Arithmetic Properties of Character Degrees and the Generalised Knutson Index

TL;DR

This paper introduces the generalized Knutson Index, computes it for specific groups, and classifies certain groups based on the relationship between their irreducible representation degrees and group order.

Contribution

It defines the generalized Knutson Index, calculates it for special linear and projective special linear groups, and classifies symmetric and alternating groups with specific properties.

Findings

Computed the generalized Knutson Index for SL(n) and PSL(2) groups.

Classified symmetric and alternating groups where the LCM of irreducible degrees equals group order.

Provided lower bounds on the generalized Knutson Indices for these groups.

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.