A Uniform Model for Kirillov-Reshetikhin Crystals I: Lifting the Parabolic Quantum Bruhat Graph

Lenart, Cristian; Naito, Seiji; Sagaki, Daisuke; Schilling, Anne; Shimozono, Mark

doi:10.1093/imrn/rnt263

Cited by 34 publications

(67 citation statements)

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“…The bijection Φ was generalized beyond type A (1) n to arbitrary non-exceptional types in [OSS03a] and to type E (1) 6 in [OS12] for tensor products of Kirillov-Reshetikhin (KR) crystals indexed by the vector representation. In the spirit of [KSS02] for type A (1) n , it is conjectured that Φ can be extended to arbitrary tensor products of KR crystals. This involves certain splitting maps (of the rectangles that index the KR crystals) and, beyond type A (1) n , also a "filling map" as first pointed out in [Sch05] and fully established in type D (1) n in [OSS13].…”

Section: Introductionmentioning

confidence: 99%

Crystal Structure on Rigged Configurations and the Filling Map

Schilling

Scrimshaw

2015

Electron. J. Combin.

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In this paper, we extend work of the first author on a crystal structure on rigged configurations of simply-laced type to all non-exceptional affine types using the technology of virtual rigged configurations and crystals. Under the bijection between rigged configurations and tensor products of Kirillov-Reshetikhin crystals specialized to a single tensor factor, we obtain a new tableaux model for Kirillov-Reshetikhin crystals. This is related to the model in terms of Kashiwara-Nakashima tableaux via a filling map, generalizing the recently discovered filling map in type D (1) n .Again let B 1 and B 2 be abstract U q (g)-crystals. A crystal morphism ψ : B 1 −→ B 2 is a map

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Section: Introductionmentioning

confidence: 99%

Crystal Structure on Rigged Configurations and the Filling Map

Schilling

Scrimshaw

2015

Electron. J. Combin.

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“…The following lemma is well known (see e.g. [LNSSS2] For example, in types A and C the quantum Bruhat graph can be explicitly described as follows (see [Len]). For type A we need the order ≺ i on 1, .…”

Section: Orr-shimozono Formulamentioning

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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“…We use Theorem 1.2,(ii) and the following result (see e.g. [LNSSS2]): the element w 0 ∈ W inverses arrows in the twisted quantum Bruhat graph.…”

Section: Properties Of W σ(λ)mentioning

confidence: 99%