Degrees of Faithful Irreducible Representations of Certain Metabelian Groups and a Question of Sim

TL;DR

This paper proves that for a specific class of metabelian groups over characteristic zero fields, all faithful irreducible representations have the same degree and Schur index, answering a question of H S Sim.

Contribution

It establishes the uniform degree and Schur index of faithful irreducible representations for certain metabelian groups in characteristic zero, extending previous results from positive characteristic.

Findings

All faithful irreducible representations have the same degree.

Schur index of faithful irreducible representations is 1 or 2.

Explicit description of the Wedderburn component in the group algebra.

Abstract

In this paper, we answer affirmatively a question of H S Sim on representations in characteristic $0$, for a class of metabelian groups. Moreover, we provide examples to point out that the analogous answer is no longer valid if the solvable group has derived length larger than 2. Let $F$ be a field of characteristic $0$ and $\overline{F}$ be its algebraic closure. We prove that if $G$ is a finite metabelian group containing a maximal abelian normal subgroup which is a p-group with abelian quotient, all possible faithful irreducible representations over $F$ have the same degree and that the Schur index of any faithful irreducible $\overline{F}$-representation with respect to $F$ is always $1$ or $2$. H S Sim had proven such a result for metacyclic groups when the characteristic of $F$ is positive and posed the question in characteristic $0$. Our result answers this question for the above class of metabelian groups affirmatively. We also determine explicitly the Wedderburn component corresponding to any faithful irreducible $\overline{F}$-representation in the group algebra $F[G]$.