Towards canonical representations of finite Heisenberg groups

TL;DR

This paper constructs a canonical irreducible representation of finite Heisenberg groups associated with finite abelian groups and symplectic forms, providing a unique and well-defined model in this mathematical setting.

Contribution

It introduces a method to construct a canonical irreducible representation of finite Heisenberg groups, resolving non-uniqueness issues in prior representations.

Findings

Constructed a canonical irreducible representation of the Heisenberg extension

Resolved non-uniqueness in the representation up to isomorphism

Provided a new framework for understanding finite Heisenberg groups

Abstract

We consider a finite abelian group $M$ of odd exponent $n$ with a symplectic form $\omega: M\times M\to \mu_n$ and the Heisenberg extension $1\to \mu_n\to H\to M\to 1$ with the commutator $\omega$. According to the Stone - von Neumann theorem, $H$ admits an irreducible representation with the tautological central character (defined up to a non-unique isomorphism). We construct such irreducible representation of $H$ defined up to a unique isomorphism, so canonical in this sense.