Hecke-Clifford algebras at roots of unity and conformal embeddings

Abstract

In this paper we give a combinatorial description of the Cauchy completion of the categories ${\mathcal{E}}_q$ and $\overline{{\mathcal{S}\mathcal{E}}_N}$ recently introduced by the first author and Snyder. This in turns gives a combinatorial description of the categories ${\overline{\mathrm{Rep}(U_q({\mathfrak{sl}}_N))}}_A$ where $A$ is the ètale algebra object corresponding to the conformal embedding ${\mathfrak{sl}}_N$ level $N$ into ${\mathfrak{so}}_{N^2-1}$ level 1. In particular we give a classification of the simple objects of these categories, a formula for their quantum dimensions, and fusion rules for tensoring with the defining object. Our method of obtaining these results is the Schur-Weyl approach of studying the representation theory of certain endomorphism algebras in ${\mathcal{E}}_q$ and ${\mathcal{S}\mathcal{E}}_N$, which are known to be subalgebras of Hecke-Clifford algebras. We build on existing literature to study the representation theory of the Hecke-Clifford algebras at roots of unity.

38 pages