Commuting Involutions of SL(n,k)

Abstract

The classification of commuting involutions of a connected, reductive algebraic group over an algebraically closed field of characteristic not two was completed by Helminck in cite{Helm88}. In this thesis, the isomorphism classes of commuting pairs of involutions of $SL(n,k)$ for arbitrary fields of characteristic not two are determined. This is done in three main parts: commuting pairs of inner involutions, commuting pairs of involutions with one inner involution and one outer involution, and commuting pairs of outer involutions. The classification is done up to (almost) inner automorphism of $SL(n,k)$ as in the case of a single involution, completed in cite{Involutions_HWD}. For commuting pairs of involutions, the classification is done in several pieces, each of which uses an isomorphism class representative of a single involution of $SL(n,k)$ as the first entry. First, it is shown that if a mapping of a certain form is invertible, it will partially zero-out the matrix representative of one of the entries in the pair of commuting involutions while fixing the other pair, and will be an inner automorphism of $SL(n,k)$. Then, it is shown that, indeed, the given mapping must be invertible for one of a few given forms. This leads to an induction argument that gives a partial classification of the commuting inner pairs. To finish the classification, whether or not the remaining pairs are isomorphic is determined. Next, using results on Quadratic forms, a 'nice' form for commuting pairs with one inner and one outer involution is found. To finish the classification of such pairs, work is completed on a field (of characteristic not two) by field basis and results are explicitly obtained for algebraically closed fields, the real numbers, finite fields, and the $p$-adic numbers. Lastly, equivalence between the classification of commuting outer pairs and commuting pairs with one inner involution and one outer involution is shown.