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# \(k\)-Schur Functions

class `sage.combinat.sf.new_kschur.``KBoundedSubspace`(Sym, k, t='t')

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

  This class implements 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 subring of the ring of symmetric functions generated by the complete homogeneous symmetric functions \(h_i\) for \(1\le i \le k\).

  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: KB = Sym.kBoundedSubspace(3); KB
      3-bounded Symmetric Functions over Univariate Polynomial Ring in t over Rational Field

  The \(k\)-Schur function basis can be constructed as follows:

      sage: ks = KB.kschur(); ks
      3-bounded Symmetric Functions over Univariate Polynomial Ring in t over Rational Field in the 3-Schur basis

  `K_kschur`()

    Returns the \(k\)-bounded basis called the K-\(k\)-Schur basis. See [Morse11] and [LamSchillingShimozono10].

    REFERENCES:

    [Morse11] | J. Morse, Combinatorics of the K-theory of affine Grassmannians, Adv. in Math., Volume 229, Issue 5, pp. 2950–2984.

    [LamSchillingShimozono10] | T. Lam, A. Schilling, M.Shimozono, K-theory Schubert calculus of the affine Grassmannian, Compositio Math. 146 (2010), 811-852.

    EXAMPLES:

        sage: kB = SymmetricFunctions(QQ).kBoundedSubspace(3,1)
        sage: g = kB.K_kschur()
        sage: g
        3-bounded Symmetric Functions over Rational Field with t=1 in the K-3-Schur basis
        sage: kB = SymmetricFunctions(QQ['t']).kBoundedSubspace(3)
        sage: g = kB.K_kschur()
        Traceback (most recent call last):
        ...
        ValueError: This basis only exists for t=1

  `khomogeneous`()

    The homogeneous basis of this algebra.

    See also

    `kHomogeneous()`

    EXAMPLES:

        sage: kh3 = SymmetricFunctions(QQ).kBoundedSubspace(3,1).khomogeneous()
        sage: TestSuite(kh3).run()

  `kschur`()

    The \(k\)-Schur basis of this algebra.

    See also

    `kSchur()`

    EXAMPLES:

        sage: ks3 = SymmetricFunctions(QQ).kBoundedSubspace(3,1).kschur()
        sage: TestSuite(ks3).run()

  `realizations`()

    A list of realizations of this algebra.

    EXAMPLES:

        sage: SymmetricFunctions(QQ).kBoundedSubspace(3,1).realizations()
        [3-bounded Symmetric Functions over Rational Field with t=1 in the 3-Schur basis also with t=1, 3-bounded Symmetric Functions over Rational Field with t=1 in the 3-bounded homogeneous basis, 3-bounded Symmetric Functions over Rational Field with t=1 in the K-3-Schur basis]

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

  `retract`(sym)

    Return the retract of `sym` from the ring of symmetric functions to `self`.

    INPUT:

    - `sym` – a symmetric function

    OUTPUT:

    - the analogue of the symmetric function in the \(k\)-bounded subspace (if possible)

    EXAMPLES:

        sage: Sym = SymmetricFunctions(QQ)
        sage: s = Sym.schur()
        sage: KB = Sym.kBoundedSubspace(3,1); KB
        3-bounded Symmetric Functions over Rational Field with t=1
        sage: KB.retract(s[2]+s[3])
        ks3[2] + ks3[3]
        sage: KB.retract(s[2,1,1])
        Traceback (most recent call last):
        ...
        ValueError: s[2, 1, 1] is not in the image

class `sage.combinat.sf.new_kschur.``KBoundedSubspaceBases`(base, t='t')

  Bases: `sage.categories.realizations.Category_realization_of_parent`

  The category of bases for the \(k\)-bounded subspace of symmetric functions.

  class `ElementMethods`
    `expand`(*args, **kwargs)

      Returns the monomial expansion of `self` in \(n\) variables.

