Dimension group of a non-negative integer irreducible matrix

In this paper we are concerned with a non-negative integer and irreducible matrix $A:{\mathbb{Z}}^d\to {\mathbb{Z}}^d$. The main contribution is to prove that if the matrix satisfies certain spectral and algebraic constraints, the cone:$C=\{v\in {\mathbb{Z}}^d/\exists n⩾0\mathrm{and}{A}^{n}v⩾0\}\subset {\mathbb{Z}}^d$is defined by linear maps ${\varphi }_0\text{,}\dots \text{,}{\varphi }_{k-1}:{\mathbb{Z}}^d\to \mathbb{R}$, in the sense that v ∈ C is equivalent to, ϕ_l(v) ⩾ 0 for all l = 0, … , k − 1 (where k is the index of cyclicity of the irreducible matrix). This result allows us to characterize the dimension group generated by the matrix, it is a subgroup of ${\mathbb{R}}^k$ endowed with an order induced by the positive cone of ${\mathbb{R}}^k$.