Basic characters of the unitriangular group (for arbitrary primes) (original) (raw)

Hecke algebras for the basic characters of the unitriangular group

Proceedings of the American Mathematical Society, 2004

Let Un(q) denote the unitriangular group of degree n over the finite field with q elements. In a previous paper we obtained a decomposition of the regular character of Un(q) as an orthogonal sum of basic characters. In this paper, we study the irreducible constituents of an arbitrary basic character ξ (ϕ) of Un(q). We prove that ξ (ϕ) is induced from a linear character of an algebra subgroup of Un(q), and we use the Hecke algebra associated with this linear character to describe the irreducible constituents of ξ (ϕ) as characters induced from an algebra subgroup of Un(q). Finally, we identify a special irreducible constituent of ξ (ϕ), which is also induced from a linear character of an algebra subgroup. In particular, we extend a previous result (proved under the assumption p ≥ n where p is the characteristic of the field) that gives a necessary and sufficient condition for ξ (ϕ) to have a unique irreducible constituent.

Counting characters of small degree in upper unitriangular groups

Journal of Pure and Applied Algebra, 2011

Let U n denote the group of upper n × n unitriangular matrices over a fixed finite field F of order q. That is, U n consists of upper triangular n × n matrices having every diagonal entry equal to 1. It is known that the degrees of all irreducible complex characters of U n are powers of q. It was conjectured by Lehrer that the number of irreducible characters of U n of degree q e is an integer polynomial in q depending only on e and n. We show that there exist recursive (for n) formulas that this number satisfies when e is one of 1, 2 and 3, and thus show that the conjecture is true in those cases.

Super-characters of finite unipotent groups of types Bn, Cn and Dn

Journal of Algebra, 2006

We define and study super-characters (over the complex field) of the classical finite unipotent groups of types B n , C n and D n. Under the assumption that the prime is sufficiently large, we extend the known results for the unitriangular group proved by the first author in the papers: [C.A.M. André, Basic characters of the unitriangular group, J. Algebra 175 (1995) 287-319], and [C.A.M. André, Basic characters of the unitriangular group (for arbitrary primes), Proc. Amer. Math. Soc. 130 (7) (2002) 1943-1954]. In particular, we prove that every irreducible (complex) character occurs as a constituent of a unique super-character. We also give a combinatorial description of all the irreducible characters of maximum degree.

On character values in finite groups

Bulletin of the Australian Mathematical Society, 1977

Let u be a nonidentity element of a finite group G and let c be a complex number. Suppose that every nonprincipal irreducible character X of G satisfies either X(l) -X(u) = c or X(u) = 0 . It is shown that c is an even positive integer and all such groups with a -8 are described.

Group elements and fields of character values

Journal of Group Theory, 2009

Let F be a subfield of the complex numbers. An element x of a finite group G is called an F-element in G if wðxÞ A F for every character w of G. We show that G has a unique largest normal subgroup N containing no nonidentity F-elements of G. Also, the canonical homomorphism G ! G=N defines a bijection from the set of classes of F-elements of G to the set of classes of F-elements of G=N.

A note on additive characters of finite fields

arXiv (Cornell University), 2020

Let Fq be the finite field with q elements, where q is a prime power and, for each integer n ≥ 1, let Fqn be the unique n-degree extension of Fq. The Fq-orders of an element in Fqn and an additive character over Fqn have been extensively used in the proof of existence results over finite fields (e.g., the Primitive Normal Basis Theorem). In this note we provide an interesting relation between these two objects.