Disconnected Reductive Groups: Classification and Representations

TL;DR

This paper classifies disconnected reductive groups over algebraically closed fields, providing explicit tabulations, algorithms, and new insights into their representation theory, including descriptions of representation rings and finiteness of the Knutson Index.

Contribution

It offers a comprehensive classification of disconnected reductive groups and introduces new results on their representation rings and the finiteness of the Knutson Index.

Findings

Explicit classification and tabulation of disconnected reductive groups.

Two descriptions of their representation rings.

Proof that the Knutson Index is finite for these groups.

Abstract

In this article, we classify disconnected reductive groups over an algebraically closed field with a few caveats. Internal parts of our result are both a classification of finite groups and a classification of integral representations of a fixed finite group. Modulo these classifications - which are impossible in different senses - our main result explicitly tabulates the groups with an efficient algorithm. Besides this, we obtain new results about the representation theory of disconnected reductive groups in characteristic zero. We give two descriptions of their representation rings and prove that their Knutson Index is finite.