      INPUT:

      - `n` – positive integer

      OUTPUT: monomial expansion of `self` in \(n\) variables

      EXAMPLES:

          sage: Sym = SymmetricFunctions(QQ)
          sage: ks = Sym.kschur(3,1)
          sage: ks[3,1].expand(2)
          x0^4 + 2*x0^3*x1 + 2*x0^2*x1^2 + 2*x0*x1^3 + x1^4
          sage: s = Sym.schur()
          sage: ks[3,1].expand(2) == s(ks[3,1]).expand(2)
          True

          sage: Sym = SymmetricFunctions(QQ['t'])
          sage: ks = Sym.kschur(3)
          sage: f = ks[3,2]-ks[1]
          sage: f.expand(2)
          t^2*x0^5 + (t^2 + t)*x0^4*x1 + (t^2 + t + 1)*x0^3*x1^2 + (t^2 + t + 1)*x0^2*x1^3 + (t^2 + t)*x0*x1^4 + t^2*x1^5 - x0 - x1

    `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 [SZ.2001].

      INPUT:

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

      REFERENCES:

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

      EXAMPLES:

          sage: Sym = SymmetricFunctions(FractionField(QQ['t']))
          sage: ks = Sym.kschur(4)
          sage: s = Sym.schur()
          sage: s(ks([3,1,1]).hl_creation_operator([1]))
          (t-1)*s[2, 2, 1, 1] + t^2*s[3, 1, 1, 1] + (t^3+t^2-t)*s[3, 2, 1] + (t^3-t^2)*s[3, 3] + (t^4+t^3)*s[4, 1, 1] + t^4*s[4, 2] + t^5*s[5, 1]
          sage: ks([3,1,1]).hl_creation_operator([1])
          (t-1)*ks4[2, 2, 1, 1] + t^2*ks4[3, 1, 1, 1] + t^3*ks4[3, 2, 1] + (t^3-t^2)*ks4[3, 3] + t^4*ks4[4, 1, 1]

          sage: Sym = SymmetricFunctions(QQ)
          sage: ks = Sym.kschur(4,t=1)
          sage: ks([3,1,1]).hl_creation_operator([1])
          ks4[3, 1, 1, 1] + ks4[3, 2, 1] + ks4[4, 1, 1]

    `is_schur_positive`(*args, **kwargs)

      Returns whether `self` is Schur positive.

      EXAMPLES:

          sage: Sym = SymmetricFunctions(QQ)
          sage: ks = Sym.kschur(3,1)
          sage: f = ks[3,2]+ks[1]
          sage: f.is_schur_positive()
          True
          sage: f = ks[3,2]-ks[1]
          sage: f.is_schur_positive()
          False

          sage: Sym = SymmetricFunctions(QQ['t'])
          sage: ks = Sym.kschur(3)
          sage: f = ks[3,2]+ks[1]
          sage: f.is_schur_positive()
          True
          sage: f = ks[3,2]-ks[1]
          sage: f.is_schur_positive()
          False

    `omega`()

      Returns the \(\omega\) operator on `self`.

      At \(t=1\), \(\omega\) maps the \(k\)-Schur function \(s^{(k)}_\lambda\) to \(s^{(k)}_{\lambda^{(k)}}\), where \(\lambda^{(k)}\) is the \(k\)-conjugate of the partition \(\lambda\).

      See also

      `k_conjugate()`.

      For generic \(t\), \(\omega\) sends \(s^{(k)}_\lambda[X;t]\) to \(t^d s^{(k)}_{\lambda^{(k)}}[X;1/t]\), where \(d\) is the size of the core of \(\lambda\) minus the size of \(\lambda\). Most of the time, this result is not in the \(k\)-bounded subspace.

      See also

      `omega_t_inverse()`.

      EXAMPLES:

          sage: Sym = SymmetricFunctions(QQ)
          sage: ks = Sym.kschur(3,1)
          sage: ks[2,2,1,1].omega()
          ks3[2, 2, 2]
          sage: kh = Sym.khomogeneous(3)
          sage: kh[3].omega()
          h3[1, 1, 1] - 2*h3[2, 1] + h3[3]

          sage: Sym = SymmetricFunctions(FractionField(QQ['t']))
          sage: ks = Sym.kschur(3)
          sage: ks[3,1,1].omega()
          Traceback (most recent call last):
          ...
          ValueError: t*s[2, 1, 1, 1] + s[3, 1, 1] is not in the image

    `omega_t_inverse`()

      Returns the map \(t\to 1/t\) composed with \(\omega\) on `self`.

