On the monomial algebra associated to the monomial characters of a finite group

TL;DR

This paper investigates the algebra generated by monomial characters of a finite group, exploring its properties, integral closure, and connections to Artin L-functions and supercharacter theory.

Contribution

It introduces the monomial algebra associated with a finite group and analyzes its integral closure and relation to holomorphic Artin L-functions, extending to supercharacter theory.

Findings

Integral closure of the monomial algebra is contained in an algebra generated by characters with holomorphic L-functions.

The study links algebraic properties to analytic behavior of Artin L-functions.

Discussion includes the supercharacter theoretic case.

Abstract

Given a finite group $G$, we study the monomial algebra $R_G$, generated by the monomial characters of $G$. In particular, we note that the integral closure of $R_G$ is contained in the algebra generated by those characters $\chi$ for which their associated Artin L-function $L(s,\chi)$ is holomorphic at $s_0\in\mathbb C\setminus\{1\}$. Also, we discuss the supercharacter theoretic case.