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Isotypic Decompositions of Lattice Determinants

✍ Scribed by Glenn Tesler

Publisher
Elsevier Science
Year
1999
Tongue
English
Weight
162 KB
Volume
85
Category
Article
ISSN
0097-3165

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✦ Synopsis

The q, t-Macdonald polynomials are conjectured by Garsia and Haiman to have a representation theoretic interpretation in terms of the S n -module M + spanned by the derivatives of a certain polynomial 2 + (x 1 , x 2 , ..., x n ; y 1 , y 2 , ..., y n ). The diagonal action of a permutation _ # S n on a polynomial P=P(x 1 , x 2 , ..., x n ; y 1 , y 2 , ..., y n ) is defined by setting _P=P(x _ 1 , x _ 2 , ..., x _ n ; y _ 1 , y _ 2 , ..., y _ n ). Since the polynomial 2 + alternates under the diagonal action, M + is S n -invariant. We analyze here the diagonal action of S n on M + and show that its decomposition into irreducibles is equivalent to a certain isotypic expansion for the translate 2 + (x 1

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