# Upper and lower bounds on Chillag table sums

## TL;DR

This paper extends Chillag's framework to $\pi$-partial characters and establishes bounds on table sums under certain conditions, with applications to Brauer, $\pi$-partial, and projective indecomposable characters.

## Contribution

It generalizes Chillag's construction to $\pi$-partial characters and derives bounds on table sums assuming an extra condition, applicable to several character types.

## Key findings

- Bounds on Chillag table sums are established under specific conditions.
- The extra condition holds for Brauer, $\pi$-partial, and projective indecomposable characters.
- Results include bounds on sums of rows and columns in character tables.

## Abstract

Chillag has showed that there is a single generalization showing that the sums of ordinary character tables, Brauer character, and projective indecomposable characters are positive integers. We show that Chillag's construction also applies to Isaacs' $\pi$-partial characters. We show that if an extra condition is assumed, then we can obtain upper and lower bounds on the Chillag's table sums. We will demonstrate that this condition holds for Brauer characters, $\pi$-partial characters, and projective indecomposable characters, and so we obtain upper and lower bounds for the table sums in those cases. We also obtain results regarding the sums of rows and columns in these tables.