Commuting Involutions of SL(n,k)

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dc.contributor.advisor Amassa Fauntleroy, Committee Member en_US
dc.contributor.advisor Naihuan Jing, Committee Member en_US
dc.contributor.advisor Aloysius Helminck, Committee Chair en_US
dc.contributor.advisor Ernest Stitzinger, Committee Member en_US Thompson, Kyle Andrew en_US 2010-04-02T19:06:11Z 2010-04-02T19:06:11Z 2010-03-15 en_US
dc.identifier.other etd-03042010-150145 en_US
dc.description.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. en_US
dc.rights I hereby certify that, if appropriate, I have obtained and attached hereto a written permission statement from the owner(s) of each third party copyrighted matter to be included in my thesis, dis sertation, or project report, allowing distribution as specified below. I certify that the version I submitted is the same as that approved by my advisory committee. I hereby grant to NC State University or its agents the non-exclusive license to archive and make accessible, under the conditions specified below, my thesis, dissertation, or project report in whole or in part in all forms of media, now or hereafter known. I retain all other ownership rights to the copyright of the thesis, dissertation or project report. I also retain the right to use in future works (such as articles or books) all or part of this thesis, dissertation, or project report. en_US
dc.subject involutions en_US
dc.subject algebraic groups en_US
dc.subject symmetric spaces en_US
dc.title Commuting Involutions of SL(n,k) en_US PhD en_US dissertation en_US Mathematics en_US

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