      Unlike the map `omega()`, the result of `omega_t_inverse()` lives in the \(k\)-bounded subspace and hence will return an element even for generic \(t\). For \(t=1\), `omega()` and `omega_t_inverse()` return the same result.

      EXAMPLES:

          sage: Sym = SymmetricFunctions(FractionField(QQ['t']))
          sage: ks = Sym.kschur(3)
          sage: ks[3,1,1].omega_t_inverse()
          1/t*ks3[2, 1, 1, 1]
          sage: ks[3,2].omega_t_inverse()
          1/t^2*ks3[1, 1, 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)

      See also

      `scalar()`

      EXAMPLES:

          sage: Sym = SymmetricFunctions(QQ['t'])
          sage: ks3 = Sym.kschur(3)
          sage: ks3[3,2,1].scalar( ks3[2,2,2] )
          t^3 + t
          sage: dks3 = Sym.kBoundedQuotient(3).dks()
          sage: [ks3[3,2,1].scalar(dks3(la)) for la in Partitions(6, max_part=3)]
          [0, 1, 0, 0, 0, 0, 0]
          sage: dks3 = Sym.kBoundedQuotient(3,t=1).dks()
          sage: [ks3[2,2,2].scalar(dks3(la)) for la in Partitions(6, max_part=3)]
          [0, t - 1, 0, 1, 0, 0, 0]
          sage: ks3 = Sym.kschur(3,t=1)
          sage: [ks3[2,2,2].scalar(dks3(la)) for la in Partitions(6, max_part=3)]
          [0, 0, 0, 1, 0, 0, 0]
          sage: kH = Sym.khomogeneous(4)
          sage: kH([2,2,1]).scalar(ks3[2,2,1])
          3

      TESTS:

          sage: Sym = SymmetricFunctions(QQ)
          sage: ks3 = Sym.kschur(3,1)
          sage: ks3(1).scalar(ks3([]))
          1

  class `KBoundedSubspaceBases.``ParentMethods`
    `an_element`()

      Return an element of `self`.

      EXAMPLES:

          sage: SymmetricFunctions(QQ['t']).kschur(3).an_element()
          2*ks3[] + 2*ks3[1] + 3*ks3[2]

    `antipode`(element)

      Return the antipode on `self` by lifting to the space of symmetric functions, computing the antipode, and then converting to `self.parent()`. This is only the antipode for \(t = 1\) and for other values of \(t\) the result may not be in the space where the \(k\)-Schur functions live.

      INPUT:

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

      EXAMPLES:

          sage: Sym = SymmetricFunctions(QQ)
          sage: ks3 = Sym.kschur(3,1)
          sage: ks3[3,2].antipode()
          -ks3[1, 1, 1, 1, 1]
          sage: ks3.antipode(ks3[3,2])
          -ks3[1, 1, 1, 1, 1]

    `coproduct`(element)

      Return the coproduct operation on `element`.

      The coproduct is first computed on the homogeneous basis if \(t=1\) and on the Hall-Littlewood `Qp` basis otherwise. The result is computed then converted to the tensor squared of `self.parent()`.

      INPUT:

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

      EXAMPLES:

          sage: Sym = SymmetricFunctions(QQ)
          sage: ks3 = Sym.kschur(3,1)
          sage: ks3[2,1].coproduct()
          ks3[] # ks3[2, 1] + ks3[1] # ks3[1, 1] + ks3[1] # ks3[2] + ks3[1, 1] # ks3[1] + ks3[2] # ks3[1] + ks3[2, 1] # ks3[]
          sage: h3 = Sym.khomogeneous(3)
          sage: h3[2,1].coproduct()
          h3[] # h3[2, 1] + h3[1] # h3[1, 1] + h3[1] # h3[2] + h3[1, 1] # h3[1] + h3[2] # h3[1] + h3[2, 1] # h3[]
          sage: ks3t = SymmetricFunctions(FractionField(QQ['t'])).kschur(3)
          sage: ks3t[2,1].coproduct()
          ks3[] # ks3[2, 1] + ks3[1] # ks3[1, 1] + ks3[1] # ks3[2] + ks3[1, 1] # ks3[1] + ks3[2] # ks3[1] + ks3[2, 1] # ks3[]
          sage: ks3t[3,1].coproduct()
          ks3[] # ks3[3, 1] + ks3[1] # ks3[2, 1] + (t+1)*ks3[1] # ks3[3] + ks3[1, 1] # ks3[2] + ks3[2] # ks3[1, 1]
          + (t+1)*ks3[2] # ks3[2] + ks3[2, 1] # ks3[1] + (t+1)*ks3[3] # ks3[1] + ks3[3, 1] # ks3[]
          sage: h3.coproduct(h3[2,1])
          h3[] # h3[2, 1] + h3[1] # h3[1, 1] + h3[1] # h3[2] + h3[1, 1] # h3[1] + h3[2] # h3[1] + h3[2, 1] # h3[]

