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.

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.

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