Exotic characters of unitriangular matrix groups

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 $\QQ(ζ_p)$. 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 $\QQ(ζ_{pr})$. 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.

24 pages, 3 figures

References in corpus (4)

• Superinduction for pattern groups
• Combinatorial methods of character enumeration for the unitriangular group
• Superclasses and supercharacters of normal pattern subgroups of the unipotent upper triangular matrix group
• Iterative character constructions for algebra groups

Cited by in corpus (5)

• Combinatorial methods of character enumeration for the unitriangular group
• Heisenberg characters, unitriangular groups, and Fibonacci numbers
• Global and local properties of finite groups with only finitely many central units in their integral group ring
• Iterative character constructions for algebra groups
• On the coadjoint orbits of maximal unipotent subgroups of reductive groups