On local fusion graphs of finite Coxeter groups

Abstract

Abstract. Given a finite group G and G-conjugacy class of involutions X, the local fusion graph F .G; X/ has X as its vertex set, with x; y 2 X joined by an edge if, and only if, x ¤ y and the product xy has odd order. In this note we investigate such graphs when G is a finite Coxeter group, addressing questions of connectedness and diameter. In particular, our results show that local fusion graphs may have an arbitrary number of connected components, each with arbitrarily large diameter.

Local fusion graphs for symmetric groups

“…This all prompts the question as to whether there are groups in which the diameter of local fusion graphs can be arbitrarily large -the answer is yes, and we direct Brought to you by | MIT Libraries Authenticated Download Date | 5/10/18 5:25 PM the reader to [4]. For further work on coprimality graphs and symmetric groups see [5], and for more recent developments on local fusion graphs see [3] and [6].…”

“…For further work on coprimality graphs and symmetric groups see [5], and for more recent developments on local fusion graphs see [3] and [6].…”

On $$\pi $$ π -Product Involution Graphs in Symmetric Groups

“…In the case when X is a G-conjugacy class of involutions, we note that P {2} (G, X ) is just a commuting involution graph. Taking π to be the set of all odd natural numbers and X a G-conjugacy class, P π (G, X ) becomes the local fusion graph F(G, X ) which has featured in [5,6].…”

Local Fusion Graphs and Sporadic Simple Groups

“…It is clear that G induces graph automorphisms (by conjugation) on F(G, X) and acts transitively on the vertices. Various properties of local fusion graphs have been investigated in [1] and [2]. In [2] local fusion graphs for finite symmetric groups are studied, the main result being that they always have diameter two, provided that the degree is at least five.…”

“…In [2] local fusion graphs for finite symmetric groups are studied, the main result being that they always have diameter two, provided that the degree is at least five. The other finite irreducible Coxeter groups are dealt with in [1], which also considers the possible diameters. There, examples are given of groups which have local fusion graphs whose diameter can be arbitrarily large.…”