Construction of irreducible representations over Khovanov-Lauda-Rouquier algebras of finite classical type

Georgia Benkart, Seok Jin Kang, Se Jin Oh, Euiyong Park

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Abstract

We give an explicit construction of irreducible modules over Khovanov-Lauda-Rouquier algebras R and their cyclotomic quotients R λ for finite classical types using a crystal basis theoretic approach. More precisely, for each element ν of the crystal B(∞) (resp. B(λ)), we first construct certain modules Δ(a; k) labeled by the adapted string a of ν. We then prove that the head of the induced module Ind(Δ(a; 1) {squared times} ⋯ {squared times} Δ(a; n)) is irreducible and that every irreducible R-module (resp. Rλ- module) can be realized as the irreducible head of one of the induced modules Ind(Δ(a; 1) {squared times} ⋯ {squared times} Δ(a; n)). Moreover, we show that our construction is compatible with the crystal structure on B(∞) (resp. B(λ).

Original languageEnglish
Pages (from-to)1312-1366
Number of pages55
JournalInternational Mathematics Research Notices
Volume2014
Issue number5
DOIs
StatePublished - Nov 2014

Bibliographical note

Funding Information:
This work was supported by KRF Grant # 2007-341-C00001 and by NRF Grant # 2010-0010753 (to S.-J.K.), and by NRF Grant # 2010-0019516 and BK21 Mathematical Sciences Division (to S.-J.O.).

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