Isocategorical groups and their Weil representations

TL;DR

This paper classifies isocategorical groups over arbitrary fields, extending previous work, and introduces a new Weil representation variant to construct concrete examples of such groups, including over finite fields.

Contribution

It extends the classification of isocategorical groups to arbitrary fields and introduces a new Weil representation variant for constructing examples.

Findings

Classified isocategorical groups over any field.

Constructed examples of non-isomorphic isocategorical groups.

Developed rational Weil representations for symplectic spaces over finite fields.

Abstract

Two groups are called isocategorical over a field $k$ if their respective categories of $k$-linear representations are monoidally equivalent. We classify isocategorical groups over arbitrary fields, extending the earlier classification of Etingof-Gelaki and Davydov for algebraically closed fields. In order to construct concrete examples of isocategorical groups a new variant of the Weil representation associated to isocategorical groups is defined. We construct examples of non-isomorphic isocategorical groups over any field of characteristic different from two and rational Weil representations associated to symplectic spaces over finite fields of characteristic two.