On the Cartan matrix of Mackey algebras

Abstract

Let k be a field of characteristic p > 0, and let 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 μ k (G) of G over k. The second one is a formula for the rank of the Cartan matrix of the cohomological Mackey algebra coμ k (G) of G over k, and a characterization of the groups G for which this matrix is nonsingular. The third result is a generalization of this rank formula and characterization to blocks of coμ k (G): in particular, if b is a block of kG, the Cartan matrix of the corresponding block coμ k (b) of coμ k (G) is nonsingular if and only if b is nilpotent with cyclic defect groups.