Proceedings of the American Mathematical Society

Author(s): Stephen Griffeth
Journal: Proc. Amer. Math. Soc. 137 (2009), 1621-1629.
MSC (2000): Primary 05E10
Posted: December 11, 2008
Retrieve article in: PDF

Abstract: We study the coinvariant ring of the complex reflection group

as a module for the corresponding rational Cherednik algebra $\mathbb{H}$ and its generalized graded affine Hecke subalgebra $\mathcal{H}$ . We construct a basis consisting of non-symmetric Jack polynomials and, using this basis, decompose the coinvariant ring into irreducible modules for $\mathcal{H}$ . The basis consists of certain non-symmetric Jack polynomials whose leading terms are the ``descent monomials'' for

recently studied by Adin, Brenti, and Roichman as well as Bagno and Biagoli. The irreducible $\mathcal{H}$ -submodules of the coinvariant ring are their ``colored descent representations''.

1.: R. M. Adin, F. Brenti, and Y. Roichman, Descent representations and multivariate statistics, Trans. Amer. Math. Soc. 357 (2005), no. 8, 3051-3082 (electronic). MR 2135735 (2006e:20019)
2.: E. Bagno and R. Biagioli, Colored-descent representations of complex reflection groups , Israel J. Math. 160 (2007), 317-347. MR 2342500 (2008g:20083)
3.: C. Dezelee, Generalized graded Hecke algebra for complex reflection group of type , arXiv:math/0605410.
4.: V.G. Drinfeld, Degenerate affine Hecke algebras and Yangians, Funktsional. Anal. i Prilozhen. 20 (1986), no. 1, 69-70. MR 831053 (87m:22044)
5.: C.F. Dunkl, Harmonic polynomials and peak sets of reflection groups, Geometriae Dedicata 32 (1989), 151-171. MR 1029672 (91h:51015)
6.: C.F. Dunkl and E.M. Opdam, Dunkl operators for complex reflection groups, Proc. London Math. Soc. (3) 86 (2003), no. 1, 70-108. MR 1971464 (2004d:20040)
7.: P. Etingof and V. Ginzburg, Symplectic reflection algebras, Calogero-Moser space, and deformed Harish-Chandra homomorphism, Invent. Math. 147 (2002), no. 2, 243-348. MR 1881922 (2003b:16021)
8.: A.M. Garsia and D. Stanton, Group actions of Stanley-Reisner rings and invariants of permutation groups, Adv. in Math. 51 (1984), no. 2, 107-201. MR 736732 (86f:20003)
9.: A.M. Garsia, Combinatorial methods in the theory of Cohen-Macaulay rings, Adv. in Math. 38 (1980), no. 3, 229-266. MR 597728 (82f:06002)
10.: I. Gordon, Baby Verma modules for rational Cherednik algebras, Bull. London Math. Soc. 35 (2003), no. 3, 321-336. MR 1960942 (2004c:16050)
11.: S. Griffeth, Rational Cherednik algebras and coinvariant rings, Ph.D. thesis, University of Wisconsin, Madison, August 2006.
12.: S. Griffeth, Orthogonal functions generalizing Jack polynomials, arXiv:0707.0251.
13.: S. Griffeth, Towards a combinatorial representation theory for the rational Cherednik algebra of type , arXiv:math/0612733.
14.: V.G. Kac, Infinite-dimensional Lie algebras, Cambridge University Press, Cambridge, 1990. MR 1104219 (92k:17038)
15.: F. Knop and S. Sahi, A recursion and a combinatorial formula for Jack polynomials, Invent. Math. 128 (1997), no. 1, 9-22. MR 1437493 (98k:33040)
16.: A. Ram and A. V. Shepler, Classification of graded Hecke algebras for complex reflection groups, Comment. Math. Helv. 78 (2003), no. 2, 308-334. MR 1988199 (2004d:20007)

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 05E10

			Journals Home Search Author Info Subscribe Tech Support Help
ISSN 1088-6826 (e) ISSN 0002-9939 (p)

Previous issue \| Table of contents \| Next issue Articles in press \| Recently posted articles \| Most recent issue \| All issues Previous Article \| Next Article


	Comments: webmaster@ams.org © Copyright 2009, American Mathematical Society Privacy Statement		Search the AMS