The Green ring and modular representations of finite groups of Lie type
type: Item
instance of: scholarly article
title: The Green ring and modular representations of finite groups of Lie type
zbMATH Open document ID: 0683.20013
DOI: 10.1016/0021-8693(89)90042-2
published in: Journal of Algebra
publication date: 1989-01-01T00:00:00Z
review text: For G a finite group with split BN-pair and \(U=O_ p(B)\) a Sylow p- subgroup of G, the permutation module \(Y=ind_{G,U}(k)\) where k is a sufficiently large field of characteristic p has first been studied by \textit{C. W. Curtis} and \textit{F. Richen} [Lect. Notes Math. 131, B1-B39 (1970; Zbl 0233.20005)]. They proved that the indecomposable summands \(Y_ i\) of Y are in one to one correspondence with the simple kG-modules and have simple head and socle as well. Later on \textit{H. Sawada} [Math. Z. 155, 29-41 (1977; Zbl 0345.20009)] and \textit{J. Green} [J. Lond. Math. Soc., II. Ser. 18, 247-252 (1978; Zbl 0403.16009)] obtained the results by use of the Hecke algebra \(End_{KG}(Y)\). \textit{N. B. Tinberg} then computed the dimensions and vertices of the \(Y_ i's\) [J. Algebra 61, 508-526 (1979; Zbl 0434.20002)] and Cabanes the Green correspondents which give rise to verify Alperin's weight conjecture for those groups. In this paper the author starts with the significant results of Curtis and Richen. Interpreting these in the Green ring and using ``Möbius inversion'' he obtains a formula for the \(Y_ i's\) from which the results of Tinberg and Cabanes follow in a quite elegant way. Moreover he discovers a natural duality on the set \(\{Y_ i\}\) which turns out to be an isometry on the subgroup of the Green ring which is spanned by the \(Y_ i's\).
reviewed by: Wolfgang Willems
Mathematics Subject Classification ID: 20C20
zbMATH DE Number: 4118578
zbMATH Keywords: groups of Lie type
zbMATH Keywords: modular representations
zbMATH Keywords: finite group with split BN- pair
zbMATH Keywords: Sylow p-subgroup
zbMATH Keywords: permutation module
zbMATH Keywords: indecomposable summands
zbMATH Keywords: simple kG-modules
zbMATH Keywords: Hecke algebra
zbMATH Keywords: dimensions
zbMATH Keywords: vertices
zbMATH Keywords: Green correspondents
zbMATH Keywords: weight conjecture
zbMATH Keywords: Green ring
Wikidata QID: Q115366270
author: Peter Sin
MaRDI profile type: Publication
full work available at URL: 0021-8693(89)90042-2
OpenAlex ID: W1987745499
cites work: Brauer morphism between modular Hecke algebras
cites work: Q3907722
cites work: Modular Representations of Finite Groups of Lie Type
cites work: Truncation and duality in the character ring of a finite group of Lie type
cites work: On a Theorem of H. Sawada
cites work: A characterization of the modular representations of finite groups with split \((B,N)\)-pairs
cites work: A decomposition of the group algebra of a finite Coxeter group
cites work: Some indecomposable modules of groups with split (B,N)-pairs
description: scientific article; zbMATH DE number 4118578