Path positivity and infinite Coxeter groups

Bapat, R. B. ; Lal, A. K. (1994) Path positivity and infinite Coxeter groups Linear Algebra and its Applications, 196 . pp. 19-35. ISSN 0024-3795

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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) = (aij), is defined to be a square matrix of order |V|, where aij = 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 Pk 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 Pk(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.

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