Characters of 2 x 2 Unitary Matrix Groups Over Quadratic Ring Extensions

Abstract

In the Journal of Algebra 323(2010) R. Barrington Leigh et al. derive the characters of the group of invertible 2 x 2 matrices over the integers modulo a power of an odd prime. We will generalize to certain local rings, and take quadratic extensions of this ring , by adjoining the root of a unit, and the root of a nilpotent element. Then we form the group of unitary 2 x 2 matrices over this ring extension. Using Clifford theory we will find the degrees and numbers of irreducible characters of these unitary groups. The overall argument is inductive, in that the local ring is indexed by some positive integral value; we assume that for all positive integral values less than the index, the requisite information is known. For the base case where the index is 1, the results were given by V. Ennola for the case of adjoining the root of a unit, and are derived in this work for the case of adjoining the root of a nilpotent element.