Path positivity and infinite Coxeter groups

Abstract

A Coxeter graph is a connected graph each of whose edges is labeled with an integer ≥ 3 or with ∞. The adjacency matrix of a Coxeter graph G, denoted by A(G) = (a_ij), is defined to be a square matrix of order |V|, where a_ij = 2 cos(π/p) if the edge (i, j) is labeled with the integer p, and 0 if there is no edge joining vertex i with vertex j. For any positive integer k, we denote by P_k the characteristic polynomial of the adjacency matrix of the path on k vertices. A Coxeter graph G is said to be path-positive if for all positive integers k the matrix P_k(A(G)) is entrywise nonnegative. It is shown that with the exception of a few cases, which are A, B, D, E, F, H, and I, any Coxeter graph is path-positive. The result can be interpreted as a new criterion for the infiniteness of a Coxeter group.