LLT symmetric functions¶
REFERENCES:
[LLT1997] | Alain Lascoux, Bernard Leclerc, Jean-Yves Thibon, Ribbon tableaux, Hall-Littlewood functions, quantum affine algebras, and unipotent varieties, J. Math. Phys. 38 (1997), no. 2, 1041-1068, Arxiv q-alg/9512031v1 [math.q.alg] |
[LT2000] | Bernard Leclerc and Jean-Yves Thibon, Littlewood-Richardson coefficients and Kazhdan-Lusztig polynomials, in: Combinatorial methods in representation theory (Kyoto) Adv. Stud. Pure Math., vol. 28, Kinokuniya, Tokyo, 2000, pp 155-220 Arxiv math/9809122v3 [math.q-alg] |
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sage.combinat.sf.llt.
LLT_class
¶ A class for working with LLT symmetric functions.
EXAMPLES:
sage: Sym = SymmetricFunctions(FractionField(QQ['t'])) sage: L3 = Sym.llt(3); L3 level 3 LLT polynomials over Fraction Field of Univariate Polynomial Ring in t over Rational Field sage: L3.cospin([3,2,1]) (t+1)*m[1, 1] + m[2] sage: HC3 = L3.hcospin(); HC3 Symmetric Functions over Fraction Field of Univariate Polynomial Ring in t over Rational Field in the level 3 LLT cospin basis sage: m = Sym.monomial() sage: m( HC3[1,1] ) (t+1)*m[1, 1] + m[2]
We require that the parameter \(t\) must be in the base ring:
sage: Symxt = SymmetricFunctions(QQ['x','t'].fraction_field()) sage: (x,t) = Symxt.base_ring().gens() sage: LLT3x = Symxt.llt(3,t=x) sage: LLT3 = Symxt.llt(3) sage: HS3x = LLT3x.hspin() sage: HS3t = LLT3.hspin() sage: s = Symxt.schur() sage: s(HS3x[2,1]) s[2, 1] + x*s[3] sage: s(HS3t[2,1]) s[2, 1] + t*s[3] sage: HS3x(HS3t[2,1]) HSp3[2, 1] + (-x+t)*HSp3[3] sage: s(HS3x(HS3t[2,1])) s[2, 1] + t*s[3] sage: LLT3t2 = Symxt.llt(3,t=2) sage: HC3t2 = LLT3t2.hcospin() sage: HS3x(HC3t2[3,1]) 2*HSp3[3, 1] + (-2*x+1)*HSp3[4]
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class
sage.combinat.sf.llt.
LLT_cospin
(llt)¶ Bases:
sage.combinat.sf.llt.LLT_generic
A class of methods for the h-cospin LLT basis of the symmetric functions.
INPUT:
self
– an instance of the LLT hcospin basisllt
– a family of LLT symmetric function bases
sage: HC3t2 = SymmetricFunctions(QQ).llt(3,t=2).hcospin() sage: TestSuite(HC3t2).run() # products are too expensive, long time (6s on sage.math, 2012)
sage: HC3x = SymmetricFunctions(FractionField(QQ['x'])).llt(3,t=x).hcospin() sage: TestSuite(HC3x).run(skip = ["_test_associativity", "_test_distributivity", "_test_prod"]) # products are too expensive, long time (5s on sage.math, 2012) sage: TestSuite(HC3x).run(elements = [HC3x.t*HC3x[1,1]+HC3x.t*HC3x[2], HC3x[1]+(1+HC3x.t)*HC3x[1,1]]) # long time (depends on previous)
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class
Element
¶
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class
sage.combinat.sf.llt.
LLT_generic
(llt, prefix)¶ Bases:
sage.combinat.sf.sfa.SymmetricFunctionAlgebra_generic
A class of methods which are common to both the hspin and hcospin of the LLT symmetric functions.
INPUT:
self
– an instance of the LLT hspin or hcospin basisllt
– a family of LLT symmetric functions
EXAMPLES:
sage: SymmetricFunctions(FractionField(QQ['t'])).llt(3).hspin() Symmetric Functions over Fraction Field of Univariate Polynomial Ring in t over Rational Field in the level 3 LLT spin basis sage: SymmetricFunctions(QQ).llt(3,t=2).hspin() Symmetric Functions over Rational Field in the level 3 LLT spin with t=2 basis sage: QQz = FractionField(QQ['z']); z = QQz.gen() sage: SymmetricFunctions(QQz).llt(3,t=z).hspin() Symmetric Functions over Fraction Field of Univariate Polynomial Ring in z over Rational Field in the level 3 LLT spin with t=z basis
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class
Element
¶ Bases:
sage.combinat.sf.sfa.SymmetricFunctionAlgebra_generic_Element
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level
()¶ Returns the level of
self
.INPUT:
self
– an instance of the LLT hspin or hcospin basis
OUTPUT:
- returns the level associated to the basis
self
.
EXAMPLES:
sage: HSp3 = SymmetricFunctions(FractionField(QQ['t'])).llt(3).hspin() sage: HSp3.level() 3
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llt_family
()¶ The family of the llt bases of the symmetric functions.
INPUT:
self
– an instance of the LLT hspin or hcospin basis
OUTPUT:
- returns an instance of the family of LLT bases associated to
self
.
EXAMPLES:
sage: HSp3 = SymmetricFunctions(FractionField(QQ['t'])).llt(3).hspin() sage: HSp3.llt_family() level 3 LLT polynomials over Fraction Field of Univariate Polynomial Ring in t over Rational Field
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class
sage.combinat.sf.llt.
LLT_spin
(llt)¶ Bases:
sage.combinat.sf.llt.LLT_generic
A class of methods for the h-spin LLT basis of the symmetric functions.
INPUT:
self
– an instance of the LLT hcospin basisllt
– a family of LLT symmetric function bases
sage: HS3t2 = SymmetricFunctions(QQ).llt(3,t=2).hspin() sage: TestSuite(HS3t2).run() # products are too expensive, long time (7s on sage.math, 2012)
sage: HS3x = SymmetricFunctions(FractionField(QQ['x'])).llt(3,t=x).hspin() sage: TestSuite(HS3x).run(skip = ["_test_associativity", "_test_distributivity", "_test_prod"]) # products are too expensive, long time (4s on sage.math, 2012) sage: TestSuite(HS3x).run(elements = [HS3x.t*HS3x[1,1]+HS3x.t*HS3x[2], HS3x[1]+(1+HS3x.t)*HS3x[1,1]]) # long time (depends on previous)
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class
Element
¶