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The dominance hierarchy in root systems of Coxeter groups

If $x$ and $y$ are roots in the root system with respect to the standard (Tits) geometric realization of a Coxeter group $W$, we say that $x$ \emph{dominates} $y$ if for all $w\in W$, $wy$ is a negative root whenever $wx$ is a negative root. We call a positive root \emph{elementary} if it does not dominate any positive root other than itself. The set of all elementary roots is denoted by $\E$. It has been proved by B. Brink and R. B. Howlett (Math. Ann. \textbf{296} (1993), 179--190) that $\E$ is finite if (and only if) $W$ is a finite-rank Coxeter group. Amongst other things, this finiteness property enabled Brink and Howlett to establish the automaticity of all finite-rank Coxeter groups. Later Brink has also given a complete description of the set $\E$ for arbitrary finite-rank Coxeter groups (J. Algebra \textbf{206} (1998)). However the set of non-elementary positive roots has received little attention in the literature. In this paper we answer a collection of questions concerning the dominance behaviour between such non-elementary positive roots. In particular, we show that for any finite-rank Coxeter group and for any non-negative integer $n$, the set of roots each dominating precisely $n$ other positive roots is finite. We give upper and lower bounds for the sizes of all such sets as well as an inductive algorithm for their computation.