Irreducible Characters of Finite Algebra Groups

TL;DR

This paper provides a parametrization of irreducible complex characters of finite algebra groups, showing they are induced from linear characters of algebra subgroups, using G-orbits on the dual space of the Jacobson radical.

Contribution

It introduces a new parametrization of irreducible characters of algebra groups via G-orbits and demonstrates their induction from algebra subgroups.

Findings

Irreducible characters are parametrized by G-orbits on the dual space of J

Every irreducible character is induced from a linear character of an algebra subgroup

Provides a structural understanding of characters in algebra groups

Abstract

Let F be a finite field with q elements, let A be a finite dimensional F-algebra and let J=J(A) be the Jacobson radical of A. Then G=1+J is a p-group, where p is the characteristic of F. We refer to G as an F-algebra group. A subgroup H of G is said to be an algebra subgroup of G if H=1+U for some multiplicatively closed F-subspace of J. In this paper, we parametrize the irreducible complex characters of G in terms of G-orbits on the dual space of J. Moreover, we prove that every irreducible complex character of G is induced from a linear character of some algebra subgroup of G.