Reflected and nonsymmetric crystal graphs

Abstract

This thesis is comprised of two projects. The first studies a certain composition of crystal operators on semistandard Young tableaux, which we term raised reflection operators, and are related to a sign-reversing involution used to prove the Littlewood--Richardson rule. In particular, we investigate the graph defined by these crystal operators. Our main result is that this graph is balanced bipartite, giving another proof of the Littlewood--Richardson rule. We do so by giving a set of local rules that this graph satisfies and showing that any graph satisfying these rules is balanced bipartite.

The second studies crystal operators on multiline queues. Certain multiline queues, called non-wrapping multiline queues, are in bijection with semistandard Young tableaux but are better equipped to study nonsymmetric polynomials called Demazure atoms. Indeed, each multiline queue comes equipped with a weak composition, called the type, and summing over all multiline queues of a fixed type yields a Demazure atom. Crystal operators on multiline queues do not preserve type. Our main result characterizes how crystal operators interact with the type of a multiline queue. In particular, we show that these operators may only change the type by a simple transposition and that the type changes if and only if the multiline queue is in a specific configuration.