    `counit`(element)

      Return the counit of `element`.

      The counit is the constant term of `element`.

      INPUT:

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

      EXAMPLES:

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

    `degree_on_basis`(b)

      Return the degree of the basis element indexed by \(b\).

      INPUT: - `b` – a partition

      EXAMPLES:

          sage: ks3 = SymmetricFunctions(QQ).kschur(3,1)
          sage: ks3.degree_on_basis(Partition([3,2]))
          5

    `one_basis`()

      Return the basis element indexing `1`.

      EXAMPLES:

          sage: ks3 = SymmetricFunctions(QQ).kschur(3,1)
          sage: ks3.one()  # indirect doctest
          ks3[]

    `transition_matrix`(other, n)

      Return the degree `n` transition matrix between `self` and `other`.

      INPUT:

      - `other` – a basis in the ring of symmetric functions
      - `n` – a positive integer

      The entry in the \(i^{th}\) row and \(j^{th}\) column is the coefficient obtained by writing the \(i^{th}\) element of the basis of `self` in terms of the basis `other`, and extracting the \(j^{th}\) coefficient.

      EXAMPLES:

          sage: Sym = SymmetricFunctions(QQ); s = Sym.schur()
          sage: ks3 = Sym.kschur(3,1)
          sage: ks3.transition_matrix(s,5)
          [1 1 1 0 0 0 0]
          [0 1 0 1 0 0 0]
          [0 0 1 0 1 0 0]
          [0 0 0 1 0 1 0]
          [0 0 0 0 1 1 1]

          sage: Sym = SymmetricFunctions(QQ['t'])
          sage: s = Sym.schur()
          sage: ks = Sym.kschur(3)
          sage: ks.transition_matrix(s,5)
          [t^2   t   1   0   0   0   0]
          [  0   t   0   1   0   0   0]
          [  0   0   t   0   1   0   0]
          [  0   0   0   t   0   1   0]
          [  0   0   0   0 t^2   t   1]

  `KBoundedSubspaceBases.``super_categories`()

    The super categories of `self`.

    EXAMPLES:

        sage: Sym = SymmetricFunctions(QQ['t'])
        sage: from sage.combinat.sf.new_kschur import KBoundedSubspaceBases
        sage: KB = Sym.kBoundedSubspace(3)
        sage: KBB = KBoundedSubspaceBases(KB); KBB
        Category of k bounded subspace bases of 3-bounded Symmetric Functions over Univariate Polynomial Ring in t over Rational Field
        sage: KBB.super_categories()
        [Category of realizations of 3-bounded Symmetric Functions over Univariate Polynomial Ring in t over Rational Field,
         Join of Category of graded modules with basis over Univariate Polynomial Ring in t over Rational Field
             and Category of coalgebras with basis over Univariate Polynomial Ring in t over Rational Field
             and Category of subobjects of sets]

class `sage.combinat.sf.new_kschur.``K_kSchur`(kBoundedRing)

  Bases: `sage.combinat.free_module.CombinatorialFreeModule`

  This class implements the basis of the \(k\)-bounded subspace called the K-\(k\)-Schur basis. See [Morse2011], [LamSchillingShimozono2010].

  REFERENCES:

  [Morse2011] | J. Morse, Combinatorics of the K-theory of affine Grassmannians, Adv. in Math., Volume 229, Issue 5, pp. 2950–2984.

  [LamSchillingShimozono2010] | T. Lam, A. Schilling, M.Shimozono, K-theory Schubert calculus of the affine Grassmannian, Compositio Math. 146 (2010), 811-852.

  `K_k_Schur_non_commutative_variables`(la)

    Returns the K-\(k\)-Schur function, as embedded inside the affine zero Hecke algebra.

