Maximal exponents of polyhedral cones (I)

Abstract

Let K be a proper (i.e., closed, pointed, full convex) cone in R n . An n × n matrix A is said to be K-primitive if there exists a positive integer k such that A k ( K ∖ { 0 } ) ⊆ int K ; 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 ) ⩽ ( m A − 1 ) ( m − 1 ) + 1 , where m A 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.