Arithmetic properties of character degrees and the generalised knutson index

TL;DR

This paper introduces a new mathematical index and calculates it for specific groups using properties of their character degrees.

Contribution

The paper introduces the generalised Knutson Index and provides new classifications for alternating and symmetric groups.

Findings

The generalised Knutson Index is computed for special linear and projective special linear groups of degree two.

Alternating and symmetric groups with specific character degree properties are classified.

A lower bound for the generalised Knutson Index of these groups is established.

Abstract

In this paper, we introduce the generalised Knutson Index and compute it for the special linear groups and projective special linear groups of degree two by computing the lowest common multiple of the degrees of their irreducible representations. We also classify all alternating and symmetric groups such that the lowest common multiple of the degrees of their irreducible representations equals the order groups, which yields a lower bound on the generalised Knutson Indices of these groups.