On idempotents and the number of simple components of semisimple group algebra

TL;DR

This paper characterizes the primitive central idempotents and Wedderburn decomposition of group algebras over number fields for finite monomial and strongly monomial groups, linking algebraic structure to number-theoretic properties.

Contribution

It provides explicit descriptions of idempotents, decompositions, and conditions for the number of simple components, along with a formula for the rank of central units in these group rings.

Findings

Primitive central idempotents described for monomial groups

Wedderburn decomposition characterized for strongly monomial groups

Formula derived for the rank of central units

Abstract

We describe the primitive central idempotents of the group algebra over a number field of finite monomial groups. We give also a description of the Wedderburn decomposition of the group algebra over a number field for finite strongly monomial groups. Further, for this class of group algebras, we describe when the number of simple components agrees with the number of simple components of the rational group algebra. Finally, we give a formula for the rank of the central units of the group ring over the ring of integers of a number field for a strongly monomial group.