Group algebras over semi-local rings
Let G be a finite group, \(\omega =\exp (2\pi i/| G|)\) and let \(\pi\) be a subset of \(\pi(G)\), the set of all prime divisors of \(| G|\). Let \(R_{\pi}=\{\alpha \in {\mathbb{Q}}[\omega]|\nu_{\rho}(\alpha)\geq 0\) whenever \(\rho\) is a prime ideal divisor of \(| G|_{\pi}{\mathbb{Z}}[\omega]\}\). Then \(R_{\pi}\) is a principal ideal domain. Definition: A subset \(B_{\pi}\) of Irr(G) such that \(| G|^{-1}\sum_{\chi \in B_{\pi}}\sum_{g\in G}\chi (1)\chi (g^{-1})g\) is a primitive idempotent of \(Z(R_{\pi}G)\) is called a \(\pi\)-block. If G is a \(\pi\)- group then G has only one \(\pi\)-block. If G is a \(\pi'\)-group then each \(\pi\)-block of G contains exactly one irreducible character of G. The main aim of this paper is to prove the second main Theorem for \(\pi\)- blocks: Th. 6. Let \(\pi\subseteq \pi(G)\), let B be a \(\pi\)-block of G, x be a \(\pi\)-element of G. Then there is a uniquely determined union of \(\pi\)-blocks of \(C_ G(x)\), say B(x) such that (i) whenever \(y,z\in C_ G(x)_{\pi'}\), \[ \sum_{\chi \in B}\chi(x^{-1}y^{-1})\chi(xz)= \sum_{\mu \in B(x)}\mu (y^{-1})\mu(z). \] (ii) Whenever \(\chi'\in B\) and y is a \(\pi\)-regular element of \(C_ G(x)\) we have \[ \chi'(xy)=\sum_{\mu \in B(x)}<\chi'_{C_ G(x)},\mu >\mu(xy). \] (iii) Whenever \(\chi\in Irr(G)\) has the property that there are complex numbers \(\{c_{\mu}|\mu\in B(x)\}\) such that \(\chi(xy)=\sum_{\mu \in B(x)}c_{\mu}\mu(xy)\) for each \(\pi\)-regular \(y\in C_ G(x)\) then either \(\chi\in B\) or \(\chi\) vanishes on \(S^ G_{\pi}(x)\).
scientific article; zbMATH DE number 4057760