Linear representation theory of cyclic group:Z2

Related notions
For the linear representation theory over fields of characteristic 2, see modular representation theory of cyclic group:Z2.

Representations
The cyclic group of two elements is a rational group: all its representations over the complex numbers are equivalent to representations over the rational numbers. In fact, all its representations over any field other than the field of two elements, can be realized as representations over the prime subfield of that field. Moreover, any representation over a prime field (other than the prime field of two elements) is completely reducible and there are only two irreducible representations, both one-dimensional:


 * The trivial representation which sends both elements to 1.
 * The sign representation which sends the identity element to $$1$$ and the non-identity element to $$-1$$. This is a faithful representation.

Character table
The character table for characteristic zero is:

There is a canonical bijection between the conjugacy classes and the irreducible representations here (unlike for bigger, more complicated groups). The trivial representation corresponds to the identity element, and the sign representation corresponds to the non-identity element.

Group ring interpretation
For any commutative unital ring $$R$$, the group ring $$R[\mathbb{Z}_2]$$ can be identified with a quotient of the polynomial ring as follows:

$$R[\mathbb{Z}_2] = R[\{ e, x \}] \cong R[t]/(t^2 - 1)$$

where the identification is as follows:

$$r_1e + r_2x \mapsto r_1 + r_2t \pmod{t^2 - 1}$$

In particular, $$e$$ is identified with the class of 1 mod $$t^2 - 1$$, and $$x$$ (the non-identity element) is identified with the class of $$t$$ mod $$t^2 - 1$$.

Case of a uniquely 2-divisible ring
Suppose $$R$$ is a uniquely 2-divisible ring, i.e., every element has a unique half. We can then rewrite the group ring using the Chinese Remainder Theorem:

$$R[t]/(t^2 - 1) \cong R[t]/(t - 1) \times R[t]/(t + 1) \cong R \times R$$

This choice of decomposition using the Chinese Remainder Theorem can also be realized using characters, as follows. We have an isomorphism:

$$R[t]/(t^2 - 1) \cong Re_1 \oplus Re_2$$

where $$e_1, e_2$$ are idempotents corresponding to the two representations, and are described as follows:

Note that we need unique 2-divisibility because constructing the idempotents requires division by 2. In particular, the group ring over $$\mathbb{Z}$$ (the ring of integers) cannot be decomposed as above.

For fields, unique 2-divisibility is equivalent to the characteristic not being equal to 2.