# Indecomposable integral representations of a group of order 2 "by hand"

This question is a duplicate of that 2010 MO question.

I am interested in classifying isomorphism classes of $$n$$-dimensional integral representations of the cyclic group $$C_2$$ of order $$2$$. Clearly, any integral representation of $$C_2$$ is a direct sum of indecomposable integral representations.

The following result is well-known:

Theorem. The group $$C_2$$ has exactly 3 isomorphism classes of indecomposable integral representations:

(1) trivial;

(2) the sign representation;

(3) the 2-dimensional representation with matrix $$\left(\begin{smallmatrix}0 & 1\\ 1 & 0\end{smallmatrix}\right).$$

In his comment Victor Protsak gives a reference. He writes: "Curtis and Reiner, Chapter 11. It's a special case of a theorem in Section 74 which classifies integral representations of cyclic groups of prime order. Naturally, this case is much easier and can be done by hand."

Question. How to prove the above theorem "by hand", without reference to the book by Curtis and Reiner?

Motivation: I am working now with algebraic $$\mathbb R$$-tori. They are classified by integral representations of the Galois group $${\rm Gal}({\mathbb C}/{\mathbb R})$$, which is a group of order $$2$$. In order to understand the well-known classification of indecomposable $$\mathbb R$$-tori, I need to understand the well-known classification of indecomposable integral representations of $${\rm Gal}({\mathbb C}/{\mathbb R})$$.

• Casselman has a nice write-up of this classification of indecomposable tori … somewhere, but I can't find it right now. Aug 15, 2020 at 14:01
• Ah, here we go. Theorem 2, p. 3, of Casselman - Computing with real tori. Let me know if this reference suffices as an answer; I leave it as a comment for now in case you're hoping for a more MO-answer-sized proof. Aug 15, 2020 at 14:07
• Excellent, thank you! Let us wait... Aug 15, 2020 at 14:57
• @LSpice: If you post this reference to Bill Casselman as an answer, I will be happy to accept it. Many thanks! Aug 16, 2020 at 13:43

In Computing with real tori, Casselman has a nice write-up of this theorem from the point of view of not just proving that these are the only indecomposable tori, but, supposing you are given an explicit integral representation of $$\operatorname C_2$$, explicitly finding/computing its decomposition into these three representations.