    INPUT:

    - `la` – A \(k\)-bounded Partition

    OUTPUT:

    - An element of the affine zero Hecke algebra.

    EXAMPLES:

        sage: g = SymmetricFunctions(QQ).kBoundedSubspace(3,1).K_kschur()
        sage: g.K_k_Schur_non_commutative_variables([2,1])
        T[3,1,0] + T[1,2,0] + T[3,2,0] + T[0,1,0] + T[2,0,1] + T[0,3,0] + T[2,0,3] + T[0,3,1] + T[2,3,2] + T[2,3,1] + T[3,1,2] + T[1,2,1] - T[2,0] - T[3,1]
        sage: g.K_k_Schur_non_commutative_variables([])
        1
        sage: g.K_k_Schur_non_commutative_variables([4,1])
        Traceback (most recent call last):
        ...
        ValueError: Partition should be 3-bounded

  `homogeneous_basis_noncommutative_variables_zero_Hecke`(la)

    Returns the homogeneous basis element indexed by `la`, viewed as an element inside the affine zero Hecke algebra. For the code, see method _homogeneous_basis.

    INPUT:

    - `la` – A \(k\)-bounded partition

    OUTPUT:

    - An element of the affine zero Hecke algebra.

    EXAMPLES:

        sage: g = SymmetricFunctions(QQ).kBoundedSubspace(3,1).K_kschur()
        sage: g.homogeneous_basis_noncommutative_variables_zero_Hecke([2,1])
        T[2,1,0] + T[3,1,0] + T[1,2,0] + T[3,2,0] + T[0,1,0] + T[2,0,1] + T[1,0,3] + T[0,3,0] + T[2,0,3] + T[0,3,2] + T[0,3,1] + T[2,3,2] + T[3,2,1] + T[2,3,1] + T[3,1,2] + T[1,2,1] - T[1,0] - 2*T[2,0] - T[0,3] - T[3,2] - 2*T[3,1] - T[2,1]
        sage: g.homogeneous_basis_noncommutative_variables_zero_Hecke([])
        1

  `lift`(x)

    Returns the lift of a \(k\)-bounded symmetric function.

    INPUT:

    - `x` – An expression in the K-\(k\)-Schur basis. Equivalently, `x` can be a

        \(k\)-bounded partition (then `x` corresponds to the basis element indexed by `x`)

    OUTPUT:

    - A symmetric function.

    EXAMPLES:

        sage: g = SymmetricFunctions(QQ).kBoundedSubspace(3,1).K_kschur()
        sage: g.lift([2,1])
        h[2] + h[2, 1] - h[3]
        sage: g.lift([])
        h[]
        sage: g.lift([4,1])
        Traceback (most recent call last):
        ...
        TypeError: do not know how to make x (= [4, 1]) an element of self (=3-bounded Symmetric Functions over Rational Field with t=1 in the K-3-Schur basis)

  `product`(x, y)

    Returns the product of the two K-\(k\)-Schur functions.

    INPUT:

    - `x`, `y` – elements of the \(k\)-bounded subspace, in the K-\(k\)-Schur basis.

    OUTPUT:

    - An element of the \(k\)-bounded subspace, in the K-\(k\)-Schur basis

    EXAMPLES:

        sage: g = SymmetricFunctions(QQ).kBoundedSubspace(3,1).K_kschur()
        sage: g.product(g([2,1]), g[1])
        -2*Kks3[2, 1] + Kks3[2, 1, 1] + Kks3[2, 2]
        sage: g.product(g([2,1]), g([]))
        Kks3[2, 1]

  `retract`(x)

    Returns the retract of a symmetric function.

    INPUT:

    - `x` – A symmetric function.

    OUTPUT:

    - A \(k\)-bounded symmetric function in the K-\(k\)-Schur basis.

    EXAMPLES:

        sage: g = SymmetricFunctions(QQ).kBoundedSubspace(3,1).K_kschur()
        sage: m = SymmetricFunctions(QQ).m()
        sage: g.retract(m[2,1])
        -2*Kks3[1] + 4*Kks3[1, 1] - 2*Kks3[1, 1, 1] - Kks3[2] + Kks3[2, 1]
        sage: g.retract(m([]))
        Kks3[]

class `sage.combinat.sf.new_kschur.``kHomogeneous`(kBoundedRing)

  Bases: `sage.combinat.free_module.CombinatorialFreeModule`

  Space of \(k\)-bounded homogeneous symmetric functions.

