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A Combinatorial Categorification of the Tensor Product of the Kirillov-Reshetikhin Crystal B 1,1 and a Fundamental Crystal

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Abstract

We use Khovanov-Lauda-Rouquier (KLR) algebras to categorify a crystal isomorphism between a fundamental 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 fundamental 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.

Keywords

KLR algebras Quiver Hecke algebras Crystals Categorification 

Mathematics Subject Classification (2010)

05E10 20C08 

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Notes

Acknowledgements

We wish to thank Peter Tingley for interesting discussions and for pointing out we were using KR crystals and not just level 1 perfect crystals. The second author would like to thank David Hill for useful discussions and some initial jump computations.

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Authors and Affiliations

  1. 1.Mathematics DepartmentDavisUSA

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