Abstract :
Let C[X, Y] denote the ring of polynomials with complex coefficients in the variables X={x1, …, xn} and Y={y1, …, yn}, let Sn denote the symmetric group of order n!, let Cm denote the cyclic group Cm={e2πij/m: 0⩽j⩽m−1} of order m, let Hk denote the subgroup of order k of Cm, and let Gn, m=Cm ≀ Sn (the wreath product of Cm with Sn). Each element ϕ of Gn, m may be represented as a generalized permutation ϕ=[ε(1) σ1, …, ε(n) σn] where ε(j)=e2πihj/m where 0⩽hj⩽m−1 and σ=[σ1, …, σn]∈Sn. Let Gn, m, k={ϕ∈Gn, m: ∏ni=1 ε(i)∈Hk}. In this paper, alternants Am, k, ν are defined in the variables X and Y and where n, m, and k are integers and ν is a partition. Setting In, m, k, ν(X, Y) to be the ideal In, m, k, ν(X, Y)={P∈C[X, Y]: P(∂x1, …, ∂xn, ∂y1, …, ∂yn) Am, k, ν=0}, where ∂xi (resp. ∂yj) is the partial differential operator with respect to xi (resp. yj), the action of Gn, m, k on the quotient ring Cn, m, k, ν=C[X, Y]/In, m, k, ν(X, Y) is isomorphic to the regular representation of Gn, m, k when ν=(1p), some ν=(p) (and k divides m) or when ν is a hook-shape and k=1. Bases are constructed for these quotient rings that exhibit the decomposition of the regular representation into irreducibles. It should be noted that the alternants Am, k, ν are generalizations of the alternants defined by A. Garsia and M. Haiman (1996, Electron. J. Combin.3, No. 2). Thus the graded characters of Cn, m, k, ν give generalizations of the q, t-Kostka coefficients.