  EXAMPLES:

      sage: Sym = SymmetricFunctions(QQ)
      sage: kH = Sym.khomogeneous(3)
      sage: kH[2]
      h3[2]
      sage: kH[2].lift()
      h[2]

class `sage.combinat.sf.new_kschur.``kSchur`(kBoundedRing)

  Bases: `sage.combinat.free_module.CombinatorialFreeModule`

  Space of \(k\)-Schur functions.

  EXAMPLES:

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

  The \(k\)-Schur function basis can be constructed as follows:

      sage: ks3 = KB.kschur(); ks3
      3-bounded Symmetric Functions over Univariate Polynomial Ring in t over Rational Field in the 3-Schur basis

  We can convert to any basis of the ring of symmetric functions and, whenever it makes sense, also the other way round:

      sage: s = Sym.schur()
      sage: s(ks3([3,2,1]))
      s[3, 2, 1] + t*s[4, 1, 1] + t*s[4, 2] + t^2*s[5, 1]
      sage: t = Sym.base_ring().gen()
      sage: ks3(s([3, 2, 1]) + t*s([4, 1, 1]) + t*s([4, 2]) + t^2*s([5, 1]))
      ks3[3, 2, 1]
      sage: s(ks3[2, 1, 1])
      s[2, 1, 1] + t*s[3, 1]
      sage: ks3(s[2, 1, 1] + t*s[3, 1])
      ks3[2, 1, 1]

  \(k\)-Schur functions are indexed by partitions with first part \(\le k\). Constructing a \(k\)-Schur function for a larger partition raises an error:

      sage: ks3([4,3,2,1]) #
      Traceback (most recent call last):
      ...
      TypeError: do not know how to make x (= [4, 3, 2, 1]) an element of self (=3-bounded Symmetric Functions over Univariate Polynomial Ring in t over Rational Field in the 3-Schur basis)

  Similarly, attempting to convert a function that is not in the linear span of the \(k\)-Schur functions raises an error:

      sage: ks3(s([4]))
      Traceback (most recent call last):
      ...
      ValueError: s[4] is not in the image

  Note that the product of \(k\)-Schur functions is not guaranteed to be in the space spanned by the \(k\)-Schurs. In general, we only have that a \(k\)-Schur times a \(j\)-Schur function is in the \((k+j)\)-bounded subspace. The multiplication of two \(k\)-Schur functions thus generally returns the product of the lift of the functions to the ambient symmetric function space. If the result happens to lie in the \(k\)-bounded subspace, then the result is cast into the \(k\)-Schur basis:

      sage: ks2 = Sym.kBoundedSubspace(2).kschur()
      sage: ks2[1] * ks2[1]
      ks2[1, 1] + ks2[2]
      sage: ks2[1] * ks2[2]
      s[2, 1] + s[3]

  Because the target space of the product of a \(k\)-Schur and a \(j\)-Schur has several possibilities, the product of a \(k\)-Schur and \(j\)-Schur function is not implemented for distinct \(k\) and \(j\). Let us show how to get around this ‘manually’:

      sage: ks3 = Sym.kBoundedSubspace(3).kschur()
      sage: ks2([2,1]) * ks3([3,1])
      Traceback (most recent call last):
      ...
      TypeError: unsupported operand parent(s) for '*': '2-bounded Symmetric Functions over Univariate Polynomial Ring in t over Rational Field in the 2-Schur basis' and '3-bounded Symmetric Functions over Univariate Polynomial Ring in t over Rational Field in the 3-Schur basis'

