Weyl modules, Demazure modules, KR-modules, crystals, fusion products and limit constructions

Fourier, Ghislain; Littelmann, Peter

doi:10.1016/j.aim.2006.09.002

Cited by 158 publications

(308 citation statements)

References 33 publications

Supporting

Mentioning

304

Contrasting

Unclassified

Order By: Relevance

“…In order to realize the crystal graph of the so called Kirillov-Reshetikhin modules KR(m, ω i , a), for i ∈ I, m ∈ Z + , we will define now the underlying combinatorial model in this paper, which we will denote by B m,i . For more details regarding KR-modules we refer to a series of papers ([Cha01], [CM06], [FL07]). …”

Section: Notation and Main Definitionsmentioning

confidence: 99%

Realization of affine type A Kirillov–Reshetikhin crystals via polytopes

Kus

2013

Journal of Combinatorial Theory, Series A

View full text Add to dashboard Cite

Abstract. On the polytope defined in [FFL11], associated to any rectangle highest weight, we define a structure of an type An-crystal. We show, by using the Stembridge axioms, that this crystal is isomorphic to the one obtained from Kashiwara's crystal bases theory. Further we define on this polytope a bijective map and show that this map satisfies the properties of a weak promotion operator. This implies in particular that we provide an explicit realization of Kirillov-Reshetikhin crystals for the affine type A (1) n via polytopes. IntroductionLet g be a affine Lie algebra and U ′ q (g) be the corresponding quantum algebra without derivation. A promotion operator pr on a crystal B of type A n is defined to be a map satisfying several conditions, namely that pr shifts the content, pr •ẽ j =ẽ j+1 • pr, pr •f j =f j+1 • pr for all j ∈ {1, · · · , n − 1} and pr n+1 = id, whereẽ j andf j respectively are the Kashiwara operators. If the latter condition is not satisfied, but pr is still bijective, then the map pr is called a weak promotion operator (see also [BST10]). The advantage of such (weak) promotion oparators are that we can associate to a given crystal B of type A n a (weak) affine crystal by settingf 0 := prOn the set of all semi-standard Young tableaux of rectangle shape, which is a realization of B(mω i ) the U q (A n )-crystal associated to the irreducible module of highest weight mω i , Schützenberger defined a promotion operator pr, called the Schützenberger's promotion operator [Sch72], which is the analogue of the cyclic Dynkin diagram automorphism i → i + 1 mod (n + 1) on the level of crystals, by using jeu-de-taquin. Given a tableaux T over the alphabet 1 ≺ 2 · · · ≺ n+1, pr(T ) is obtained from T by removing all letters n+1, adding one to each entry in the remaining tableaux, using jeu-de-taquin to slide all letters up and finally filling the holes with 1's (see also Section 6). One of the combinatorial descriptions of KR m,i in the affine A was shown that, as a {1, · · · , n}-crystal, KR m,i is isomorphic to B(mω i ) and the affine crystal constructed from B(mω i ) using Schützenberger's promotion operator is isomorphic to the Kirillov-Reshetikhin crystal KR m,i . The two ways of computing the affine crystal structure, one given by [KKM + 92] and the other by [Shi02], are shown to be equivalent in [OSS03]. Another combinatorial model in this type without using a promotion operator is described in [Kwo]. In this paper, we introduce a new realization of Kirillov-Reshetikhin crystals of type A (1) n . In [FFL11] the authors have constructed for all dominant integral A n weights λ a polytope in R n n−1 2 and a basis of the irreducible A n module of highest weight λ and have shown that the basis elements are parametrized by the integral points. For λ = mω i we can understand this polytope in R i(n−i+1) and denote the intersection of this polytope with Z i(n−i+1) + by B m,i . We define certain maps on B m,i and show that this becomes a crystal of type A n . As a set, we can identify B m,i with certain ...

show abstract

Section: Notation and Main Definitionsmentioning

confidence: 99%

Realization of affine type A Kirillov–Reshetikhin crystals via polytopes

Kus

2013

Journal of Combinatorial Theory, Series A

View full text Add to dashboard Cite

show abstract

“…These modules are invariant with respect to the current algebra g ⊗ C[t] and provide a filtration on L by finite-dimensional spaces: [18,19]; some special cases are also contained in [12,23]). Let F m (N ) = D(N θ) ∩ F m be an intersection of the Demazure module with the m-th space of the PBW filtration.…”

Section: Introductionmentioning

confidence: 99%

“…This gives a filtration on D(N θ). In order to describe the filtration F • (N ) we use a notion of the fusion product of g ⊗ C[t] modules (see [17,11]) and the Fourier-Littelmann results [19].…”

Section: Introductionmentioning

confidence: 99%

The PBW Filtration, Demazure Modules and Toroidal Current Algebras

Feigin¹

2008

SIGMA

View full text Add to dashboard Cite

Abstract. Let L be the basic (level one vacuum) representation of the affine Kac-Moody Lie algebra g. The m-th space F m of the PBW filtration on L is a linear span of vectors of the form x 1 · · · x l v 0 , where l ≤ m, x i ∈ g and v 0 is a highest weight vector of L. In this paper we give two descriptions of the associated graded space L gr with respect to the PBW filtration. The "top-down" description deals with a structure of L gr as a representation of the abelianized algebra of generating operators. We prove that the ideal of relations is generated by the coefficients of the squared field e θ (z) 2 , which corresponds to the longest root θ. The "bottom-up" description deals with the structure of L gr as a representation of the current algebra g ⊗ C[t]. We prove that each quotient F m /F m−1 can be filtered by graded deformations of the tensor products of m copies of g.

show abstract

“…as conjectured in [5] (for simply laced g this had been proved previously in [4,9]). Both [26, Theorem A] and (2.4.2) were extended to the positive characteristic case in [2,10].…”

Section: 3mentioning

confidence: 50%

“…1 when g is simply laced [4,9]. Furtheremore, it was shown in [26] that, for any g, W (λ) admits a level-1 Demazure flag, i.e., a filtration whose subsequent quotients are isomorphic to Demazure modules of level 1.…”

Section: Introductionmentioning

confidence: 99%

Limits of Multiplicities in Excellent Filtrations and Tensor Product Decompositions for Affine Kac-Moody Algebras

Jakelic

Moura

2017

Algebr Represent Theor

View full text Add to dashboard Cite

Abstract. We express the multiplicities of the irreducible summands of certain tensor products of irreducible integrable modules for an affine Kac-Moody algebra over a simply laced Lie algebra as sums of multiplicities in appropriate excellent filtrations (Demazure flags). As an application, we obtain expressions for the outer multiplicities of tensor products of two fundamental modules for sl2 in terms of partitions with bounded parts, which subsequently lead to certain partition identities.

show abstract

Weyl modules, Demazure modules, KR-modules, crystals, fusion products and limit constructions

Cited by 158 publications

References 33 publications

Realization of affine type A Kirillov–Reshetikhin crystals via polytopes

Realization of affine type A Kirillov–Reshetikhin crystals via polytopes

The PBW Filtration, Demazure Modules and Toroidal Current Algebras

Limits of Multiplicities in Excellent Filtrations and Tensor Product Decompositions for Affine Kac-Moody Algebras

Contact Info

Product

Resources

About