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 Φ, 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, κ(w). Writing w=c1⋯cr as a product of disjoint cycles we associate to each cycle ci a `crossing number' κ(ci), which is the number of positive roots α in ci for which w⋅α is negative. Let Seqk(w) be the sequence of κ(ci) written in increasing order, and let κ(w) = max Seqk(w). The length of w can be retrieved from this sequence, but Seqk(w) provides much more information. For a conjugacy class X of W let kmin(X)=min{κ(w)|w∈X} and let κ(W) be the maximum value of kmin across all conjugacy classes of W. We call κ(w) and κ(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 Seqk(u) = Seqk(v) and kmin(X)=κ(u)=κ(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.