TITLE: Involutions in GL_n(Z)
QUESTION [12 upvotes]: Is there a classification of involutions in $\text{GL}_n(\mathbb{Z})$?
Here's some more details about what I mean.  Consider $f \in \text{GL}_n(\mathbb{Z})$ such that $f^2=1$.  Regard $f$ as an automorphism of $\mathbb{Z}^n$.  Extend $f$ to an automorphism $g$ of $\mathbb{Q}^n$.  Then we can write $\mathbb{Q}^n = E_1 \oplus E_{-1}$, where $g$ acts as the identity on $E_1$ and as $-1$ on $E_{-1}$.  Restricting this decomposition to $\mathbb{Z}^n$, we obtain a finite-index subgroup $A$ of $\mathbb{Z}^n$ and a decomposition $A = F_1 \oplus F_{-1}$ such that $f$ acts as the identity on $F_1$ and as $-1$ on $F_{-1}$.
However, we definitely cannot assume that $A = \mathbb{Z}^n$.  For instance, the matrix whose first row is $(0 1)$ and whose second row is $(1 0)$ (by the way, I can't figure out how to get my matrices to display correctly) is an involution in $\text{GL}_n(\mathbb{Z})$ that can be diagonalized over $\mathbb{Q}$ but not over $\mathbb{Z}$.
What else can be said here?

REPLY [15 votes]: The problem is equivalent to classifying isomorphism classes of $n$-dimensional integral representations of the cyclic group $C_2$ of order 2, or $\mathbb{Z}[C_2]$-modules on $\mathbb{Z}^n$. This group has exactly 3 isomorphism classes of indecomposable free $\mathbb{Z}$-modules: 
(1) trivial 
(2) sign representation 
(3) 2-dimensional with matrix $\begin{bmatrix}0 & 1\\ 1 & 0\end{bmatrix}.$
Every $n$-dimensional $\mathbb{Z}[C_2]$-module is a direct sum of (1), (2), (3) with uniquely determined multiplicities. Thus any involution is conjugate over $\mathbb{Z}$ to a block diagonal matrix with blocks [1], [-1], $\begin{bmatrix}0 & 1\\ 1 & 0\end{bmatrix}$ whose sizes are uniquely determined.