  The workaround:

      sage: f = s(ks2([2,1])) * s(ks3([3,1])); f # Convert to Schur functions first and multiply there.
      s[3, 2, 1, 1] + s[3, 2, 2] + (t+1)*s[3, 3, 1] + s[4, 1, 1, 1]
      + (2*t+2)*s[4, 2, 1] + (t^2+t+1)*s[4, 3] + (2*t+1)*s[5, 1, 1]
      + (t^2+2*t+1)*s[5, 2] + (t^2+2*t)*s[6, 1] + t^2*s[7]

  or:

      sage: f = ks2[2,1].lift() * ks3[3,1].lift()
      sage: ks5 = Sym.kBoundedSubspace(5).kschur()
      sage: ks5(f) # The product of a 'ks2' with a 'ks3' is a 'ks5'.
      ks5[3, 2, 1, 1] + ks5[3, 2, 2] + (t+1)*ks5[3, 3, 1] + ks5[4, 1, 1, 1]
      + (t+2)*ks5[4, 2, 1] + (t^2+t+1)*ks5[4, 3] + (t+1)*ks5[5, 1, 1] + ks5[5, 2]

  For other technical reasons, taking powers of \(k\)-Schur functions is not implemented, even when the answer is still in the \(k\)-bounded subspace:

      sage: ks2([1])^2
      Traceback (most recent call last):
      ...
      TypeError: unsupported operand type(s) for ** or pow(): 'kSchur_with_category.element_class' and 'int'

  Todo

  Get rid of said technical “reasons”.

  However, at \(t=1\), the product of \(k\)-Schur functions is in the span of the \(k\)-Schur functions always. Below are some examples at \(t=1\)

      sage: ks3 = Sym.kBoundedSubspace(3, t=1).kschur(); ks3
      3-bounded Symmetric Functions over Univariate Polynomial Ring in t over Rational Field with t=1 in the 3-Schur basis also with t=1
      sage: s = SymmetricFunctions(ks3.base_ring()).schur()
      sage: ks3(s([3]))
      ks3[3]
      sage: s(ks3([3,2,1]))
      s[3, 2, 1] + s[4, 1, 1] + s[4, 2] + s[5, 1]
      sage: ks3([2,1])^2    # taking powers works for t=1
      ks3[2, 2, 1, 1] + ks3[2, 2, 2] + ks3[3, 1, 1, 1]

  TESTS:

  Check that trac ticket #13743 is fixed:

      sage: ks3 = SymmetricFunctions(QQ).kschur(3, 1)
      sage: f = ks3[2,1]
      sage: f.coefficient(f.support()[0])
      1

  `product`(left, right)

    Take the product of two \(k\)-Schur functions.

    If \(t \neq 1\), then take the product by lifting to the Schur functions and then retracting back into the the \(k\)-bounded subspace (if possible).

    If \(t=1\), then the product is done using `_product_from_combinatorial_algebra_multiply()` and this method calls `_multiply_basis()`.

    INPUT:

    - `left`, `right` – elements of the \(k\)-Schur functions

    OUTPUT:

    - an element of the \(k\)-Schur functions

    EXAMPLES:

        sage: Sym = SymmetricFunctions(QQ['t'])
        sage: ks3 = Sym.kschur(3,1)
        sage: kH = Sym.khomogeneous(3)
        sage: ks3(kH[2,1,1])
        ks3[2, 1, 1] + ks3[2, 2] + ks3[3, 1]
        sage: ks3([])*kH[2,1,1]
        ks3[2, 1, 1] + ks3[2, 2] + ks3[3, 1]
        sage: ks3([3,3,3,2,2,1,1,1])^2
        ks3[3, 3, 3, 3, 3, 3, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1]
        sage: ks3([3,3,3,2,2,1,1,1])*ks3([2,2,2,2,2,1,1,1,1])
        ks3[3, 3, 3, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1]
        sage: ks3([2,2,1,1,1,1])*ks3([2,2,2,1,1,1,1])
        ks3[2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1] + ks3[2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1]
        sage: ks3[2,1]^2
        ks3[2, 2, 1, 1] + ks3[2, 2, 2] + ks3[3, 1, 1, 1]
        sage: ks3 = Sym.kschur(3)
        sage: ks3[2,1]*ks3[2,1]
        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]

    TESTS:

        sage: Sym = SymmetricFunctions(QQ['t'])
        sage: ks3 = Sym.kschur(3,1)
        sage: kH = Sym.khomogeneous(3)
        sage: ks3.product( ks3([]), ks3([]) )
        ks3[]
        sage: ks3.product( ks3([]), kH([]) )
        ks3[]
        sage: ks3 = Sym.kschur(3)
        sage: ks3.product( ks3([]), ks3([]) )
        ks3[]

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