Title : On the Cartan matrix of Mackey algebras

Author : Serge Bouc

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.