Some applications of CHEVIE to the theory of algebraic groups.
type: Item
instance of: scholarly article
zbMATH Open document ID: 1258.20038
author: Meinolf Geck
publication date: 2012-10-04T00:00:00Z
full work available at URL: https://arxiv.org/abs/1008.1464
Mathematics Subject Classification ID: 20G05
Mathematics Subject Classification ID: 20-04
Mathematics Subject Classification ID: 20F55
Mathematics Subject Classification ID: 20E45
Mathematics Subject Classification ID: 20G15
Mathematics Subject Classification ID: 20C40
Mathematics Subject Classification ID: 20C20
Mathematics Subject Classification ID: 20C33
zbMATH DE Number: 6090227
zbMATH Keywords: Coxeter groups
zbMATH Keywords: Weyl groups
zbMATH Keywords: computer algebra systems
zbMATH Keywords: algorithms
zbMATH Keywords: irreducible characters
zbMATH Keywords: conjugacy classes
zbMATH Keywords: unipotent classes
zbMATH Keywords: reductive algebraic groups
zbMATH Keywords: Green functions
describes a project that uses: CHEVIE
describes a project that uses: GAP
describes a project that uses: Coxeter
MaRDI profile type: Publication
arXiv ID: 1008.1464
arXiv classification: math.RT
title: Some applications of CHEVIE to the theory of algebraic groups.
published in: Carpathian Journal of Mathematics
review text: The main goal of this article is to provide three examples of how the computer algebra package CHEVIE has played a key role in recent developments in the theory of algebraic groups and their representation theory. The package has been used directly to verify special cases of theorems as well as indirectly through examples which have suggested new general results.NEWLINENEWLINE Let \(G\) be a connected reductive algebraic group over an algebraically closed field \(k\) of prime characteristic \(p\), and let \(W\) denote the associated Weyl group. The first application of CHEVIE discussed is the recent work of \textit{G. Lusztig} [Represent. Theory 15, 494-530 (2011; Zbl 1263.20045)]. In that work, under the assumption that the prime \(p\) is good relative to the root system of \(G\), Lusztig constructed a surjective map from conjugacy classes of \(W\) to unipotent classes of \(G\). In addition to explaining the result and the role played by CHEVIE, the author also provides some computations suggesting that the result may also hold in bad characteristic. Lusztig's proof involves the notion of an excellent element in a conjugacy class. In the more general setting of \(W\) being a finite Coxeter group, the author presents a new result on the general existence of excellent elements.NEWLINENEWLINE The second example is in the context of irreducible characters of \(W\), information about which plays a key role in the representation theory of \(G\). In much earlier work, \textit{G. Lusztig} [Indag. Math. 41, 323-335 (1979; Zbl 0435.20021)], Lusztig introduced the notion of \(a\)-invariants associated to irreducible characters of \(W\). These were originally defined through the use of an associated Iwahori-Hecke algebra. In this paper, the author shows how \(a\)-invariants can be defined directly in terms of the irreducible characters themselves. The author then discusses Lusztig's families of irreducible characters.NEWLINENEWLINE The last example presented is in connection w…
reviewed by: Christopher P. Bendel
description: scientific article; zbMATH DE number 6090227