Séminaire Lotharingien de Combinatoire, 78B.34 (2017), 12 pp.

Ben Salisbury and Travis Scrimshaw

Using Rigged Configurations to Model B(\infty)

Abstract. Crystal bases provide a rich environment for one to study quantized universal enveloping algebras and their representation theory for any symmetrizable Kac-Moody algebra by elucidating the underlying combinatorics. While the definition of a crystal basis involves complicated algebra, the combinatorial nature allows these crystals to be modeled using combinatorial objects. In this work, the underlying combinatorial model consists of rigged configurations, which allow for a uniform description of these crystals across all symmetrizable Kac-Moody types. Their flexibility is exhibited by the fact that the combinatorial isomorphism to crystals of tableaux is understood and that the star-crystal structure is easily computable directly from the rigged configurations. These results are summarized in this abstract.

Received: November 14, 2016. Accepted: February 17, 2017. Final version: April 1, 2017.

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