Centralizer algebras of the primitive unitary reflection group of order $96$

TL;DR

This paper investigates the structure of the centralizer algebra associated with the primitive unitary reflection group of order 96, focusing on its irreducible representations and semi-simple algebraic structure.

Contribution

It provides a detailed analysis of the irreducible representations and the semi-simple structure of the centralizer algebra for this specific reflection group, a topic not extensively covered before.

Findings

Explicit description of irreducible representations

Analysis of the semi-simple structure of the centralizer algebra

Relevance to coding theory and number theory

Abstract

Among the unitary reflection groups, the one on the title is singled out by its importance in, for example, coding theory and number theory. In this paper we start with describing the irreducible representations of this group and then examine the semi-simple structure of the centralizer algebra in the tensor representation.