Isotypic Decompositions of Lattice Determinants

The q; t-Macdonald polynomials are conjectured by Garsia and Haiman to have a representation theoretic interpretation in terms of the Sn-module M spanned by the derivatives of a certain polynomial (x1; x2; : : : ; xn; y1; y2; : : : ; yn). The diagonal action of a permutation 2 Sn on a polynomial P = P (x1; x2; : : : ; xn; y1; y2; : : : ; yn) is de ned by setting P = P (x 1 ; x 2 ; : : : ; x n; y 1 ; y 2 ; : : : ; y n). Since the polynomial alternates under the diagonal action,M is Sn-invariant. We analyze here the diagonal action of Sn onM and show that its decomposition into irreducibles is equivalent to a certain isotypic expansion for the translate (x1+x 0 1 ; x2+x 0 2 ; : : : ; xn+x 0 n ; y1+y 0 1 ; y2+y 0 2 ; : : : ; yn+y 0 n ) of the polynomial .