arXiv subjects
Publications and source records attributed to Diego Martín Duro.
In this article, we classify (up to isomorphism) disconnected reductive groups over an algebraically closed field. For the fixed dimension and number of components, our results show that there are only finitely many reductive groups of that dimension with that many components. Besides this, we obtain new results about the structure and representation theory of disconnected reductive groups.
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.
Global representation rings were discovered by Sarah Witherspoon in 1995 and the Knutson Index was introduced by the second author in 2022. In the present paper we introduce the Knutson Index for general commutative rings and study it for Burnside rings and global representation rings. We also introduce the global table of a finite group, that encompasses both the character table and the Burnside table of marks. We discuss what properties of a group can be recovered from its global table.
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.