Regular Article
Isotypic Decompositions of Lattice Determinants

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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 permutationσSnon 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+x1,x2+x2, …, xn+xn;y1+y1,y2+y2, …, yn +yn) of the polynomialΔμ.

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Work carried out under NSF Grant support.