Representation Theory

Author(s): Cédric Bonnafé; Lacrimioara Iancu
Journal: Represent. Theory 7 (2003), 587-609.
MSC (2000): Primary 20C08; Secondary 20C15
Posted: November 19, 2003
Retrieve article in: PDF DVI PostScript

Abstract: Kazhdan and Lusztig have shown that the partition of the symmetric group ${\mathfrak{S}}_n$ into left cells is given by the Robinson-Schensted correspondence. The aim of this paper is to provide a similar description of the left cells in type

for a special class of choices of unequal parameters. This is based on a generalization of the Robinson-Schensted correspondence in type

. We provide an explicit description of the left cell representations and show that they are irreducible and constructible.

1.: S. Ariki, Robinson-Schensted correspondence and left cells. Combinatorial methods in representation theory (Kyoto, 1998), 1-20, Adv. Stud. Pure Math., 28, Kinokuniya, Tokyo, 2000. MR 2002i:20003
2.: R. Dipper, G. D. James and G. E. Murphy, Hecke algebras of type ${B}_n$ at roots of unity, Proc. London Math. Soc. (3) 70 (1995), 505-528. MR 96b:20004
3.: W. Fulton, Young tableaux, London Math. Soc. Stud. Texts, vol. 35, Cambridge University Press, 1997. MR 99f:05119
4.: M. Geck, Constructible characters, leading coefficients and left cells for finite Coxeter groups with unequal parameters, Represent. Theory, vol. 6, (2002), 1-30. MR 2003d:20009
5.: M. Geck, On the induction of Kazhdan-Lusztig cells, Bull. London Math. Soc. 35 (2003), 608-614.
6.: M. Geck, G. Hiss, F. Lübeck, G. Malle and G. Pfeiffer, CHEVIE -- A system for computing and processing generic character tables, Appl. Algebra Engrg. Comm. Comput. 7 (1996), 175-210. MR 99m:20017
7.: M. Geck and S. Kim, Bases for the Bruhat-Chevalley order on all finite Coxeter groups, J. Algebra 197 (1997), 278-310. MR 98k:20066
8.: M. Geck and G. Pfeiffer, Characters of finite Coxeter groups and Iwahori-Hecke algebras, London Math. Soc. Monographs, New Series 21, Oxford University Press, 2000. MR 2002k:20017
9.: J. J. Graham and G. I. Lehrer, Cellular algebras, Invent. Math. 123 (1996), 1-34. MR 97h:20016
10.: L. Iancu, Cellules de Kazhdan-Lusztig et correspondance de Robinson-Schensted, C. R. Acad. Sci. Paris, Sér. I 336 (2003), 791-794.
11.: D. A. Kazhdan and G. Lusztig, Representations of Coxeter groups and Hecke algebras, Invent. Math. 53 (1979), 165-184. MR 81j:20066
12.: D. E. Knuth, The art of computer programming, volume 3: Sorting and Searching, Addison-Wesley, Second Edition, 1998.
13.: G. Lusztig, Left cells in Weyl groups, Lie Group Representations, I (R. L. R. Herb and J. Rosenberg, eds.), Lecture Notes in Math., vol. 1024, Springer-Verlag, 1983, pp. 99-111. MR 85f:20035
14.: G. Lusztig, Characters of reductive groups over a finite field, Ann. of Math. Stud., vol. 107, Princeton University Press, 1984. MR 86j:20038
15.: G. Lusztig, Sur les cellules gauches des groupes de Weyl, C. R. Acad. Sci. Paris 302 (1986), 5-8. MR 87e:20089
16.: G. Lusztig, Intersection cohomology methods in representation theory, Proceedings of the International Congress of Mathematics, Kyoto, Japan, 1990 (I. Satake, ed.), Springer-Verlag, 1991, pp. 155-174. MR 93e:20059
17.: G. Lusztig, Hecke algebras with unequal parameters, in CRM Monogr. Ser., Vol. 18, American Mathematical Society, 2003.
18.: I. G. Macdonald, Symmetric functions and Hall polynomials, Second edition, Oxford Math. Monographs, Clarendon Press, Oxford, 1995. MR 96h:05207
19.: S. Okada, Wreath products by the symmetric groups and product posets of Young's lattices, J. Combin. Theory Ser. A 55 (1990) 14-32. MR 91i:20014
20.: D. E. Taylor, The geometry of the classical groups, Sigma Series in Pure Mathematics, Volume 9, Heldermann Verlag Berlin, 1992. MR 94d:20028

Cédric Bonnafé
Affiliation: Laboratoire de Mathématiques de Besançon, Université de Franche-Comté, 16 Route de Gray, 25030 Besançon Cedex, France
Email: bonnafe@math.univ-fcomte.fr

Lacrimioara Iancu
Affiliation: Laboratoire de Mathématiques de Besançon, Université de Franche-Comté, 16 Route de Gray, 25030 Besançon Cedex, France --- and --- Facultatea de Stiinte, Universitatea de Nord Baia Mare, Victoriei 76, RO-4800 Baia Mare, Romania
Email: iancu@math.univ-fcomte.fr

DOI: 10.1090/S1088-4165-03-00188-2
PII: S 1088-4165(03)00188-2
Received by editor(s): February 4, 2003
Received by editor(s) in revised form: September 17, 2003
Posted: November 19, 2003
Copyright of article: Copyright 2003, American Mathematical Society

			Journals Home Search Author Info Subscribe Tech Support Help
ISSN 1088-4165

Previous volume \| Table of contents \| Next volume Articles in press \| Most recent volume \| All volumes Previous Article \| Next Article


	Comments: webmaster@ams.org © Copyright 2008, American Mathematical Society Privacy Statement		Search the AMS