On the minimal norm of a non-regular generalized character of an arbitrary finite group

TL;DR

This paper establishes a lower bound on the sum of squared absolute values of certain generalized characters of finite groups, identifying cases where this bound is tight, thus advancing understanding of character theory.

Contribution

It provides a new lower bound for sums of squared values of non-regular generalized characters in finite groups and characterizes when this bound is achieved.

Findings

Lower bound of |G|/d - 1 for sums of squared generalized characters

Identification of cases where the minimum is attained

Extension of character theory to non-regular generalized characters

Abstract

We prove that for any finite group G, the sum across non-identity elements of the squared absolute value of any generalized character of G which does not vanish on all non-identity elements of G is at least |G|/d -1, where d is the maximal degree of a complex irreducible character of G, and we identify all cases where this minimum possible value is attained.