Loewy Series of Weyl Modules and the Poincaré Polynomials of Quiver Varieties

Kodera, Ryosuke; Naoi, Katsuyuki

doi:10.2977/prims/77

Cited by 28 publications

(29 citation statements)

References 8 publications

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“…For this action, the weight decomposition is H * (M(W )) ∼ = ⊕ µ H * (M(m, W )) (where as usual µ and m are related by λ − µ = i m i α i ). By a theorem of Kodera-Naoi [KN12] we have that H * (M(W )) is isomorphic to a dual Weyl module. By Fourier-Littelmann [FL06] the dual module is isomorphic to Γ(Gr λ , O(1)), so combining these results we have:…”

Section: Consider the Mapsmentioning

confidence: 96%

Highest weights for truncated shifted Yangians and product monomial crystals

et al. 2019

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Truncated shifted Yangians are a family of algebras which are natural quantizations of slices in the affine Grassmannian. We study the highest weight representations of these algebras. In particular, we conjecture that the possible highest weights for these algebras are described by product monomial crystals, certain natural subcrystals of Nakajima's monomials. We prove this conjecture in type A. We also place our results in the context of symplectic duality and prove a conjecture of Hikita in this situation.

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Section: Consider the Mapsmentioning

confidence: 96%

Highest weights for truncated shifted Yangians and product monomial crystals

et al. 2019

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“…The study of the category of graded finite-dimensional representations of the current algebra has been of interest in recent years for a variety of reasons. The work of [27] relates graded characters of certain representations to the Poincare polynomials of quiver varieties. The homological properties of the category are similar to those of the BGG category O for the simple Lie algebra.…”

Section: Introductionmentioning

confidence: 99%

Demazure Modules, Fusion Products and Q-Systems

Chari

Venkatesh

2014

Commun. Math. Phys.

113

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In this paper, we introduce a family of indecomposable finite-dimensional graded modules for the current algebra associated to a simple Lie algebra. These modules are indexed by an |R + |-tuple of partitions ξ = (ξ α ), where α varies over a set R + of positive roots of g and we assume that they satisfy a natural compatibility condition. In the case when the ξ α are all rectangular, for instance, we prove that these modules are Demazure modules in various levels. As a consequence we see that the defining relations of Demazure modules can be greatly simplified. We use this simplified presentation to relate our results to the fusion products, defined in [15], of representations of the current algebra. We prove that the Q-system of [22] extends to a canonical short exact sequence of fusion products of representations associated to certain special partitions ξ. Finally, in the last section we deal with the case of sl2 and prove that the modules we define are just fusion products of irreducible representations of the associated current algebra and give monomial bases for these modules.

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“…It is understood that the sum of the entries in the zeroth row is zero. The weight of the pattern in (6), for instance, is (5,6,4).…”

Section: 23mentioning

confidence: 99%

On Chari–Loktev bases for local Weyl modules in type A

Raghavan

Ravinder

Viswanath

2018

Journal of Combinatorial Theory, Series A

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This paper is a study of the bases introduced by Chari-Loktev in [1] for local Weyl modules of the current algebra associated to a special linear Lie algebra. Partition overlaid patterns, POPs for shortwhose introduction is one of the aims of this paper-form convenient parametrizing sets of these bases. They play a role analogous to that played by (Gelfand-Tsetlin) patterns in the representation theory of the special linear Lie algebra.The notion of a POP leads naturally to the notion of area of a pattern. We observe that there is a unique pattern of maximal area among all those with a given bounding sequence and given weight. We give a combinatorial proof of this and discuss its representation theoretic relevance.We then state a conjecture about the "stability", i.e., compatibility in the long range, of Chari-Loktev bases with respect to inclusions of local Weyl modules. In order to state the conjecture, we establish a certain bijection between colored partitions and POPs, which may be of interest in itself. The stability conjecture has been proved in [6] in the rank one case.2010 Mathematics Subject Classification. 17B67 (05E10). Key words and phrases. Gelfand-Tsetlin pattern, Chari-Loktev basis, Demazure module, partition overlaid pattern, POP, area of a pattern, stability of Chari-Loktev bases.The second named author acknowledges support from CSIR under the SPM doctoral fellowship scheme. The first and third named authors acknowledge support from DAE under a XII plan project. 1 One of our aims in the present paper is to clarify the parametrizing set of this Chari-Loktev basis. Towards this end we introduce the notion of a partition overlaid pattern or POP (see §2.8). Dominant integral weights for g = sl r+1 may be identified with non-increasing sequences of non-negative integers of length r + 1 with the last element of the sequence being 0. Gelfand-Tsetlin patterns (or just patterns-see §2 for the precise definition) with bounding sequence (corresponding to) λ parametrize the Gelfand-Tsetlin basis for V (λ). Analogously, as we show in §4.6, POPs with bounding sequence λ parametrize the Chari-Loktev basis for W (λ). The weight of the underlying pattern of a POP equals the h-weight of the corresponding basis element and the number of boxes in the partition overlay determines the grade.The notion of a POP leads naturally to the notion of the area of a pattern (see §2.3). For a weight µ of V (λ), it turns out that the piece of highest grade in the µ-weight space of the local Weyl module W (λ) is one dimensional (the h-weights of W (λ) are precisely those of V (λ)). We give a representation theoretic proof of this fact in §4. This suggests-even proves, albeit circuitously-that there must be a unique pattern of highest area among all those with bounding sequence λ and weight µ. We give a direct, elementary, and purely combinatorial proof of this in §3.In §6, we state a conjecture about the "stability" of the Chari-Loktev basis. To describe what is meant by stability, let θ be the highest root of g. We then have nat...

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Loewy Series of Weyl Modules and the Poincaré Polynomials of Quiver Varieties

Cited by 28 publications

References 8 publications

Highest weights for truncated shifted Yangians and product monomial crystals

Highest weights for truncated shifted Yangians and product monomial crystals

Demazure Modules, Fusion Products and Q-Systems

On Chari–Loktev bases for local Weyl modules in type A

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