Linear representation theory of group of integers

This article discusses the linear representation theory of the group of integers.

Essentially, the linear representation theory of the group of integers boils down to the problem of classifying similarity types of matrices. Explicitly, the image of the generator $$1 \in \mathbb{Z}$$ can be viewed as the matrix of interest. Two representations of $$\mathbb{Z}$$, both of degree $$n$$, over a field $$K$$ are equivalent as linear representations if the images of 1 under the two representations are conjugate in $$GL(n,K)$$.

The similarity type of a matrix can be determined by converting it to rational canonical form. This also relates to the structure theorem for finitely generated modules over PIDs. The PID in question is $$K[t]$$, the polynomial ring of one variable in $$K$$.