<%@ Page Language="C#" MasterPageFile="~/Main.master" AutoEventWireup="true" Title="Volume 11 Issue 2" %>
Maximally Clustered Elements and Schubert Varieties
Jozsef Losonczy
Mathematics Department, Long Island University, Brookville, NY 11548, USA
losonczy@liu.edu
Annals of Combinatorics 11 (2) p.195-212 June, 2007
AMS Subject Classification: 20F55, 14M15
Abstract:
We introduce and study a class of “maximally clustered” elements for simply laced Coxeter groups. Such elements include as a special case the freely braided elements of Green and the author, which in turn constitute a superset of the i ji-avoiding elements of Fan. We show that any reduced expression for a maximally clustered element is short-braid equivalent to a “contracted” expression, which can be characterized in terms of certain subwords called “braid clusters”. We establish some properties of contracted reduced expressions and apply these to the study of Schubert varieties in the simply laced setting. Specifically, we give a smoothness criterion for Schubert varieties indexed by maximally clustered elements.
Keywords: braid relation, Coxeter group, root system, Schubert variety

References:

1. N. Bourbaki, Groupes et Algèbres de Lie, Chapitres IV-VI, Masson, Paris, 1981.

2. S.C. Billey and A. Postnikov, Smoothness of Schubert varieties via patterns in root systems, Adv. Appl. Math. 34 (2005) 447-466.

3. S.C. Billey and V. Lakshmibai, Singular Loci of Schubert Varieties, Progress in Mathematics, Vol. 182, Birkhäuser, Boston, 2000.

4. C.K. Fan, A Hecke algebra quotient and properties of commutative elements of a Weyl group, Ph.D. thesis, M.I.T., 1995.

5. C.K. Fan, Schubert varieties and short braidedness, Transform. Groups 3 (1998) 51-56.

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

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

8. R.M. Green and J. Losonczy, Schubert varieties and free braidedness, Transform. Groups 9 (2004) 327-336.

9. J.E. Humphreys, Reection Groups and Coxeter Groups, Cambridge Studies in Advanced Mathematics, Vol. 29, Cambridge Univ. Press, Cambridge, 1990.

10. S. Kumar, The nil Hecke ring and singularity for Schubert varieties, Invent. Math. 123 (1996) 471-506.

11. V. Lakshmibai and B. Sandhya, Criterion for smoothness of Schubert varieties in SL(n)=B, Proc. Indian Acad. Sci. Math. Sci. 100 (1990) 45-52.

12. T. Mansour, On an open problem of Green and Losonczy: exact enumeration of freely braided elements, Discrete Math. Theor. Comput. Sci. 6 (2004) 461-470.

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

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

15. B.E. Tenner, Reduced decompositions and permutation patterns, J. Algebraic Combin. 24 (2006) 263-284.

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

17. V. Vatter, Enumeration schemes for restricted permutations, Combin. Probab. Comput., to appear.