Maximal Exponents of K-primitive Matrices: the Polyhedral Cone Case

Let K be a proper (i.e., closed, pointed, full convex) cone in R. An n × n matrix A is said to be K-primitive if there exists a positive integer k such that A(K \ {0}) ⊆ intK; 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 denoted by γ(K). It is proved that for any positive integers m,n, 3 ≤ n ≤ m, the maximum value of γ(K), as K runs through all n-dimensional polyhedral cones with m extreme rays, equals (n− 1)(m− 1) + 1 when m is even or m and n are both odd, and is at least (n−1)(m−1) and at most (n−1)(m−1)+1 when m is odd and n is even. For the cases when m = n,m = n+1 or n = 3, the cones K and the corresponding K-primitive matrices A such that γ(K) and γ(A) attain the maximum value are identified up to respectively linear isomorphism and cone-equivalence modulo positive scalar multiplication. AMS classification: 15A48; 05C50; 47A06.