Nilpotent elements in the Green ring
type: Item
instance of: scholarly article
title: Nilpotent elements in the Green ring
zbMATH Open document ID: 0612.20005
DOI: 10.1016/0021-8693(86)90219-X
published in: Journal of Algebra
publication date: 1986-01-01T00:00:00Z
review text: Let k be a field of characteristic p and G a finite group. Theorem 2.1 is a generalization of a theorem of Landrock for the absolutely irreducible case: Suppose that M and N are absolutely indecomposable kG-modules. Then \(M\otimes N\) has the trivial module k as a direct summand if and only if the following two conditions are satisfied: (i) \(M\cong N^*\), (ii) \(p\nmid \dim N\). Moreover if k is a component of \(N^*\times N\) then it has multiplicity one. Section 2 also gives a generalization of 2.1 to Scott modules. M is absolutely p-divisible if, for every extension field \(k_ 1\) of k, every direct summand of \(k_ 1\otimes_ kM\) has k-dimension divisible by p. Let a(G;p) be the linear span in the Green ring a(G) \((A(G)={\mathbb{C}}\otimes_{{\mathbb{Z}}}a(G))\) of such module classes M. Then 2.1 readily implies that a(G;p) is an ideal. Furthermore A(G)/A(G;p) has no nilpotent elements. A new proof that A(G,Cyc) has no nilpotent elements is given. Finally for an indecomposable M a short exact sequence \(0\to \Omega (M)\to Y\to \Omega^{-1}(M)\to 0\) is obtained, which is split if and only if \(p| \dim M.\) In Section 3 a general technique is given for producing an infinity of nilpotent and idempotent elements in the Green ring. Previous methods of Zemanek are replaced by calculations in the cohomology ring of certain modules. In Section 4 the conditions required for the techniques of Section 3 are verified for \(G\cong {\mathbb{Z}}/(p^ r)\times {\mathbb{Z}}/(p^ s)\) by detailed calculations. These are given more scope by realizing G as a quotient of a p-group, etc.
Mathematics Subject Classification ID: 20C20
Mathematics Subject Classification ID: 20C05
zbMATH DE Number: 3989621
zbMATH Keywords: absolutely indecomposable kG-modules
zbMATH Keywords: direct summand
zbMATH Keywords: Scott modules
zbMATH Keywords: Green ring
zbMATH Keywords: nilpotent and idempotent elements
zbMATH Keywords: cohomology ring
Wikidata QID: Q105419657
author: David Benson
author: Jon F. Carlson
MaRDI profile type: Publication
cites work: Almost-split sequences and group rings
cites work: The Green ring of a finite group
cites work: Scott modules and lower defect groups
cites work: The varieties and the cohomology ring of a module
cites work: The variety of an indecomposable module is connected
cites work: Structure in representation algebras
cites work: Certain representation algebras
cites work: Relative components of representations
cites work: Construction of Auslander-Reiten quivers for a class of group rings
cites work: On the indecomposable representations of a finite group
cites work: A transfer theorem for modular representations
cites work: Q5343374
cites work: On the semisimplicity of the modular representation algebra of a finite group
cites work: The integral representation ring of a finite group
cites work: Nilpotent Elements in Rings of Integral Representations
cites work: K-theory of finite groups and orders. Notes by E. Graham Evans
cites work: Nilpotent elements in representation rings
cites work: Nilpotent elements in representation rings over fields of characteristic 2
full work available at URL: 0021-8693(86)90219-x
OpenAlex ID: W2078041522
description: scientific article; zbMATH DE number 3989621