Canonical left cells in affine Weyl groups
type: Item
instance of: scholarly article
title: Canonical left cells in affine Weyl groups
zbMATH Open document ID: 0664.20028
DOI: 10.1016/0001-8708(88)90031-X
author: G. Lusztig
author: Nanhua Xi
published in: Advances in Mathematics
publication date: 1988-01-01T00:00:00Z
review text: Let (W,S) be an irreducible affine Weyl group with S the set of Coxeter generators. Define the sets \({\mathcal R}(w)=\{s\in S|\) \(ws<w\}\) and \(Y_ s=\{y\in W|\) \({\mathcal R}(y)\subset \{s\}\}\) for any \(w\in W\) and \(s\in S\). An element \(s\in S\) is said to be special if the subgroup of W generated by S-\(\{\) \(s\}\) has maximum possible order. Then in the present paper, the authors show that for any \(s\in S\), the intersection of the set \(Y_ s\) with any two-sided cell \(\Omega\) of W is nonempty and is a union of some left cells of W. This result becomes more accurate in two cases. One is that when s is special, \(Y_ s\cap \Omega\) is exactly one left cell of W. The other is that when \(\Omega\) is the lowest two-sided cell of W, \(Y_ s\cap \Omega\) consists of n(s) left cells of W, where \(n(s)=\)order of subgroup generated by \(S-\{s_ 0\}/\)order of subgroup generated by \(S-\{s\}\) and \(s_ 0\) is a special element of S.
Mathematics Subject Classification ID: 20H15
Mathematics Subject Classification ID: 20G15
Mathematics Subject Classification ID: 20G05
zbMATH DE Number: 4085906
zbMATH Keywords: irreducible affine Weyl group
zbMATH Keywords: Coxeter generators
zbMATH Keywords: two-sided cell
zbMATH Keywords: left cells
zbMATH Keywords: special element
MaRDI profile type: Publication
cites work: Representations of Coxeter groups and Hecke algebras
cites work: Q3686887
cites work: Cells in affine Weyl groups. II
cites work: Q3804915
cites work: A Two-Sided Cell in an Affine Weyl Group
cites work: A Two-Sided Cell in an Affine Weyl Group, II
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description: scientific article; zbMATH DE number 4085906