Twisted Orthogonality Relations For Certain $\mathbb{Z}/m\mathbb{Z}$-graded Algebras

TL;DR

This paper investigates Frobenius $ ext{star}$-algebras, establishing orthogonality relations for their irreducible characters, and extends these results to $ ext{Z}/m ext{Z}$-graded Frobenius $ ext{star}$-algebras with twisted orthogonality relations.

Contribution

It introduces twisted orthogonality relations for $ ext{Z}/m ext{Z}$-graded Frobenius $ ext{star}$-algebras, expanding the understanding of their character theory.

Findings

Established orthogonality relations for Frobenius $ ext{star}$-algebras.

Proved twisted orthogonality relations for $ ext{Z}/m ext{Z}$-graded Frobenius $ ext{star}$-algebras.

Extended classical character theory to graded algebra settings.

Abstract

In this paper we will study the notion of a Frobenius $\star$-algebra and prove some orthogonality relations for the irreducible characters of a Frobenius $\star$-algebra. Then we will study $\mathbb{Z}/m\mathbb{Z}$-graded Frobenius $\star$- algebras and prove some twisted orthogonality relations for them.