Combinatorics of finite abelian groups and Weil representations

TL;DR

This paper explores the structure of Weil representations of symplectic groups over finite abelian groups, revealing their multiplicity-free decomposition, parametrization of irreducibles, and polynomial behavior of dimensions as the prime varies.

Contribution

It provides a detailed analysis of the irreducible components of Weil representations for finite abelian groups, including explicit calculations and polynomial parametrizations.

Findings

Weil representation decomposes multiplicity-free for finite abelian groups of odd order.

Irreducible representations are parametrized by a p-independent partially ordered set.

Representation dimensions are polynomial functions in p, explicitly calculated.

Abstract

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, 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.