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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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