Abstract
We use Khovanov-Lauda-Rouquier (KLR) algebras to categorify a crystal isomorphism between a funda-mental crystal and the tensor product of a Kirillov-Reshetikhin crystal and another fundamental crystal, all in affine type. The nodes of the Kirillov-Reshetikhin crystal correspond to a family of “trivial” modules. The nodes of the fun-damental crystal correspond to simple modules of the corresponding cyclotomic KLR algebra. The crystal operators correspond to socle of restriction and behave compatibly with the rule for tensor product of crystal graphs.
Highlights
Kang-Kashiwara [9] and Webster [23] show the cyclotomic Khovanov-Lauda-Rouquier (KLR) algebra RΛ categorifies the highest weight representation V (Λ) in arbitrary symmetrizable type. (KLR algebras are known in the literature as quiver Hecke algebras.) By a slight abuse of language, we will say the combinatorial version of this statement is that RΛ categorifies the crystal B(Λ), where simple modules correspond to nodes, and functors that take socle of restriction correspond to arrows, i.e. the Kashiwara crystal operators [16]
Our main theorems give a purely module-theoretic construction of this crystal isomorphism. (One must modify the form of the crystal isomorphism in the case B1,1 is not perfect or when Λi is not of level 1.) Each node of B1,1 corresponds to an infinite family of “trivial” modules, but note this does not give a categorification of B
For a construction of simple modules related to the crystal B(∞) for finite type KLR algebras see [2]
Summary
Kang-Kashiwara [9] and Webster [23] show the cyclotomic Khovanov-Lauda-Rouquier (KLR) algebra RΛ categorifies the highest weight representation V (Λ) in arbitrary symmetrizable type. (KLR algebras are known in the literature as quiver Hecke algebras.) By a slight abuse of language, we will say the combinatorial version of this statement is that RΛ categorifies the crystal B(Λ), where simple modules correspond to nodes, and functors that take socle of restriction correspond to arrows, i.e. the Kashiwara crystal operators [16]. (One must modify the form of the crystal isomorphism in the case B1,1 is not perfect or when Λi is not of level 1.) Each node of B1,1 corresponds to an infinite family of “trivial” modules, but note this does not give a categorification of B. These “trivial” modules Tp;k are the KLR analogues of the nodes in highest weight crystals studied in [21]. For a construction of simple modules related to the crystal B(∞) for finite type KLR algebras see [2]. This paper generalizes the theorems and constructions from [22] for type A affine
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