Exotic characters of unitriangular matrix groups

Abstract

Let UT n ( q ) denote the unitriangular group of unipotent n × 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 Q ( ζ p ) . However, all proofs of this fact to date have been both non-constructive and computer dependent. In the preliminary work Marberg (2010) [15], 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 > 6 r , then UT n ( q ) has an irreducible character of degree q 5 r 2 − 2 r which takes values outside Q ( ζ p r ) . 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.