Abstracts
for the Seminar

Spring 2017

**Speaker: **Patricia Hersh, NCSU

**Title: **Crystal graphs and SB-labelings

**Time:** 2:30 PM, Monday, Jan 30, 2017

**Place:** Malott 206

**Abstract:** Crystal graphs are an important tool to study the representation theory of Kac-Moody algebras. The crystal graphs arising from highest weight representations are in fact partially ordered sets. In joint work with Cristian Lenart, we study these crystal posets. We prove two types of positive results for those poset intervals which includes the highest weight vector of the representation: (1) a crystal operator analogue of the statement that any two reduced expressions for the same Coxeter group element are connected by braid moves, and (2) poset topological and Moebius function results analogous to those of weak Bruhat order. We also provide examples demonstrating that both types of results fail arbitrarily badly for arbitrary intervals in crystal posets, even in type A. The first such examples were found by computer, specifically by Moebius function computations. To our surprise, the first example we discovered with unexpectedly large Moebius function was also the first example where (1) failed. Very recently this link between (1) and (2) has been explained using the theory of SB-labelings, a theory we developed recently in separate joint work with Karola Meszaros. The talk will tell this story, providing background along the way.

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