Lambda and psi operations on Green rings
type: Item
instance of: scholarly article
title: Lambda and psi operations on Green rings
zbMATH Open document ID: 0539.20011
DOI: 10.1016/0021-8693(84)90142-X
published in: Journal of Algebra
publication date: 1984-01-01T00:00:00Z
review text: A(G) denotes the Green ring formed from (\({\mathbb{C}}\)-combinations of) finite-dimensional (right) kG-modules, where k is a field of characteristic p and G is a finite group. A species is a \({\mathbb{C}}\)- algebra epimorphism \(s:A(G)\to {\mathbb{C}},\quad x\mapsto(s,x).\) For instance if g is a p'-element of G, the evaluation map \(b_ g\) of the Brauer character value \((b_ g,M)\) for each module M on the element g is a species. For each \(n\in {\mathbb{N}}\), a \({\mathbb{C}}\)-algebra endomorphism \(\psi^ n\) of A(G) is constructed; these \(\psi^ n\) are analogous to Adams operators defined in other contexts; they satisfy \((b_{g^ n},x)=(b_ g,\psi^ n(x))\) for Brauer species \(b_ g\) and \(b_{g^ n}.\) More generally the power \(s^ n\) (again a species) of s is defined as \((s^ n,n)= (s,\psi^ n(x)). \lambda^ n(x)\) is also defined in terms of \(\psi\) and for \(n<p\), \(\lambda^ n(x)\) is the exterior nth power of x. This makes A(G) into a special \(\lambda\)-ring (coefficients of A(G) taken in \({\mathbb{Z}}[1/p])\). An examination of the vertices and origins of powers of a species is given.
Mathematics Subject Classification ID: 20C20
Mathematics Subject Classification ID: 20C05
zbMATH DE Number: 3857358
zbMATH Keywords: Green ring
zbMATH Keywords: Brauer character
zbMATH Keywords: Adams operators
zbMATH Keywords: Brauer species
zbMATH Keywords: vertices
author: David Benson
MaRDI profile type: Publication
cites work: Q3950715
cites work: \(\lambda\)-rings and the representation theory of the symmetric group
cites work: Q5628258
cites work: Lambda-rings, binomial domains, and vector bundles over cp(∞)
full work available at URL: 0021-8693(84)90142-x
OpenAlex ID: W1979706700
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description: scientific article; zbMATH DE number 3857358