Blocks of cyclotomic Hecke algebras and Khovanov-Lauda algebras

Brundan, Jonathan; Kleshchev, Alexander

doi:10.1007/s00222-009-0204-8

Cited by 254 publications

(494 citation statements)

References 20 publications

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“…This conjecture has been proven by Kleshchev and Brundan in type A [2,3]. They construct an isomorphism…”

Section: Introductionmentioning

confidence: 69%

“…The relations in R(ν) for identically colored strands imply The following Proposition appears in an algebraic form in the work of Brundan and Kleshchev [2]. Consider i ∈ Seq(ν) with i = i i i and i ∈ Seq(ν ), i ∈ Seq(ν ), i ∈ Seq(ν ),…”

Section: Local Relations For Cyclotomic Quotientsmentioning

confidence: 99%

“…Sliding dots through the crossings in i 1 j using (2) shows that ν . Furthermore, if after applying antigravity moves to conf(i) a configuration of the above form appears, then it is not hard to check that 1 i = 0 in R Λ ν .…”

Section: J S a (R)mentioning

confidence: 99%

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Nilpotency in type A cyclotomic quotients

Hoffnung

Lauda

2010

J Algebr Comb

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“…This conjecture has been proven by Kleshchev and Brundan in type A [2,3]. They construct an isomorphism…”

Section: Introductionmentioning

confidence: 69%

Section: Local Relations For Cyclotomic Quotientsmentioning

confidence: 99%

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Nilpotency in type A cyclotomic quotients

Hoffnung

Lauda

2010

J Algebr Comb

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“…and a somewhat complicated set of defining relations, which may be found in [3] or [10], for example. These relations allow one to write down a basis for R n : to do this, choose and fix a reduced expression w = t i 1 .…”

Section: The Quiver Hecke Algebramentioning

confidence: 99%

Dyck tilings and the homogeneous Garnir relations for graded Specht modules

Fayers

2016

J Algebr Comb

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Suppose λ and μ are integer partitions with λ ⊇ μ. Kenyon and Wilson have introduced the notion of a cover-inclusive Dyck tiling of the skew Young diagram λ\μ, which has applications in the study of double-dimer models. We examine these tilings in more detail, giving various equivalent conditions and then proving a recurrence which we use to show that the entries of the transition matrix between two bases for a certain permutation module for the symmetric group are given by counting coverinclusive Dyck tilings. We go on to consider the inverse of this matrix, showing that its entries are determined by what we call cover-expansive Dyck tilings. The fact that these two matrices are mutual inverses allows us to recover the main result of Kenyon and Wilson. We then discuss the connections with recent results of Kim et al., who give a simple expression for the sum, over all μ, of the number of cover-inclusive Dyck tilings of λ\μ. Our results provide a new proof of this result. Finally, we show how to use our results to obtain simpler expressions for the homogeneous Garnir relations for the universal Specht modules introduced by Kleshchev, Mathas and Ram for the cyclotomic quiver Hecke algebras.

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“…Unfortunately, as [5, (2.5)] shows, this duality is usually not cyclic; in particular, it is not for a generic choice of parameters, or for the choice which is most important in geometric and representation theoretic applications, such as [4,15].…”

mentioning

confidence: 99%

Cyclicity for categorified quantum groups

et al. 2016

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Abstract. We equip the categorified quantum group attached KLR algebra and an arbitrary choice of scalars with duality functor which is cyclic, that is, such that f = f * * for all 2-morphisms f . This is accomplished via a modified diagrammatic formalism.Consider the 2-category U Q (g) (as defined in [3,5,13]) which categorifies a Kac-Moody Lie algebra g. In this 2-category, every 1-morphism possesses a left dual and a right dual. Standard arguments show that the left and right duals (which are isomorphic) are well-defined up to unique isomorphism; however they are not well-defined as 1-morphisms. That is, the operations of left and right dual are anafunctors, not functors. In particular, every 1-morphism is isomorphic to its double-dual, but there is no fixed isomorphism intrinsic to the 2-category structure.However, we can choose a right duality functor, which allows us to fix a left and right dual. In fact, this duality will be strict: any 1-morphism will be equal to its double dual. One such functor is supplied by the functor τ defined in [6, 3.46], that is by rightward rotation by 180 • . However, in the graphical calculus defined in [5], this functor lacks one of the basic properties we expect from a duality: while a 1-morphism is equal to its double dual, for a 2-morphism f : u → v, we will not necessarily have that f and f * * are equal.If the equality f = f * * does hold, we call the resulting duality cyclic. In this case, the functor of double dual is isomorphic to the identity; thus, the identity map defines a (strictly) pivotal structure on this 2-category. This property is also equivalent to the induced biadjunction on the objects (u, u * ) being cyclic. In diagrammatic terms, this means that a morphism is unchanged by a full 360 • rotation. It follows that the string diagram calculus which uses a cyclic duality functor enjoys a topological invariance that greatly simplifies computations. Furthermore, a pivotal structure is necessary for defining convolution in the Hochschild cohomology of a representation of this 2-category; in particular, it plays a key role in the authors' previous work on traces of categorified quantum groups, and related work of Shan, Varagnolo and Vasserot [1,14,17].Our work is motivated by work of Brundan [3], which shows that the 2-category introduced in [5] can be defined as in [13], where the functor F i is the right dual of E i (by definition), but there is no a priori connection between F i and the left dual of E i . In order to define a duality functor, we must thus choose an isomorphism of E i to the right dual of F i (that is, of F i with the left dual of E i ).In [3, 1.2], one such isomorphism is defined, which matches the choice implicit in [5]. Unfortunately, as [5, (2.5)] shows, this duality is usually not cyclic; in particular, it is not for a generic choice of parameters, or for the choice which is most important in geometric and representation theoretic applications, such as [4,15].In this paper, we introduce a 2-category U cyc Q (g) for an arbitrary choice of...

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Blocks of cyclotomic Hecke algebras and Khovanov-Lauda algebras

Cited by 254 publications

References 20 publications

Nilpotency in type A cyclotomic quotients

Nilpotency in type A cyclotomic quotients

Dyck tilings and the homogeneous Garnir relations for graded Specht modules

Cyclicity for categorified quantum groups

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