Hostname: page-component-848d4c4894-r5zm4 Total loading time: 0 Render date: 2024-06-28T05:54:59.164Z Has data issue: false hasContentIssue false
  • English
  • Français

Proof of Cellini’s conjecture on self-avoiding paths in Coxeter groups

Part of: Special aspects of infinite or finite groups Representation theory of groups

Published online by Cambridge University Press:  30 November 2011

Matthew Dyer
Matthew Dyer*
Affiliation:
Department of Mathematics, 255 Hurley Building, University of Notre Dame, Notre Dame, Indiana 46556, USA (email: dyer.1@nd.edu)
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

This note proves Cellini’s conjecture that, in a Coxeter system (W,S) with reflections T, the T-increasing paths in W are self-avoiding. Here, a T-increasing path is a sequence v,t1v,…,tn⋯t1v in W with ti∈T and t1≺⋯≺tn in a reflection order ⪯ of T.

Keywords

Coxeter grouproot systemHecke algebraBruhat order

MSC classification

Secondary: 20F55: Reflection and Coxeter groups 20C08: Hecke algebras and their representations
Type
Research Article
Information
Compositio Mathematica , Volume 148 , Issue 2 , March 2012 , pp. 548 - 554
Copyright
Copyright © Foundation Compositio Mathematica 2011

References

[BB07]Billera, L. J. and Brenti, F., Quasisymmetric functions and Kazhdan–Lusztig polynomials, arXiv:0710.3965v1 [math.CO] (2007).Google Scholar
[BB05]Björner, A. and Brenti, F., Combinatorics of Coxeter groups, Graduate Texts in Mathematics, vol. 231 (Springer, New York, 2005); MR 2133266.Google Scholar
[Bou68]Bourbaki, N., Éléments de mathématique. Fasc. XXXIV. Groupes et algèbres de Lie., Actualités Scientifiques et Industrielles, vol. 1337 (Hermann, Paris, 1968), ch. 4–ch. 6;  MR 0240238(39#1590).Google Scholar
[BI06]Brenti, F. and Incitti, F., Lattice paths, lexicographic correspondence and Kazhdan–Lusztig polynomials, J. Algebra 303 (2006), 742–762; MR 2255134(2007m:05228).CrossRefGoogle Scholar
[Cel00]Cellini, P., T-increasing paths on the Bruhat graph of affine Weyl groups are self-avoiding, J. Algebra 228 (2000), 107–118; MR 1760958(2001e:20033).CrossRefGoogle Scholar
[Dye87]Dyer, M. J., Hecke algebras and reflections in Coxeter groups, PhD thesis, University of Sydney (1987).Google Scholar
[Dye90]Dyer, M. J., Reflection subgroups of Coxeter systems, J. Algebra 135 (1990), 57–73; MR 1076077(91j:20100).CrossRefGoogle Scholar
[Dye91]Dyer, M. J., On the Bruhat graph of a Coxeter system, Compositio Math. 78 (1991), 185–191; MR 1104786(92c:20076).Google Scholar
[Dye92]Dyer, M. J., Hecke algebras and shellings of Bruhat intervals. II. Twisted Bruhat orders, in Kazhdan–Lusztig theory and related topics (Chicago, IL, 1989), Contemporary Mathematics, vol. 139 (American Mathematical Society, Providence, RI, 1992), 141–165; MR 1197833(94c:20072).CrossRefGoogle Scholar
[Dye93]Dyer, M. J., Hecke algebras and shellings of Bruhat intervals, Compositio Math. 89 (1993), 91–115; MR 1248893(95c:20053).Google Scholar
[Dye94]Dyer, M. J., Quotients of twisted Bruhat orders, J. Algebra 163 (1994), 861–879; MR 1265869(95c:20054).CrossRefGoogle Scholar
[Hum90]Humphreys, J. E., Reflection groups and Coxeter groups, Cambridge Studies in Advanced Mathematics, vol. 29 (Cambridge University Press, Cambridge, 1990); MR 1066460(92h:20002).CrossRefGoogle Scholar