Virtually irreducible lattices and Brauer correspondence of blocks
type: Item
instance of: scholarly article
title: Virtually irreducible lattices and Brauer correspondence of blocks
zbMATH Open document ID: 0818.20008
author: Harald Ellers
published in: Journal of Algebra
publication date: 1995-08-14T00:00:00Z
review text: Let \(G\) be a finite group, \(R\) a complete discrete valuation ring of characteristic zero with valuation \(\nu\), field of fractions \(K\) and residue field \(\overline {R}\) of characteristic \(p>0\), which is assumed to be algebraically closed, and \(K\) is a splitting field for each subgroup of \(G\). An \(RG\)-lattice \(M\) is called virtually irreducible if for all \(\alpha\in \text{End}_{RG} (M)\), we have that \(\nu(\text{tr } \alpha)\geq \nu(\text{rk } M)\), and equality holds if and only if \(\alpha\) is invertible. This concept was introduced by \textit{R. Knörr} [Proc. Lond. Math. Soc., III. Ser. 59, 99-132 (1989; Zbl 0676.20006)], and has some nice properties, since a source or a Green correspondent of a virtually irreducible lattice is virtually irreducible, and a lattice affording an absolutely irreducible character is virtually irreducible. The principal aim of this paper is to prove analogues for virtually irreducible lattices of some classical theorems which relate the Brauer correspondence of blocks to irreducible characters. One of these classical results is the following: if the block \(b\) of a subgroup of \(G\) contains an irreducible character \(\zeta\) such that \(\zeta^ G\) is defined, then \(b^ G\) is defined and it contains \(\zeta^ G\). Its analogue for virtually irreducible lattices says that: if \(H\) is a subgroup of \(G\), \(b\) is a block of \(RH\), \(M\) is a virtually irreducible lattice in \(b\) such that \(M^ G= V\oplus W\) with \(V\) virtually irreducible and \(M\nmid W_ H\), then \(B^ G\) is defined and it contains \(V\), and moreover \(\nu(\text{rk } V)= \nu(\text{rk } M)+\nu | G: M|\). This theorem is applied to obtain a new proof of Brauer's first main theorem, and also to extend many of Knörr's results. Another main result is the following generalization of Brauer's third main theorem. Let \(B\) be a block of \(RG\), \(M\) an indecomposable \(RG\)-lattice in \(B\) of height zero, \(H\leq G\) and \(b\) a block of \(RH\) w…
reviewed by: Andrei Marcus
Mathematics Subject Classification ID: 20C20
Mathematics Subject Classification ID: 20C05
zbMATH DE Number: 554266
zbMATH Keywords: blocks of finite groups
zbMATH Keywords: Green correspondent
zbMATH Keywords: virtually irreducible lattices
zbMATH Keywords: Brauer correspondence
zbMATH Keywords: irreducible characters
zbMATH Keywords: Brauer's first main theorem
zbMATH Keywords: generalization of Brauer's third main theorem
zbMATH Keywords: block
zbMATH Keywords: defect group
zbMATH Keywords: indecomposable summand
MaRDI profile type: Publication
full work available at URL: jabr.1993.1209
OpenAlex ID: W1986018758
DOI: 10.1006/JABR.1993.1209
description: scientific article; zbMATH DE number 554266