Quotient of symmetric function space by ideal generated by Hall-Littlewood symmetric functions¶
The quotient of symmetric functions by the ideal generated by the Hall-Littlewood P symmetric functions indexed by partitions with first part greater than \(k\). When \(t=1\) this space is the quotient of the symmetric functions by the ideal generated by the monomial symmetric functions indexed by partitions with first part greater than \(k\).
AUTHORS:
- Chris Berg (2012-12-01)
- Mike Zabrocki - \(k\)-bounded Hall Littlewood P and dual \(k\)-Schur functions (2012-12-02)
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sage.combinat.sf.k_dual.
AffineSchurFunctions
¶ This basis is dual to the \(k\)-Schur functions at \(t=1\). This realization follows the monomial expansion given by Lam [Lam2006].
REFERENCES:
[Lam2006] T. Lam, Schubert polynomials for the affine Grassmannian, J. Amer. Math. Soc., 21 (2008), 259-281.
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sage.combinat.sf.k_dual.
DualkSchurFunctions
¶ This basis is dual to the \(k\)-Schur functions. The expansion is given in Section 4.12 of [LLMSSZ]. When \(t=1\) this basis is equal to the
AffineSchurFunctions
and that basis is more efficient in this case.REFERENCES:
[LLMSSZ] T. Lam, L. Lapointe, J. Morse, A. Schilling, M. Shimozono, M. Zabrocki, k-Schur functions and affine Schubert calculus.
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class
sage.combinat.sf.k_dual.
KBoundedQuotient
(Sym, k, t='t')¶ Bases:
sage.structure.unique_representation.UniqueRepresentation
,sage.structure.parent.Parent
Initialization of the ring of Symmetric functions modulo the ideal of monomial symmetric functions which are indexed by partitions whose first part is greater than \(k\).
INPUT:
Sym
– an element of classsage.combinat.sf.sf.SymmetricFunctions
k
– a positive integerR
– a ring
EXAMPLES:
sage: Sym = SymmetricFunctions(QQ) sage: Q = Sym.kBoundedQuotient(3,t=1) sage: Q 3-Bounded Quotient of Symmetric Functions over Rational Field with t=1 sage: km = Q.km() sage: km 3-Bounded Quotient of Symmetric Functions over Rational Field with t=1 in the 3-bounded monomial basis sage: F = Q.affineSchur() sage: F(km(F[3,1,1])) == F[3,1,1] True sage: km(F(km([3,2]))) == km[3,2] True sage: F[3,2].lift() m[1, 1, 1, 1, 1] + m[2, 1, 1, 1] + m[2, 2, 1] + m[3, 1, 1] + m[3, 2] 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] sage: F[1,2] Traceback (most recent call last): ... ValueError: [1, 2] is not a valid partition sage: F[4,2] Traceback (most recent call last): ... ValueError: Partition is not 3-bounded 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: HLPk = Q.kHallLittlewoodP() sage: HLPk[2,1]*HLPk[2,1] 4*HLP3[2, 2, 1, 1] + 6*HLP3[2, 2, 2] + 2*HLP3[3, 2, 1] + 2*HLP3[3, 3] sage: dks = Q.dual_k_Schur() sage: dks[2,1]*dks[2,1] 2*dks3[1, 1, 1, 1, 1, 1] + 4*dks3[2, 1, 1, 1, 1] + 4*dks3[2, 2, 1, 1] + 4*dks3[2, 2, 2] + 2*dks3[3, 1, 1, 1] + 4*dks3[3, 2, 1] + 2*dks3[3, 3]
sage: Q = Sym.kBoundedQuotient(3) Traceback (most recent call last): ... TypeError: unable to convert 't' to a rational sage: Sym = SymmetricFunctions(QQ['t'].fraction_field()) sage: Q = Sym.kBoundedQuotient(3) sage: km = Q.km() sage: F = Q.affineSchur() sage: F(km(F[3,1,1])) == F[3,1,1] True sage: km(F(km([3,2]))) == km[3,2] True sage: dks = Q.dual_k_Schur() sage: HLPk = Q.kHallLittlewoodP() sage: dks(HLPk(dks[3,1,1])) == dks[3,1,1] True sage: km(dks(km([3,2]))) == km[3,2] True sage: dks[2,1]*dks[2,1] (t^3+t^2)*dks3[1, 1, 1, 1, 1, 1] + (2*t^2+2*t)*dks3[2, 1, 1, 1, 1] + (t^2+2*t+1)*dks3[2, 2, 1, 1] + (t^2+2*t+1)*dks3[2, 2, 2] + (t+1)*dks3[3, 1, 1, 1] + (2*t+2)*dks3[3, 2, 1] + (t+1)*dks3[3, 3]
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AffineGrothendieckPolynomial
(la, m)¶ Returns the affine Grothendieck polynomial indexed by the partition
la
. Because this belongs to the completion of the algebra, and hence is an infinite sum, it computes only up to those symmetric functions of degree at mostm
. See_AffineGrothendieckPolynomial()
for the code.INPUT:
la
– A \(k\)-bounded partitionm
– An integer
EXAMPLES:
sage: Q = SymmetricFunctions(QQ).kBoundedQuotient(3,t=1) sage: Q.AffineGrothendieckPolynomial([2,1],4) 2*m3[1, 1, 1] - 8*m3[1, 1, 1, 1] + m3[2, 1] - 3*m3[2, 1, 1] - m3[2, 2]
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F
()¶ The affine Schur basis of the \(k\)-bounded quotient of symmetric functions, indexed by \(k\)-bounded partitions. This is also equal to the affine Stanley symmetric functions (see
