Categorification of based modules over the complex representation ring of $S_4$

TL;DR

This paper classifies irreducible based modules of small rank over the complex representation ring of S_4 and explores their categorification via module categories related to projective representations.

Contribution

It provides a complete classification of irreducible based modules of rank up to 5 over r(S_4) and discusses their categorification through module categories of Rep(S_4).

Findings

16 inequivalent irreducible based modules classified

Classification applies to modules of rank up to 5

Categorification linked to projective representations of subgroups

Abstract

The complex representation rings of finite groups are the fundamental class of fusion rings, categorified by the corresponding fusion categories of complex representations. The category of $\mathbb{Z}_+$-modules of finite rank over such a representation ring is also semisimple. In this paper, we classify the irreducible based modules of rank up to 5 over the complex representation ring $r(S_4)$ of the symmetric group $S_4$. Totally 16 inequivalent irreducible based modules are obtained. Based on such a classification result, we further discuss the categorification of based modules over $r(S_4)$ by module categories over the complex representation category ${\rm Rep}(S_4)$ of $S_4$ arisen from projective representations of certain subgroups of $S_4$.