The \(p\)-blocks of the Mackey algebra
Let \(G\) be a finite group, \(p\) a prime number, and \(\mathcal O\) a complete discrete valuation ring with residue field of characteristic \(p\). Denote by \(G^c\) the set \(G\) on which \(G\) acts by conjugation. By definition, a crossed \(G\)-set is a \(G\)-set over \(G^c\), and the Burnside group \(B^c(G)\) is the Grothendieck group of the category of crossed \(G\)-sets for relation given by disjoint union decomposition. The direct product of crossed \(G\)-sets induces a product on \(B^c(G)\), and \(B^c(G)\) becomes a commutative ring with identity element. More generally, if \(R\) is a commutative ring, let \(B^c_R(G)=R\otimes_{\mathbb Z}B^c(G)\). The author shows that there is an action of crossed \(G\)-sets on Mackey functors, which induces a ring homomorphism from \(B^c_R(G)\) to the center of the Mackey algebra \(\mu_R(G)\). Recall that this algebra was introduced by \textit{J. Thévenaz} and \textit{P. Webb} [Trans. Am. Math. Soc. 347, 1865--1961 (1995; Zbl 0834.20011)], who proved that the category of Mackey functors for \(G\) over \(R\) is equivalent to the category of \(\mu_R(G)\)-modules. In the paper under review, the author also introduces the \(p\)-local Mackey algebra \(\mu_{\mathcal O}^1(G)\), and by giving a description of the primitive idempotents and of the prime spectrum of \(B^c_{\mathcal O}(G)\), he states explicit formulae for the block idempotents of \(\mu_{\mathcal O}^1(G)\) in terms of the corresponding blocks of the group algebra \({\mathcal O}G\). Several consequences and applications of these formulae are given.
scientific article; zbMATH DE number 2037026
scientific article