Generic Representation Theory of the Heisenberg Group

TL;DR

This paper generalizes a known representation result from the additive group to the Heisenberg group, showing that for large enough prime characteristic, all representations decompose into products derived from Lie algebra representations.

Contribution

It extends the representation theory framework from the additive group to the Heisenberg group, revealing a factorization structure for large prime characteristics.

Findings

Representations factor into commuting products from Lie algebra representations

All representations resemble those of direct powers over characteristic zero fields

Valid for prime p greater than twice the dimension d

Abstract

In this paper we extend a result for representations of the Additive group $G_a$ given in [3] to the Heisenberg group $H_1$. Namely, if $p$ is greater than 2d then all $d$-dimensional characteristic $p$ representations for $H_1$ can be factored into commuting products of representations, with each factor arising from a representation of the Lie algebra of $H_1$, one for each of the the representation's Frobenius layers. In this sense, for a fixed dimension and large enough $p$, all representations for $H_1$ look generically like representations for direct powers of it over a field of characteristic zero. The reader may consult chapter 13 of [1] for a fuller account of what follows.