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
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
The homogeneous basis of this algebra.
See also
EXAMPLES:
sage: kh3 = SymmetricFunctions(QQ).kBoundedSubspace(3,1).khomogeneous()
sage: TestSuite(kh3).run()
The \(k\)-Schur basis of this algebra.
See also
EXAMPLES:
sage: ks3 = SymmetricFunctions(QQ).kBoundedSubspace(3,1).kschur()
sage: TestSuite(ks3).run()
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]
Return the retract of sym from the ring of symmetric functions to self.
INPUT:
OUTPUT:
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
Bases: sage.categories.realizations.Category_realization_of_parent
The category of bases for the \(k\)-bounded subspace of symmetric functions.
Returns the monomial expansion of self in \(n\) variables.
INPUT:
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
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:
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]
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
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
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
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
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]
Return standard scalar product between self and x.
INPUT:
See also
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
Return an element of self.
EXAMPLES:
sage: SymmetricFunctions(QQ['t']).kschur(3).an_element()
2*ks3[] + 2*ks3[1] + 3*ks3[2]
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:
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]
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:
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[]
Return the counit of element.
The counit is the constant term of element.
INPUT:
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
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
Return the basis element indexing 1.
EXAMPLES:
sage: ks3 = SymmetricFunctions(QQ).kschur(3,1)
sage: ks3.one() # indirect doctest
ks3[]
Return the degree n transition matrix between self and other.
INPUT:
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]
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]
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. |
Returns the K-\(k\)-Schur function, as embedded inside the affine zero Hecke algebra.
INPUT:
OUTPUT:
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
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:
OUTPUT:
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
Returns the lift of a \(k\)-bounded symmetric function.
INPUT:
\(k\)-bounded partition (then x corresponds to the basis element indexed by x)
OUTPUT:
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)
Returns the product of the two K-\(k\)-Schur functions.
INPUT:
OUTPUT:
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]
Returns the retract of a symmetric function.
INPUT:
OUTPUT:
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[]
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]
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
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:
OUTPUT:
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[]