The Knutson Index of the Representation Ring

TL;DR

This paper investigates the existence of virtual modules that tensor with simple modules to produce the regular module in Hopf algebras, introduces the Knutson Index to measure conjecture failure, and presents counterexamples to prior conjectures.

Contribution

It introduces the Knutson Index as a new measure of conjecture failure and provides counterexamples to Savitskii's refined conjecture.

Findings

Counterexamples to Savitskii's Conjecture

Properties of the Knutson Index

Relation to tensor products in Hopf algebras

Abstract

In this paper, we study if, for a given simple module over a Hopf algebra, there exists a virtual module such that their tensor product is the regular module. This is related to a conjecture by Donald Knutson, later disproved and refined by Savitskii, stating that for every irreducible character of a finite group, there exists a virtual character such that their tensor product is the regular character. We also introduce the Knutson Index as a measure of Knutson's Conjecture failure, discuss its algebraic properties and present counter-examples to Savitskii's Conjecture.