Jack polynomials and the coinvariant ring of 𝐺(𝑟,𝑝,𝑛)

Griffeth, Stephen

doi:10.1090/S0002-9939-08-09697-4

Jack polynomials and the coinvariant ring of $G(r,p,n)$
HTML articles powered by AMS MathViewer

by Stephen Griffeth

Proc. Amer. Math. Soc. 137 (2009), 1621-1629

DOI: https://doi.org/10.1090/S0002-9939-08-09697-4

Published electronically: December 11, 2008

PDF | Request permission

Abstract:

We study the coinvariant ring of the complex reflection group $G(r,p,n)$ 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 $G(r,p,n)$ 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”.

References

Ron M. Adin, Francesco Brenti, and Yuval Roichman, Descent representations and multivariate statistics, Trans. Amer. Math. Soc. 357 (2005), no. 8, 3051–3082. MR 2135735, DOI 10.1090/S0002-9947-04-03494-4
Eli Bagno and Riccardo Biagioli, Colored-descent representations of complex reflection groups $G(r,p,n)$, Israel J. Math. 160 (2007), 317–347. MR 2342500, DOI 10.1007/s11856-007-0065-z
C. Dezelee, Generalized graded Hecke algebra for complex reflection group of type $G(r,1,n)$, arXiv:math/0605410.
V. G. Drinfel′d, Degenerate affine Hecke algebras and Yangians, Funktsional. Anal. i Prilozhen. 20 (1986), no. 1, 69–70 (Russian). MR 831053
Charles F. Dunkl, Harmonic polynomials and peak sets of reflection groups, Geom. Dedicata 32 (1989), no. 2, 157–171. MR 1029672, DOI 10.1007/BF00147428
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, DOI 10.1112/S0024611502013825
Pavel Etingof and Victor Ginzburg, Symplectic reflection algebras, Calogero-Moser space, and deformed Harish-Chandra homomorphism, Invent. Math. 147 (2002), no. 2, 243–348. MR 1881922, DOI 10.1007/s002220100171
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, DOI 10.1016/0001-8708(84)90005-7
Adriano M. Garsia, Combinatorial methods in the theory of Cohen-Macaulay rings, Adv. in Math. 38 (1980), no. 3, 229–266. MR 597728, DOI 10.1016/0001-8708(80)90006-7
Iain Gordon, Baby Verma modules for rational Cherednik algebras, Bull. London Math. Soc. 35 (2003), no. 3, 321–336. MR 1960942, DOI 10.1112/S0024609303001978
S. Griffeth, Rational Cherednik algebras and coinvariant rings, Ph.D. thesis, University of Wisconsin, Madison, August 2006.
S. Griffeth, Orthogonal functions generalizing Jack polynomials, arXiv:0707.0251.
S. Griffeth, Towards a combinatorial representation theory for the rational Cherednik algebra of type $G(r,p,n)$, arXiv:math/0612733.
Victor G. Kac, Infinite-dimensional Lie algebras, 3rd ed., Cambridge University Press, Cambridge, 1990. MR 1104219, DOI 10.1017/CBO9780511626234
Friedrich Knop and Siddhartha Sahi, A recursion and a combinatorial formula for Jack polynomials, Invent. Math. 128 (1997), no. 1, 9–22. MR 1437493, DOI 10.1007/s002220050134
Arun Ram and Anne V. Shepler, Classification of graded Hecke algebras for complex reflection groups, Comment. Math. Helv. 78 (2003), no. 2, 308–334. MR 1988199, DOI 10.1007/s000140300013