Representations of Finite Unipotent Linear Groups by the Method of Clusters
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.
Keywords
Cite
@article{arxiv.1004.2674,
title = {Representations of Finite Unipotent Linear Groups by the Method of Clusters},
author = {Ning Yan},
journal= {arXiv preprint arXiv:1004.2674},
year = {2010}
}
Comments
The first version of this paper dates back to 2001, and served as my Ph.D. dissertation. The current version was perpared in 2006. A more updated version is likely to appear some time after the AIM workshop on Supercharacter Theory and Combinatorial Hopf Algebra, to be held in Palo Alto in May, 2010.