The $p$-blocks of the Mackey algebra

Serge Bouc


Abstract :

Let $p$ be a prime number, and ${\cal O}$ be a complete discrete valuation  ring of characteristic 0 whith residue field of characteristic $p$. Let $G$ be a finite group, and denote by $\mu_{\cal O}(G)$ the Mackey algebra of $G$ over ${\cal O}$.\par
Formulae for the primitive idempotents in the center of $\mu_{\cal O}(G)$ have been given by Yoshida (and slightly corrected by Oda). However, those formulae are expressed in terms of ordinary irreducible characters of the centralizers of subgroups of $G$. The aim of this article is to state explicit formulae for the block idempotents of $\mu_{\cal O}(G)$, in terms of the blocks of the group algebra ${\cal O}G$. \par
The proof uses the natural ring homomorphism from the crossed Burnside ring $B_{\cal O}^c(G)$ to the center of the Mackey algebra, and a description of the prime spectrum and block idempotents of $B_{\cal O}^c(G)$. \par