Exotic characters of unitriangular matrix groups

TL;DR

This paper explicitly constructs exotic irreducible characters of unitriangular groups over finite fields, demonstrating their values lie outside certain cyclotomic fields and providing new insights into their character theory.

Contribution

It offers a constructive, hand-derived description of exotic characters of $ ext{UT}_n(q)$, advancing understanding beyond previous non-constructive proofs.

Findings

Existence of irreducible characters outside $ ext{QQ}( ext{ extmu}_p)$ for large $n$

Explicit construction of characters with values in larger cyclotomic fields

Identification of Kirillov functions that are not characters for certain parameters

Abstract

Let $\UT_n(q)$ denote the unitriangular group of unipotent $n\times n$ upper triangular matrices over a finite field with cardinality $q$ and prime characteristic $p$. It has been known for some time that when $p$ is fixed and $n$ is sufficiently large, $\UT_n(q)$ has ``exotic'' irreducible characters taking values outside the cyclotomic field $\QQ(\zeta_p)$. However, all proofs of this fact to date have been both non-constructive and computer dependent. In a preliminary work, we defined a family of orthogonal characters decomposing the supercharacters of an arbitrary algebra group. By applying this construction to the unitriangular group, we are able to derive by hand an explicit description of a family of characters of $\UT_n(q)$ taking values in arbitrarily large cyclotomic fields. In particular, we prove that if $r$ is a positive integer power of $p$ and $n>6r$, then $\UT_n(q)$ has an irreducible character of degree $q^{5r^2-2r}$ which takes values outside $\QQ(\zeta_{pr})$. By the same techniques, we are also able to construct explicit Kirillov functions which fail to be characters of $\UT_n(q)$ when $n>12$ and $q$ is arbitrary.