The Excess Zero Graph of a Coxeter Group

TL;DR

This paper analyzes the excess zero graph $\mathcal{E}_0(W)$ of a Coxeter group, whose vertices are non-identity involutions and where edges encode length-additive products $\ell(xy)=\ell(x)+\ell(y)$. It establishes sharp connectivity and diameter properties for the component not containing the longest element $w_0$, showing an overall diameter bound of at most $3$ in many families and giving precise values in finite, affine, and compact hyperbolic cases, as well as infinite-rank scenarios. It then derives valency results for involutions, proving a symmetry that yields $|\Delta_1(r)|=(|\mathcal{I}(W)|-1)/2$ for fundamental reflections and introducing the recursive quantity $\delta(m,n)$ for type $A_n$, along with a classification of pendant elements (valency $1$) across finite irreducible types and explicit descriptions for $A_n$, $D_n$ (odd $n$), $E_6$, and $I_2(m)$ (odd $m$). The paper also presents a complete pendant-element picture in these families and multiplies computational data for small exceptional groups to illustrate the rich structure of $\mathcal{E}_0(W)$. Overall, the work links length-combinatorics in Coxeter groups with involution structure, offering both exact classifications and practical data that may inform the ancestor-property conjecture in prefixes.

Abstract

For a Coxeter group $W$ with length function $\ell$, the excess zero graph $\mathcal{E}_0(W)$ has vertex set the non-identity involutions of $W$, with two involutions $x$ and $y$ adjacent whenever $\ell(xy)=\ell(x)+\ell(y)$. Properties of this graph such as connectivity, diameter and valencies of certain vertices of $\mathcal{E}_0(W)$ are explored.

Figures (1)

• Figure 1: Labelling of Dynkin diagrams

Theorems & Definitions (63)

• Definition 1.1
• Theorem 1.2
• Theorem 1.3
• Corollary 1.4
• Theorem 1.5
• Theorem 1.6
• Corollary 1.7
• Lemma 2.1: Humph
• Lemma 2.2
• proof