WeylGroups.ElementMethods.stanley_symmetric_function()
) indexed by an affine Grassmannian permutation.EXAMPLES:
sage: SymmetricFunctions(QQ).kBoundedQuotient(2,t=1).affineSchur() 2-Bounded Quotient of Symmetric Functions over Rational Field with t=1 in the 2-bounded affine Schur basis
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a_realization
()¶ Returns a particular realization of
self
(the basis of \(k\)-bounded monomials if \(t=1\) and the basis of \(k\)-bounded Hall-Littlewood functions otherwise).EXAMPLES:
sage: Sym = SymmetricFunctions(QQ) sage: Q = Sym.kBoundedQuotient(3,t=1) sage: Q.a_realization() 3-Bounded Quotient of Symmetric Functions over Rational Field with t=1 in the 3-bounded monomial basis sage: Q = Sym.kBoundedQuotient(3,t=2) sage: Q.a_realization() 3-Bounded Quotient of Symmetric Functions over Rational Field with t=2 in the 3-bounded Hall-Littlewood P basis
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affineSchur
()¶ The affine Schur basis of the \(k\)-bounded quotient of symmetric functions, indexed by \(k\)-bounded partitions. This is also equal to the affine Stanley symmetric functions (see
WeylGroups.ElementMethods.stanley_symmetric_function()
) indexed by an affine Grassmannian permutation.EXAMPLES:
sage: SymmetricFunctions(QQ).kBoundedQuotient(2,t=1).affineSchur() 2-Bounded Quotient of Symmetric Functions over Rational Field with t=1 in the 2-bounded affine Schur basis
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ambient
()¶ Returns the Symmetric Functions over the same ring as
self
. This is needed to realize our ring as a quotient.
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an_element
()¶ Returns an element of the quotient ring of \(k\)-bounded symmetric functions. This method is here to make the TestSuite run properly.
EXAMPLES:
sage: Q = SymmetricFunctions(QQ).kBoundedQuotient(3,t=1) sage: Q.an_element() 2*m3[] + 2*m3[1] + 3*m3[2]
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dks
()¶ The dual \(k\)-Schur basis of the \(k\)-bounded quotient of symmetric functions, indexed by \(k\)-bounded partitions. At \(t=1\) this is also equal to the affine Schur basis and calculations will be faster using elements in the
affineSchur()
basis.EXAMPLES:
sage: SymmetricFunctions(QQ['t'].fraction_field()).kBoundedQuotient(2).dual_k_Schur() 2-Bounded Quotient of Symmetric Functions over Fraction Field of Univariate Polynomial Ring in t over Rational Field in the dual 2-Schur basis
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dual_k_Schur
()¶ The dual \(k\)-Schur basis of the \(k\)-bounded quotient of symmetric functions, indexed by \(k\)-bounded partitions. At \(t=1\) this is also equal to the affine Schur basis and calculations will be faster using elements in the
affineSchur()
basis.EXAMPLES:
sage: SymmetricFunctions(QQ['t'].fraction_field()).kBoundedQuotient(2).dual_k_Schur() 2-Bounded Quotient of Symmetric Functions over Fraction Field of Univariate Polynomial Ring in t over Rational Field in the dual 2-Schur basis
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kHLP
()¶ The Hall-Littlewood P basis of the \(k\)-bounded quotient of symmetric functions, indexed by \(k\)-bounded partitions. At \(t=1\) this basis is equal to the \(k\)-bounded monomial basis and calculations will be faster using elements in the \(k\)-bounded monomial basis (see
kmonomial()
).EXAMPLES:
sage: SymmetricFunctions(QQ['t'].fraction_field()).kBoundedQuotient(2).kHallLittlewoodP() 2-Bounded Quotient of Symmetric Functions over Fraction Field of Univariate Polynomial Ring in t over Rational Field in the 2-bounded Hall-Littlewood P basis
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kHallLittlewoodP
()¶ The Hall-Littlewood P basis of the \(k\)-bounded quotient of symmetric functions, indexed by \(k\)-bounded partitions. At \(t=1\) this basis is equal to the \(k\)-bounded monomial basis and calculations will be faster using elements in the \(k\)-bounded monomial basis (see
kmonomial()
).EXAMPLES:
sage: SymmetricFunctions(QQ['t'].fraction_field()).kBoundedQuotient(2).kHallLittlewoodP() 2-Bounded Quotient of Symmetric Functions over Fraction Field of Univariate Polynomial Ring in t over Rational Field in the 2-bounded Hall-Littlewood P basis
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km
()¶ The monomial basis of the \(k\)-bounded quotient of symmetric functions, indexed by \(k\)-bounded partitions.
