Combinatorics of finite abelian groups and Weil representations

Details Combinatorics of finite abelian groups and Weil representations Combinatorics of finite abelian groups and Weil representations

2010-10-18

arxiv.org/abs/1010.3528v1

Mathematics

Representation Theory

24 pages, 3 figures

Scientific paper

The Weil representation of the symplectic group associated to a finite abelian group of odd order is shown to have a multiplicity-free decomposition. When the abelian group is p-primary of type $\lambda$, the irreducible representations occurring in the Weil representation are parametrized by a partially ordered set which is independent of p. As p varies, the dimension of the irreducible representation corresponding to each parameter is shown to be a polynomial in p which is calculated explicitly. The commuting algebra of the Weil representation has a basis indexed by another partially ordered set which is independent of p. The expansions of the projection operators onto the irreducible invariant subspaces in terms of this basis are calculated. The coefficients are again polynomials in p. These results remain valid in the more general setting of finitely generated torsion modules over a Dedekind domain.

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