Hecke-Clifford algebras at roots of unity and conformal embeddings

In this paper we give a combinatorial description of the Cauchy completion of the categories $\mathcal{E}_q$ and $\overline{\mathcal{SE}_N}$ recently introduced by the first author and Snyder. This in turns gives a combinatorial description of the categories $\overline{\operatorname{Rep}(U_q(\mathfrak{sl}_N))}_{A}$ where $A$ is the \`etale 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 $\mathcal{SE}_N$, 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.