Celebratio Mathematica — Benkart

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G. Benkart, S.-J. Kang, S.-j. Oh, and E. Park: “Construction of irreducible representations over Khovanov–Lauda–Rouquier algebras of finite classical type,” Int. Math. Res. Not. 2014 : 5 (January 2014), pp. 1312–1366. In memory of Professor Hyo Chul Myung. MR 3178600 Zbl 1355.17009 ArXiv 1108.1048 article

We give an explicit construction of irreducible modules over Khovanov–Lauda–Rouquier algebras \( R \) and their cyclotomic quotients \( R^{\lambda} \) for finite classical types using a crystal basis theoretic approach. More precisely, for each element \( \nu \) of the crystal \( B(\infty) \) (resp. \( B(\lambda) \)), we first construct certain modules \( \Delta(\mathbf{a};k) \) labeled by the adapted string \( \mathbf{a} \) of \( \nu \). We then prove that the head of the induced module \[ \operatorname{Ind}(\Delta(\mathbf{a};1) \boxtimes \cdots \boxtimes \Delta(\mathbf{a};n)) \] is irreducible and that every irreducible \( R \)-module (resp. \( R^{\lambda} \)-module) can be realized as the irreducible head of one of the induced modules \[ \operatorname{Ind}(\Delta(\mathbf{a};1) \boxtimes \cdots \boxtimes \Delta(\mathbf{a};n)) .\] Moreover, we show that our construction is compatible with the crystal structure on \( B(\infty) \) (resp. \( B(\lambda) \)).

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