Blocks of endomorphism algebras
Let \(G\) be a finite group, let \(\mathcal O\) be a suitable ring of coefficients, let \(A\) be an \({\mathcal O} G\)-lattice, let \({\mathcal H} (A) := \text{End}_{{\mathcal O} G} (A)\), and let \({\mathcal F} (A) := \text{End}_{{\mathcal H} (A)} (A)\). Then \({\mathcal F} (A)\) is a \(G\)-algebra, and \({\mathcal F} (A)^G = Z({\mathcal H} (A))\). Thus every block (idempotent) \(\alpha\) of \({\mathcal H} (A)\) has a defect group \(P\). The author shows that \(P\) is contained in a defect group of the block of \({\mathcal O} G\) containing \(A_\alpha\), and that \(P\) contains a vertex of every indecomposable direct summand of \(A_\alpha\). Let \(N_G(P) \leq H \leq G\). Then it is possible to associate to \(\alpha\) a unique block \(\text{ass}_H(\alpha)\) of \({\mathcal H} (\text{Res}^G_H (A))\). Conversely, given a block \(\beta\) of \({\mathcal H} (B)\) with defect group \(P\), for an \({\mathcal O} H\)-lattice \(B\), it is possible to associate to \(\beta\) a block \(\text{ass}_G (\beta)\) of \({\mathcal H} (\text{Ind}^G_H (B))\). The author studies the behaviour of these maps on blocks with respect to defect groups and the Brauer and the Green correspondence.
scientific article; zbMATH DE number 645102
scientific article