Combinatorics of finite abelian groups and Weil representations

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.

26 pages, 3 figures Revised version, to appear in Pacific Journal of Mathematics

References in corpus (5)

• On Representations of General Linear Groups over Principal Ideal Local Rings of Length Two
• On Cuspidal Representations of General Linear Groups over Discrete Valuation Rings
• An easy proof of the Stone-von Neumann-Mackey theorem
• Locally compact abelian groups with symplectic self-duality
• Degenerations and orbits in finite abelian groups

Cited by in corpus (2)

• Clifford group is not a semidirect product in dimensions $N$ divisible by four
• Weil Representation of a Generalized Linear Group over a Ring of Truncated Polynomials over a Finite Field Endowed with a Second Class Involution