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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.

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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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  1. Mathematics Department, One Shields Ave, Davis, CA, 95616, USA

    Henry Kvinge & Monica Vazirani

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  1. Henry Kvinge
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  2. Monica Vazirani
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Correspondence to Monica Vazirani.

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Presented by Peter Littelmann.

The second author was partially supported by NSA grant H98230-12-1-0232, ICERM, and the Simons Foundation.

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Kvinge, H., Vazirani, M. A Combinatorial Categorification of the Tensor Product of the Kirillov-Reshetikhin Crystal B 1,1 and a Fundamental Crystal. Algebr Represent Theor 21, 1277–1331 (2018). https://doi.org/10.1007/s10468-017-9747-3

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  • DOI: https://doi.org/10.1007/s10468-017-9747-3

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