On Brauer configuration algebras induced by finite groups

TL;DR

This paper investigates the representation theory of Brauer configuration algebras induced by finite groups, focusing on their Cartan matrices and module lengths, and introduces subgroup-occurrence to establish combinatorial identities.

Contribution

It computes key representation-theoretic invariants of Brauer configuration algebras associated with finite groups and introduces the novel concept of subgroup-occurrence for combinatorial analysis.

Findings

Calculated the Cartan matrix of the algebra.

Determined the module length of indecomposable projective modules.

Established combinatorial equalities involving subgroup-occurrence.

Abstract

In this article we calculate two aspects of the representation theory of a Brauer configuration algebra: its Cartan matrix, and the module length of its associated indecomposable projective modules. Then we introduce the concept of subgroup-occurrence of an element in a group and use the previous aspects to demonstrate combinatorial equalities satisfied for any finite group.