Dimension group of a non-negative integer irreducible matrix

In this paper we are concerned with a non-negative integer and irreducible matrix A:Zd→ZdA:Zd→Zd. The main contribution is to prove that if the matrix satisfies certain spectral and algebraic constraints, the cone:C={v∈Zd/∃n⩾0andAnv⩾0}⊂Zdis defined by linear maps ϕ0,…,ϕk-1:Zd→Rϕ0,…,ϕk-1:Zd→R, in the sense that v ∈ C is equivalent to, ϕl(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 RkRk endowed with an order induced by the positive cone of RkRk.