Primitive decompositions of idempotents of the group algebras of dihedral groups and generalized quaternion groups

TL;DR

This paper presents a method for computing primitive decompositions of idempotents in semisimple finite group algebras, with applications to dihedral and quaternion groups, revealing new trigonometric identities and an isomorphism between specific group algebras.

Contribution

The paper introduces a general method for primitive decomposition of idempotents in semisimple group algebras and applies it to dihedral and quaternion groups, discovering new identities and an algebra isomorphism.

Findings

Derived explicit primitive idempotents for dihedral and quaternion group algebras.

Established new trigonometric identities from character orthogonality relations.

Described an isomorphism between the group algebras of D8 and Q8.

Abstract

In this paper, we introduce a method computing the primitive decomposition of idempotents of any semisimple finite group algebra based on its matrix representations and Wedderburn decomposition. Particularly, we use this method to calculate the examples of the dihedral group algebras $\mathbb{C}[D_{2n}]$ and generalized quaternion group algebras $\mathbb{C}[Q_{4m}]$. Inspired by the orthogonality relations of the character tables of these two families of groups, we obtain two sets of trigonometric identities. Furthermore, a group algebra isomorphism between $\mathbb{C}[D_{8}]$ and $\mathbb{C}[Q_{8}]$ is described, under which the two complete sets of primitive orthogonal idempotents of these two group algebras we find correspond to each other bijectively.