Maximal exponents of polyhedral cones (III)

Abstract:

Let $K$ be a proper (i.e., closed, pointed, full, convex) cone in $\mathbb {R}^n$. An $n\times n$ matrix $A$ is said to be $K$-primitive if $AK\subseteq K$ and there exists a positive integer $k$ such that $A^k(K \setminus \{ 0 \}) \subseteq$ int $K$; the least such $k$ is referred to as the exponent of $A$ and is denoted by $\gamma (A)$. For a polyhedral cone $K$, the maximum value of $\gamma (A)$, taken over all $K$-primitive matrices $A$, is denoted by $\gamma (K)$. It is proved that for any positive integers $m,n, 3 \le n \le m$, the maximum value of $\gamma (K)$, as $K$ runs through all $n$-dimensional polyhedral cones with $m$ extreme rays, equals $(n-1)(m-1)+\frac {1}{2}\left (1+(-1)^{(n-1)m}\right )$. For the $3$-dimensional case, the cones $K$ and the corresponding $K$-primitive matrices $A$ such that $\gamma (K)$ and $\gamma (A)$ attain the maximum value are identified up to respectively linear isomorphism and cone-equivalence modulo positive scalar multiplication.