Non-vanishing elements and complex group algebras

TL;DR

This paper investigates the properties of rational non-vanishing elements in finite groups, establishing a lower bound on the size of their centralizers based on the group's character sum and structure.

Contribution

It proves that for rational non-vanishing elements, the centralizer order exceeds a specific bound related to the group's character sum, linking element properties to group algebra invariants.

Findings

The order of the centralizer of rational non-vanishing elements is at least the group's weight.

The paper connects element rationality and non-vanishing conditions to algebraic bounds.

Provides a new inequality relating element centralizers to the group's character sum.

Abstract

Let $G$ be a finite group, and let $\mathrm{Irr}(G)$ denote the set of irreducible complex characters of $G$. An element $x$ of $G$ is said to be vanishing, if for some $\chi$ in $\mathrm{Irr}(G)$, we have $\chi(x)=0$. Also the element $x$ is called rational if $x$ is conjugate to $x^i$ for every integer $i$ co-prime to the order of $x$. We define the weight of $G$ as $\omega(G):=(\sum_{\chi\in \mathrm{Irr}(G)}\chi(1))^2/|G|$. In this paper, we show that for every rational non-vanishing element $x\in G$, the order of $C_G(x)$ is at least $\omega(G)$.