A combinatorial realization of Schur-Weyl duality via crystal graphs and dual equivalence graphs

TL;DR

This paper provides a combinatorial framework linking crystal graphs and dual equivalence graphs to realize Schur-Weyl duality, enriching the understanding of representation theory through combinatorial structures.

Contribution

It introduces a novel combinatorial construction that embeds dual equivalence graphs into the 0-weight space of crystal graphs, illustrating Schur-Weyl duality.

Findings

Established a dual equivalence graph structure on crystal graphs' 0-weight space

Connected crystal edges with dual equivalence graph edges combinatorially

Extended Stembridge's and author's local characterizations to this setting

Abstract

For any polynomial representation of the special linear group, the nodes of the corresponding crystal may be indexed by semi-standard Young tableaux. Under certain conditions, the standard Young tableaux occur, and do so with weight 0. Standard Young tableaux also parametrize the vertices of dual equivalence graphs. Motivated by the underlying representation theory, in this paper, we explainthis connection by giving a combinatorial manifestation of Schur-Weyl duality. In particular, we put a dual equivalence graph structure on the 0-weight space of certain crystal graphs, producing edges combinatorially from the crystal edges. The construction can be expressed in terms of the local characterizations given by Stembridge for crystal graphs and the author for dual equivalence graphs.