Representations of Finite Unipotent Linear Groups by the Method of Clusters

TL;DR

This paper introduces a novel 'method of clusters' to better understand the complex representation theory of the unipotent upper triangular subgroup of the general linear group over finite fields, providing new structural insights.

Contribution

It develops a variant of the orbit method using clusters of coadjoint orbits, constructing a structured subring in the representation ring of U(n, F_q).

Findings

Constructs a subring in the representation ring of U(n, F_q)

Provides a new perspective on the orbit method in finite fields

Enhances understanding of the representation theory of unipotent groups

Abstract

The general linear group GL(n, K) over a field K contains a particularly prominent subgroup U(n, K), consisting of all the upper triangular unipotent elements. In this paper we are interested in the case when K is the finite field F_q, and our goal is to better understand the representation theory of U(n, F_q). The complete classification of the complex irreducible representations of this group has long been known to be a difficult task. The orbit method of Kirillov, famous for its success when K has characteristic 0, is a natural source of intuition and conjectures, but in our case the relation between coadjoint orbits and complex representations is still a mystery. Here we introduce a natural variant of the orbit method, in which the central role is played by certain clusters of coadjoint orbits. This "method of clusters" leads to the construction of a subring in the representation ring of U(n, F_q) that is rich in structure but pleasantly comprehensible. The cluster method also has many of the major features one would expect from the philosophy of orbit method.