On the Cartan matrix of Mackey algebras
Group Theory
2010-09-07 v2 Rings and Algebras
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.
Keywords
Cite
@article{arxiv.1001.0321,
title = {On the Cartan matrix of Mackey algebras},
author = {Serge Bouc},
journal= {arXiv preprint arXiv:1001.0321},
year = {2010}
}