On the Cartan matrix of Mackey algebras

TL;DR

This paper provides explicit formulas for the determinants and ranks of Cartan matrices of Mackey and cohomological Mackey algebras over fields of positive characteristic, including characterizations of when these matrices are non-singular.

Contribution

It introduces new explicit formulas for the determinant and rank of Cartan matrices of Mackey and cohomological Mackey algebras, and characterizes blocks with non-singular matrices.

Findings

Determinant formula for Cartan matrix of mu_k(G).

Rank formula and non-singularity characterization for comu_k(G).

Non-singularity of block Cartan matrices characterized by nilpotent blocks with cyclic defect groups.

Abstract

Let k be a field of characteristic p>0, and G be a finite group. The first result of this paper is an explicit formula for the determinant of the Cartan matrix of the Mackey algebra mu_k(G) of G over k. The second one is a formula for the rank of the Cartan matrix of the cohomological Mackey algebra comu_k(G) of G over k, and a characterization of the groups G for which this matrix is non singular. The third result is a generalization of this rank formula and characterization to blocks of comu_k(G) : in particular, if b is a block of kG, the Cartan matrix of the corresponding block comu_k(b) of comu_k(G) is non singular if and only if b is nilpotent with cyclic defect groups.