Extremal part of the PBW-filtration and nonsymmetric Macdonald polynomials

Cherednik, Ivan; Feigin, Evgeny

doi:10.1016/j.aim.2015.06.014

Cited by 15 publications

(18 citation statements)

References 29 publications

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“…Let us consider the special case where g is a simple complex Lie algebra with triangular decomposition g = n + ⊕ h ⊕ n and M = V (λ) = U (n).v λ , a simple finite-dimensional highest weight module for g. Then V (λ) a is a cyclic module for the deformed Lie algebra b ⊕ n a . These graded modules V (λ) a have been studied under various aspects quite a lot in recent years [Fei12,Gor11,FFL11a,FFL11b,FFL13b,FFL13a,CF13,Fou14,BD14]. For example in [FFL11a] the annihilating ideal for V (λ) a as a S(n)-module has been computed in the case sl n+1 .…”

Section: Introductionmentioning

confidence: 99%

PBW-degenerated Demazure modules and Schubert varieties for triangular elements

Fourier

2016

Journal of Combinatorial Theory, Series A

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We study certain faces of the normal polytope introduced by Feigin, Littelmann and the author whose lattice points parametrize a monomial basis of the PBW-degenerated of simple modules for sln+1. We show that lattice points in these faces parametrize monomial bases of PBW-degenerated Demazure modules associated to Weyl group elements satisfying a certain closure property, for example Kempf elements. These faces are again normal polytopes and their Minkowski sum is compatible with tensor products, which implies that we obtain flat degenerations of the corresponding Schubert varieties to PBW degenerated and toric varieties.

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Section: Introductionmentioning

confidence: 99%

PBW-degenerated Demazure modules and Schubert varieties for triangular elements

Fourier

2016

Journal of Combinatorial Theory, Series A

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“…The nonsymmetric Macdonald polynomials proved to play an important role in representation theory: the specializations E λ (x, q, 0) were identified with the characters of the level one Demazure modules of the corresponding affine Kac-Moody Lie algebras (see [S, I]). It has been demonstrated recently ([CO2,CF,OS]) that for anti-dominant weights λ the specialization t = ∞ is also very meaningful. In particular, the functions E λ (x, q −1 , ∞) turned out to be polynomials in x, q with nonnegative integer coefficients [OS]; these polynomials were conjectured in [CO2] to coincide with the PBW twisted characters of the level one Demazure modules (see also [CF,FM1,FM2]).…”

Section: Introductionmentioning

confidence: 99%

Generalized Weyl modules, alcove paths and Macdonald polynomials

Feigin

Makedonskyi

2017

Sel. Math. New Ser.

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Classical local Weyl modules for a simple Lie algebra are labeled by dominant weights. We generalize the definition to the case of arbitrary weights and study the properties of the generalized modules. We prove that the representation theory of the generalized Weyl modules can be described in terms of the alcove paths and the quantum Bruhat graph. We make use of the Orr-Shimozono formula in order to prove that the t = ∞ specializations of the nonsymmetric Macdonald polynomials are equal to the characters of certain generalized Weyl modules.1 Remark 1.5. The coroots β k (w) comprise the set of all positive affine coroots which are mapped to the negative roots by w. We note also that {β k (w)} is the sequence of labels of walls crossed by a shortest walk from the alcove w −1 to the initial alcove of the current sheet (see example on page 6 in [RY]). Letb = (b 1 , . . . , b l ) ∈ 0, 1 l be a binary word and let J = {i|b i = 0}, J = {j 1 < · · · < j r }. We call J the set of foldings. For an element u ∈ W a we set z 0 = uw, z k+1 = z k s β j k+1 , k = 0, . . . , r − 1.We denote this data by an alcove path p J , so p J can be written as z 0 β j 1

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“…This framework of PBW filtrations has been introduced in [15], and various aspects of this construction have been studied in recent times. It gained a lot of attention due to its connection to different subjects such as degenerations of flag varieties [13], toric degenerations of flag varieties [17], Newton-Okounkov bodies [17], poset polytopes [1,20], quiver Grassmannians [7,8], Schubert varieties [5,9,10,21], graded characters [3,11], non-symmetric Macdonald polynomials [11,19], to name but a few.…”

Section: Introductionmentioning

confidence: 99%

PBW-type filtration on quantum groups of type A

2016

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Abstract. We will introduce an N-filtration on the negative part of a quantum group of type An, such that the associated graded algebra is a q-commutative polynomial algebra. This filtration is given in terms of the representation theory of quivers, by realizing the quantum group as the Hall algebra of a quiver. We show that the induced associated graded module of any simple finite-dimensional module (of type 1) is isomorphic to a quotient of this polynomial algebra by a monomial ideal, and we provide a monomial basis for this associated graded module. This construction can be viewed as a quantum analog of the classical PBW framework, and in fact, by considering the classical limit, this basis is the monomial basis provided by Feigin, Littelmann and the second author in the classical setup.

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Extremal part of the PBW-filtration and nonsymmetric Macdonald polynomials

Cited by 15 publications

References 29 publications

PBW-degenerated Demazure modules and Schubert varieties for triangular elements

PBW-degenerated Demazure modules and Schubert varieties for triangular elements

Generalized Weyl modules, alcove paths and Macdonald polynomials

PBW-type filtration on quantum groups of type A

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