Combinatorics of finite abelian groups and Weil representations

Kunal Dutta and Amritanshu Prasad

Vol. 275 (2015), No. 2, 295–324

DOI: 10.2140/pjm.2015.275.295

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.

Keywords

Weil representation, Heisenberg group, Clifford group, finite abelian group

Mathematical Subject Classification 2010

Primary: 05E10, 11F27

Secondary: 81R05

Milestones

Received: 15 May 2013

Revised: 21 May 2014

Accepted: 23 May 2014

Published: 15 May 2015