Knutson's Conjecture on the Representation Ring

Donald Knutson’s Conjecture, later refined by Savitskii, stated that for every irreducible character of a finite group, there exists a virtual character such that their tensor product results in the regular character. The original conjecture was disproven by Savitskii, and we revisit the arguments and counterexamples provided. We then disprove the refinement proposed by Savitskii and also introduce the Knutson Index as a measure of failure of the Knutson’s Conjecture, examine its algebraic properties, and present new counterexamples that highlight the limitations of Knutson Conjecture.