Isotypic Decompositions of Lattice Determinants

Abstract

Theq,t-Macdonald polynomials are conjectured by Garsia and Haiman to have a representation theoretic interpretation in terms of theSn-moduleMμspanned by the derivatives of a certain polynomialΔμ(x1,x2,…,xn;y1,y2,…,yn). The diagonal action of a permutationσ∈Snon a polynomialP=P(x1,x2,…,xn;y1 ,y2,…,yn) is defined by settingσP=P(xσ1,xσ2< F,…,xσn;yσ1,yσ2,…,yσn). Since the polynomialΔμalternates under the diagonal action,MμisSn-invariant. We analyze here the diagonal action ofSnonMμand show that its decomposition into irreducibles is equivalent to a certain isotypic expansion for the translateΔμ(x1+x′1,x2+x ′2,…,xn+x′n;y1+y′1,y2+y′2,…,yn +y′n) of the polynomialΔμ.