EXAMPLES:
sage: SymmetricFunctions(QQ).kBoundedQuotient(2,t=1).kmonomial() 2-Bounded Quotient of Symmetric Functions over Rational Field with t=1 in the 2-bounded monomial basis
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kmonomial
()¶ The monomial basis of the \(k\)-bounded quotient of symmetric functions, indexed by \(k\)-bounded partitions.
EXAMPLES:
sage: SymmetricFunctions(QQ).kBoundedQuotient(2,t=1).kmonomial() 2-Bounded Quotient of Symmetric Functions over Rational Field with t=1 in the 2-bounded monomial basis
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lift
(la)¶ Gives the lift map from the quotient ring of \(k\)-bounded symmetric functions to the symmetric functions. This method is here to make the TestSuite run properly.
INPUT:
la
– A \(k\)-bounded partition
OUTPUT:
- The monomial element or a Hall-Littlewood P element of the symmetric functions
- indexed by the partition
la
.
EXAMPLES:
sage: Q = SymmetricFunctions(QQ).kBoundedQuotient(3,t=1) sage: Q.lift([2,1]) m[2, 1] sage: Q = SymmetricFunctions(QQ['t'].fraction_field()).kBoundedQuotient(3) sage: Q.lift([2,1]) HLP[2, 1]
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one
()¶ Returns the unit of the quotient ring of \(k\)-bounded symmetric functions. This method is here to make the TestSuite run properly.
EXAMPLES:
sage: Q = SymmetricFunctions(QQ).kBoundedQuotient(3,t=1) sage: Q.one() m3[]
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realizations
()¶ A list of realizations of the \(k\)-bounded quotient.
EXAMPLES:
sage: kQ = SymmetricFunctions(QQ['t'].fraction_field()).kBoundedQuotient(3) sage: kQ.realizations() [3-Bounded Quotient of Symmetric Functions over Fraction Field of Univariate Polynomial Ring in t over Rational Field in the 3-bounded monomial basis, 3-Bounded Quotient of Symmetric Functions over Fraction Field of Univariate Polynomial Ring in t over Rational Field in the 3-bounded Hall-Littlewood P basis, 3-Bounded Quotient of Symmetric Functions over Fraction Field of Univariate Polynomial Ring in t over Rational Field in the 3-bounded affine Schur basis, 3-Bounded Quotient of Symmetric Functions over Fraction Field of Univariate Polynomial Ring in t over Rational Field in the dual 3-Schur basis] sage: HLP = kQ.ambient().hall_littlewood().P() sage: all( rzn(HLP[3,2,1]).lift() == HLP[3,2,1] for rzn in kQ.realizations()) True sage: kQ = SymmetricFunctions(QQ).kBoundedQuotient(3,1) sage: kQ.realizations() [3-Bounded Quotient of Symmetric Functions over Rational Field with t=1 in the 3-bounded monomial basis, 3-Bounded Quotient of Symmetric Functions over Rational Field with t=1 in the 3-bounded Hall-Littlewood P basis, 3-Bounded Quotient of Symmetric Functions over Rational Field with t=1 in the 3-bounded affine Schur basis, 3-Bounded Quotient of Symmetric Functions over Rational Field with t=1 in the dual 3-Schur basis] sage: m = kQ.ambient().m() sage: all( rzn(m[3,2,1]).lift() == m[3,2,1] for rzn in kQ.realizations()) True
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retract
(la)¶ Gives the retract map from the symmetric functions to the quotient ring of \(k\)-bounded symmetric functions. This method is here to make the TestSuite run properly.
INPUT:
la
– A partition
OUTPUT:
- The monomial element of the \(k\)-bounded quotient indexed by
la
.
EXAMPLES:
sage: Q = SymmetricFunctions(QQ).kBoundedQuotient(3,t=1) sage: Q.retract([2,1]) m3[2, 1]
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sage.combinat.sf.k_dual.
KBoundedQuotientBases
¶ The category of bases for the \(k\)-bounded subspace of symmetric functions.
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sage.combinat.sf.k_dual.
KBoundedQuotientBasis
¶ Abstract base class for the bases of the \(k\)-bounded quotient.
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sage.combinat.sf.k_dual.
kMonomial
¶ The basis of monomial symmetric functions indexed by partitions with first part less than or equal to \(k\).
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sage.combinat.sf.k_dual.
kbounded_HallLittlewoodP
¶ The basis of P Hall-Littlewood symmetric functions indexed by partitions with first part less than or equal to \(k\).