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Glenn Tesler,
Isotypic Decompositions of Lattice Determinants,
Journal of Combinatorial Theory, Series A,85 (1999), 208-227.

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Abstract

The q,t-Macdonald polynomials are conjectured by Garsia and Haiman to have a representation theoretic interpretation in terms of the Sn-module M\mu spanned by the derivatives of a certain polynomial \Delta\mu(x1,x2,..., xn;y1,y2,...,yn). The diagonal action of a permutation \sigma\in Sn on a polynomial P=P(x1,x2,...,xn;y1, y2,...,yn) is defined by setting \sigma P=P(x\sigma1,x\sigma2, ...,x\sigman; y\sigma1,y\sigma2,..., y\sigman). Since the polynomial \Delta\mu alternates under the diagonal action, M\mu is Sn-invariant. We analyze here the diagonal action of Sn on M\mu and show that its decomposition into irreducibles is equivalent to a certain isotypic expansion for the translate \Delta\mu(x1+x1',x2+x2',...,xn+xn';y1+y1',y2+y2',...,yn+yn') of the polynomial \Delta\mu.