Primitive Central Idempotents of the Group Algebra

TL;DR

This paper presents a character-free approach to find primitive central idempotents of finite group algebras, enabling the construction of irreducible representations and isomorphisms with matrix algebras for various finite groups.

Contribution

It introduces a novel method to derive primitive central idempotents without character theory, applicable to a broad class of finite groups including meta-abelian groups.

Findings

Primitive central idempotents obtained as eigenbasis of the centre

Irreducible representations constructed without character theory

Isomorphisms between simple blocks and matrix algebras established

Abstract

An approach to representations of finite groups is presented without recourse to character theory. Considering the group algebra C[G] as an algebra of linear maps on C[G] (by left multiplication), we derive the primitive central idempotents as a simultaneous eigenbasis of the centre, Z(C[G]). We apply this framework to obtain the irreducible representations of a class of finite meta-abelian groups. In particular, we give a general construction of the isomorphism between simple blocks of C[G] and the corresponding matrix algebra where G can be any finite group.