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 dominates y if for all w ∈ W , wy is a negative root whenever wx is a negative root. We call a positive root 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 [B. Brink, R.B. Howlett, A finiteness property and an automatic structure of Coxeter groups, Math. Ann. 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 [B. Brink, The set of dominance-minimal roots, J. Algebra 206 (1998) 371–412]. 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 behavior 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.