Maximal exponents of polyhedral cones (I)

Let K be a proper (i.e., closed, pointed, full convex) cone in Rn. An n×n matrix A is said to be K-primitive if there exists a positive integer k such that ; the least such k is referred to as the exponent of A and is denoted by γ(A). For a polyhedral cone K, the maximum value of γ(A), taken over all K-primitive matrices A, is called the exponent of K and is denoted by γ(K). It is proved that if K is an n-dimensional polyhedral cone with m extreme rays then for any K-primitive matrix A, γ(A)⩽(mA−1)(m−1)+1, where mA denotes the degree of the minimal polynomial of A, and the equality holds only if the digraph (E,P(A,K)) associated with A (as a cone-preserving map) is equal to the unique (up to isomorphism) usual digraph associated with an m×m primitive matrix whose exponent attains Wielandt's classical sharp bound. As a consequence, for any n-dimensional polyhedral cone K with m extreme rays, γ(K)⩽(n−1)(m−1)+1. Our work answers in the affirmative a conjecture posed by Steve Kirkland about an upper bound of γ(K) for a polyhedral cone K with a given number of extreme rays.