Generic Representation Theory of the Unipotent Upper Triangular Groups

TL;DR

This paper demonstrates that for unipotent upper triangular groups over prime characteristic fields, Lie theory can fully determine their representation theory under certain conditions relating the characteristic, group size, and representation dimension.

Contribution

It establishes that Lie theory suffices to understand the representation theory of unipotent upper triangular groups in prime characteristic, given specific hypotheses on the characteristic.

Findings

Lie theory fully determines representations under the hypothesis p ≥ max(n, 2d)

Identifies analogies between characteristic zero and positive characteristic theories

Provides a framework connecting Lie theory and unipotent group representations in prime characteristic

Abstract

It is generally believed (and for the most part is probably true) that Lie theory, in contrast to the characteristic zero case, is insufficient to tackle the representation theory of algebraic groups over prime characteristic fields. However, in this paper we show that, for a large and important class of unipotent algebraic groups (namely the unipotent upper triangular groups $U_n$), and under a certain hypothesis relating the characteristic $p$ to both $n$ and the dimension $d$ of a representation (specifically, $p \geq \text{max}(n,2d)$), Lie theory is completely sufficient to determine the representation theory of these groups. To finish, we mention some important analogies (both functorial and cohomological) between the characteristic zero theories of these groups and their `generic' representation theory in characteristic $p$.