On generalized cells in affine Weyl groups
type: Item
instance of: scholarly article
title: On generalized cells in affine Weyl groups
zbMATH Open document ID: 0942.20019
author: Kirsten Bremke
published in: Journal of Algebra
publication date: 2000-05-29T00:00:00Z
review text: The concept of cells for an arbitrary Coxeter system \((W,S)\) was introduced by \textit{D. Kazhdan} and \textit{G. Lusztig} [in Invent. Math. 53, 165-184 (1979; Zbl 0499.20035)]. They define left, right, and two-sided cells, which play an important role in the study of the representations of the corresponding Hecke algebra. \textit{G. Lusztig} [Lect. Notes Math. 1024, 99-111 (1983; Zbl 0537.20019)], extends the concept of cells to the case in which there are integers \(C_s\geq 1\) associated to the simple reflections \(s\in S\) satisfying \(C_s=C_t\) if \(s,t\in S\) are conjugate. Integers subject to these conditions are referred to as parameters and the cells as generalized cells. The original cells are obtained if all \(C_s\) are equal. For equal parameters, cells in affine Weyl groups have been intensively studied. They have been explicitly described for type \(\widetilde A_n\), \(n\in\mathbb{N}\), rank 2, 3 and 4. Much less is known for unequal parameters. In this paper, the author determines the lowest generalized two-sided cell for affine Weyl groups. The author shows that it consists of at most \(|W_0|\) generalized left cells, where \(W_0\) denotes the corresponding finite Weyl group. For parameters coming from graph automorphisms, the author proves that this bound is exact. For such parameters, the author also characterizes all generalized left cells for finite and affine Weyl groups.
reviewed by: Chen Chengdong
Mathematics Subject Classification ID: 20F55
Mathematics Subject Classification ID: 20G05
Mathematics Subject Classification ID: 20G15
Mathematics Subject Classification ID: 20H15
zbMATH DE Number: 1019689
zbMATH Keywords: affine Weyl groups
zbMATH Keywords: generalized cells
zbMATH Keywords: Coxeter systems
zbMATH Keywords: simple reflections
zbMATH Keywords: finite Weyl groups
MaRDI profile type: Publication
full work available at URL: jabr.1996.6889
OpenAlex ID: W2004468766
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DOI: 10.1006/JABR.1996.6889
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description: scientific article; zbMATH DE number 1019689