G-complete reducibility in non-connected groups

TL;DR

This paper introduces an algorithm to determine G-complete reducibility of subgroups within non-connected reductive groups by reducing the problem to the connected case through a series of operations.

Contribution

It provides a novel algorithm that reduces the G-complete reducibility problem in non-connected groups to the connected case, enhancing computational methods in group theory.

Findings

Algorithm effectively reduces the problem to connected groups

Enables systematic checking of G-complete reducibility

Improves understanding of subgroup structure in non-connected groups

Abstract

In this paper we present an algorithm for determining whether a subgroup H of a non-connected reductive group G is G-completely reducible. The algorithm consists of a series of reductions; at each step, we perform operations involving connected groups, such as checking whether a certain subgroup of G^0 is G^0 -cr. This essentially reduces the problem of determining G-complete reducibility to the connected case.