How large is the character degree sum compared to the character table sum for a finite group?

TL;DR

This paper investigates the ratio of the character table sum to the character degree sum in finite groups, establishing bounds, conjectures, and explicit formulas, with a focus on Coxeter and symmetric groups, revealing connections to involutions and roots.

Contribution

It extends bounds and conjectures about the character table sum ratio, proves these for Coxeter and generalized symmetric groups, and provides explicit generating functions for these sums.

Findings

The ratio of character table sum to degree sum is at most two for many groups.

The ratio is at least one, with equality iff the group is abelian, proved for Coxeter and generalized symmetric groups.

Asymptotics of character sums match the count of involutions in certain groups.

Abstract

In 1961, Solomon gave upper and lower bounds for the sum of all the entries in the character table of a finite group in terms of elementary properties of the group. In a different direction, we consider the ratio of the character table sum to the sum of the entries in the first column, also known as the character degree sum, in this work. First, we propose that this ratio is at most two for many natural groups. Secondly, we extend a conjecture of Fields to postulate that this ratio is at least one with equality if and only if the group is abelian. We establish the validity of this property and conjecture for all finite irreducible Coxeter groups. In addition, we prove the conjecture for generalized symmetric groups. The main tool we use is that the sum of a column in the character table of an irreducible Coxeter group (resp. generalized symmetric group) is given by the number of square roots (resp. absolute square roots) of the corresponding conjugacy class representative. As a byproduct of our results, we show that the asymptotics of character table sums is the same as the number of involutions in symmetric, hyperoctahedral and demihyperoctahedral groups. We also derive explicit generating functions for the character table sums for these latter groups as infinite products of continued fractions. In the same spirit, we prove similar generating function formulas for the number of square roots and absolute square roots in $n$ for the generalized symmetric groups $G(r,1,n)$.