Root Cycles in Coxeter Groups

Abstract

For an element $w$ of a Coxeter group $W$ there are two important attributes, namely its length, and its expression as a product of disjoint cycles in its action on $\Phi$, the root system of $W$. This paper investigates the interaction between these two features of $w$, introducing the notion of the crossing number of $w$, $\kappa(w)$. Writing $w = c_1 \cdots c_r$ as a product of disjoint cycles we associate to each cycle $c_i$ a `crossing number' $\kappa(c_i)$, which is the number of positive roots $\alpha$ in $c_i$ for which $w\cdot \alpha$ is negative. Let Seq$_k(w)$ be the sequence of $\kappa(c_i)$ written in increasing order, and let $\kappa(w)$ = max Seq$_k(w)$. The length of $w$ can be retrieved from this sequence, but Seq$_k(w)$ provides much more information. For a conjugacy class $X$ of $W$ let $k_{\min}(X)=\min \{\kappa(w) \;|\;w \in X\}$ and let $\kappa(W)$ be the maximum value of $k_{\min}$ across all conjugacy classes of $W$. We call $\kappa(w)$ and $\kappa(W)$, respectively, the crossing numbers of $w$ and $W$. Here we determine the crossing numbers of all finite Coxeter groups and of all universal Coxeter groups. We also show, among other things, that for finite irreducible Coxeter groups if $u$ and $v$ are two elements of minimal length in the same conjugacy class $X$, then Seq$_k(u)$ = Seq$_k(v)$ and $k_{\min}(X)=\kappa(u)=\kappa(v)$. Also it is shown that the crossing number of an arbitrary Coxeter group is bounded below by the crossing number of a standard parabolic subgroup. Finally, examples are given to show that crossing numbers can be arbitrarily large for finite and infinite irreducible Coxeter groups.