A counter-example to a conjecture of Lusztig.
type: Item
instance of: scholarly article
title: A counter-example to a conjecture of Lusztig.
zbMATH Open document ID: 1197.20033
author: Jian-yi Shi
published in: Journal of Algebra
publication date: 2010-06-02T00:00:00Z
review text: In [Adv. Stud. Pure Math. 6, 255-287 (1985; Zbl 0569.20032)], \textit{G. Lusztig} defined a function \(a\) on a Coxeter group \(W\). This function plays an important role in the study of the Hecke algebra of \(W\); in particular, Lusztig proved that for \(W\) affine or finite, the function \(a\) is constant on the two-sided cells of \(W\). Later on, in the book [\textit{G. Lusztig}, Hecke algebras with unequal parameters. CRM Monograph Series 18. Providence: AMS (2003; Zbl 1051.20003)], he defined another function \(a'\) on \(W\), related to the function \(a\), and conjectured that these two functions \(a\) and \(a'\) are in fact equal. Together with some other conjecture, this would imply that there are only finitely many two-sided cells in \(W\). This paper disproves this conjecture of Lusztig, by exhibiting a counter-example. The group \(W\) in this example is the affine Coxeter group \(\widetilde A_{10}\), which is seen as a group of permutations of \(\mathbb{Z}\), as in [\textit{J.-y. Shi}, The Kazhdan-Lusztig cells in certain affine Weyl groups. Lecture Notes in Mathematics 1179. Berlin etc.: Springer-Verlag (1986; Zbl 0582.20030)]. The author describes explicitly an element \(w\in W\) for which \(a(w)\neq a'(w)\). Finally, the author suggests another conjecture about the structure on two-sided cells of Coxeter groups. Assuming some other conjectures of Lusztig, this new conjecture is weaker than the previous one. Then he checks this new conjecture on Coxeter groups of type \(\widetilde A_n\) and \(\widetilde D_4\).
reviewed by: Jean Lécureux
Mathematics Subject Classification ID: 20F55
Mathematics Subject Classification ID: 20C08
zbMATH DE Number: 5716403
zbMATH Keywords: affine Weyl groups
zbMATH Keywords: two-sided cells
zbMATH Keywords: \(a\)-function
zbMATH Keywords: Lusztig conjecture
zbMATH Keywords: affine Coxeter groups
Wikidata QID: Q123352087
MaRDI profile type: Publication
full work available at URL: j.jalgebra.2010.01.018
OpenAlex ID: W2049920279
cites work: Cells in the affine Weyl group of type \(\tilde D_ 4\)
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cites work: Cells in affine Weyl groups. II
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cites work: A Two-Sided Cell in an Affine Weyl Group
cites work: Some results relating two presentations of certain affine Weyl groups
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cites work: Left-connectedness of some left cells in certain Coxeter groups of simply-laced type.
DOI: 10.1016/J.JALGEBRA.2010.01.018
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description: scientific article; zbMATH DE number 5716403