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Hecke-Clifford algebras at roots of unity and conformal embeddings

Cain Edie-Michell, Hans Wenzl

Abstract

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 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 $\mathcal{SE}_N$, 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.

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 and recently introduced by the first author and Snyder. This in turns gives a combinatorial description of the categories where is the ètale algebra object corresponding to the conformal embedding level into 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 and , 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.

Paper Structure

This paper contains 27 sections, 52 theorems, 127 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Cauchy Completion, Negligibles, and Semisimplification
  4. The Categories $\mathcal{E}_q$ and $\mathcal{SE}_N$.
  5. The Hecke-Clifford algebras
  6. The algebras $G_n[0]$
  7. Strict Young diagrams and strict tableaux
  8. The isomeric Lie super algebra ${\mathfrak q}(N)$
  9. Characters for ${\mathfrak q}(N)$
  10. Formal characters for ${\mathfrak q}(\infty)$
  11. Matrix coefficients
  12. Representations of $G_n$
  13. Classical limit $q\to 1$
  14. Representations of $G_n$ at $q\in \mathbb{C}$
  15. Representations of $G_n$ at $q$ a primitive $4N$-th root of unity
  16. ...and 12 more sections

Key Result

Theorem 1.2

The isomorphism classes of simple objects in $\operatorname{Ab}(\overline{\mathcal{SE}_{N}})$ are parameterised by the set We have the following decomposition formulae for the tensor product of the simple $\square$ with any other simple: Furthermore, the quantum dimensions of the simple objects are given by where the rational functions $q_\lambda$ are evaluated at $q=e^{2\pi i \frac{1}{4N}}$.

Theorems & Definitions (126)

  • Definition 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Remark 1.4
  • Definition 2.1
  • Definition 2.2
  • Remark 2.3
  • Lemma 2.4
  • Corollary 2.5
  • Definition 2.6
  • ...and 116 more