Categorifying the tensor product of a level 1 highest weight and perfect crystal in type A

2015/08/16 by Monica Vazirani, Vazirani, Monica
Mathematics · #05E10 #20C08 #Advanced Topics in Algebra #Algebraic structures and combinatorial models #Combinatorics (math.CO) #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #Representation Theory (math.RT)

paper · pdf · doi:10.48550/arxiv.1508.03802

openalex publication_date 2015/08/16 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We use Khovanov-Lauda-Rouquier algebras to categorify a crystal isomorphism between a highest weight crystal and the tensor product of a perfect crystal and another highest weight crystal, all in level 1 type A affine. The nodes of the perfect crystal correspond to a family of trivial modules and the nodes of the highest weight crystal correspond to simple modules, which we may also parameterize by ℓ-restricted partitions. In the case ℓ is a prime, one can reinterpret all the results for the symmetric group in characteristic ℓ. The crystal operators correspond to socle of restriction and behave compatibly with the rule for tensor product of crystal graphs.

Citations

• 2-Kac-Moody algebras

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