Affine Coxeter groups, involution classes and commuting involution graphs

Sbeiti Clarke, Amal (2018) Affine Coxeter groups, involution classes and commuting involution graphs. Doctoral thesis, Birkbeck, University of London.

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Abstract

For a group G and X a subset of G the commuting graph of G on X, denoted C(G,X), is the graph whose vertex set is X where there is an edge joining x,y Є X whenever x commutes with y and x≠y. If the elements of X are involutions, then C(G,X) is called a commuting involution graph. In this thesis, we investigate conjugacy classes of involutions, studying the connectedness of the commuting involution graph and determining the size of the diameter of the connected C(G,X), where X is a conjugacy class of involutions of G and G is an affine Coxeter group. We show that if G is of type ~ Cn, ~Bn or ~Dn and C(G,X) is connected, then Diam C(G;X) is at most n+2. If G is of type ~G2, then C(G,X) is disconnected. If G is of type ~ F4, then C(G,X) is connected when X is a conjugacy class of (r2r3)2 or r3r5. Otherwise, it is disconnected. Finally, we examine the connectedness of C(W,X) where W is an arbitrary Coxeter group, R is its set of simple reflections, and X = RW. In this case we call C(W,X) the commuting reflection graph of W.

Item Type: Thesis (Doctoral)
Copyright Holders: The copyright of this thesis rests with the author, who asserts his/her right to be known as such according to the Copyright Designs and Patents Act 1988. No dealing with the thesis contrary to the copyright or moral rights of the author is permitted.
School/Department: School of Business, Economics & Informatics > Economics, Mathematics & Statistics
Depositing User: ORBIT Editor
Date Deposited: 21 Dec 2018 15:01
Last Modified: 21 Dec 2018 15:59
URI: http://bbktheses.da.ulcc.ac.uk/id/eprint/364

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