  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.
