On the Cartan matrix of Mackey algebras

Abstract

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.

Equivalences between blocks of cohomological Mackey algebras

“…However the determinant of the Cartan matrix of a block of a cohomological Mackey algebra can be zero. Bouc in [6] proved that the Cartan matrix of coµ k (b) is non singular if and only if the block b is a nilpotent block with cyclic defect group. This proof is based on a combinatorial approach, and it may be surprising that nilpotent blocks and cyclic defect groups appear in that situation.…”

“…Although the determinant of the Cartan Matrix of a block b of kG is a power of p, for the corresponding blocks of the Mackey algebra, it is much more complicated (see [6]). By the results of [16] this determinant is non zero.…”

The primitive idempotents of the p-permutation ring

“…Our formulae are an essential tool in [2], where Cartan matrices of Mackey algebras are considered, and some invariants of these matrices (determinant, rank) are explicitly computed.…”

Equivalences between blocks of p-local Mackey algebras

“…Remark 3.6. By Section 4.4 of [6], the sub-matrix indexed by the ordinary characters of G, and the (isomorphism classes of) indecomposable p-permutation kG-modules is the decomposition matrix of the cohomological Mackey algebra coµ O (G). The sub-matrix indexed by the ordinary characters of G and the isomorphism classes of indecomposable projective kG-modules is the decomposition matrix of OG.…”