Generic Representation Theory of the Additive and Heisenberg Groups

TL;DR

This paper explores the deep connection between the representation theories of the Additive and Heisenberg groups over fields of characteristic zero and large prime characteristic p, revealing a factorization structure that simplifies understanding their representations.

Contribution

It establishes a link between characteristic zero and characteristic p representations for these groups, showing that large p representations decompose into products of Lie algebra representations.

Findings

Representations in large p can be factored into commuting products.

Each factor corresponds to a Lie algebra representation.

Representations resemble those of direct powers over characteristic zero fields.

Abstract

In this paper we give an intimate connection between the characteristic zero representation theories of the Additive and Heisenberg groups, and their characteristic p >0 theories when p is much larger than the dimension a representation. In particular, if p >> dimension, then all characteristic p representations for these groups can be factored into commuting products of representations, with each factor arising from a representation of the Lie algebra of the group, one for each of the representation's Frobenius layers. In this sense, for a fixed dimension and large enough p, all representations for these groups look generically like representations for direct powers of themselves over a field of characteristic zero.