `codomain`.

  This is a temporary quick hack to wrap around the existing symmetrica conversions, without changing their specs.

  EXAMPLES:

      sage: Sym = SymmetricFunctions(QQ[x])
      sage: p = Sym.p(); s = Sym.s()
      sage: def t(x) : [(p,c)] = x; return [ (p,2*c), (p.conjugate(), c) ]
      sage: f = sage.combinat.sf.sf.SymmetricaConversionOnBasis(t, p, s)
      sage: f(Partition([3,1]))
      s[2, 1, 1] + 2*s[3, 1]

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