<%@ Page Language="C#" MasterPageFile="~/Main.master" AutoEventWireup="true" Title="Volume 13 Issue 3" %>
On the Maximally Clustered Elements of Coxeter Groups
R. M. Green
Department of Mathematics, University of Colorado, Campus Box 395, Boulder, CO 80309-0395, USA
Annals of Combinatorics 14 (4) pp.467-478 December, 2010
AMS Subject Classification: 20F55
We continue the study of the maximally clustered elements for simply laced Coxeter groups which were recently introduced by Losonczy. Such elements include as a special case the freely braided elements introduced by Losonczy and the author, which in turn constitute a superset of the iji-avoiding elements of Fan. Our main result is to classify the MC-finite Coxeter groups, namely, those Coxeter groups having finitely many maximally clustered elements. Remarkably, any simply laced Coxeter group having finitely many $iji$-avoiding elements also turns out to
be MC-finite.
Keywords: contractible triples, highly connected graph


1. Björner, A., Brenti, F.: Combinatorics of Coxeter Groups. Springer, New York (2005)

2. Bourbaki, N.: Groupes et Algèbres de Lie, Chapitres IV--VI. Masson, Paris (1981)

3. Fan, C.K.: A Hecke algebra quotient and properties of commutative elements of a Weyl group. Ph.D. thesis, MIT, Cambridge (1995)

4. Green, R.M., Losonczy, J.: Freely braided elements in Coxeter groups. Ann. Combin. 6, 337--348 (2002)

5. Green, R.M., Losonczy, J.: Freely braided elements in Coxeter groups, II. Adv. Appl. Math. 33(1), 26--39 (2004)

6. Humphreys, J.E.: Reflection Groups and Coxeter Groups. Cambridge University Press, Cambridge (1990)

7. Losonczy, J.: Maximally clustered elements and Schubert varieties. Ann. Combin. 11(2), 195--212 (2007)

8. Matsumoto, H.: Générateurs et relations des groupes de Weyl généralisés. C. R. Acad. Sci. Paris 258, 3419--3422 (1964)

9. Stembridge, J.R.: On the fully commutative elements of Coxeter groups. J. Algebraic Combin. 5(4), 353--385 (1996)

10. Tits, J.: Le probléme des mots dans les groupes de Coxeter, In: Symposia Mathematica, Vol. 1, pp. 175--185. Academic Press, London (1969)