nLab characters are cyclotomic integers

Statement

Proposition

Let $G$ be a finite group, $k$ a field, and $V \in G Rep_k$ a finite-dimensional $k$-linear representation of $G$.

Then the character $\chi_{V} \colon G \to k$ of $V$ takes values inside the cyclotomic integers for some root of unity.

(e.g. Naik)

It follows that

1. characters take values in algebraic integers;

2. if the ground field $k$ has characteristic zero, then a character with values in the rational numbers in fact already takes values in the integers;

3. in particular if the ground field $k =\mathbb{Q}$ is the rational numbers, then all characters take values in the actual integers.

   (e.g. Yang, lemma 2)

Examples

For cyclic groups

The following example ovbserves that for cyclic groups the general fact that characters are cyclotomic integers reduces to the trigonometric statement that if the cosine of a rational angle is itself rational, then it is in fact integer:

Example

(rational characters for cyclic groups)

For $n \in \mathbb{N}$, $n \geq 2$, consider the cyclic group $G = \mathbb{Z}/n$. Its irreducible linear representations over the real numbers are, up to isomorphism,

1. the 1-dimensional trivial representation $\mathbf{1}$;

2. the 1-dimensional sign representation $\mathbf{1}_{sgn}$;

3. the 2-dimensional rotation-representation $\mathbf{2}_k$ for $0 \lt k \lt n/2$, generated by the matrix

   $$\rho_{\mathbf{2}_k}(1) \;=\; \left( \array{ cos(\theta) & -sin(\theta) \\ sin(\theta) & \phantom{-}cos(\theta) } \right) \phantom{AA} \text{with} \; \theta \coloneqq 2 \pi \tfrac{k}{n} \,,$$

(by this Example).

Now the characters $\chi_{\mathbf{1}}$ and $\chi_{\mathbf{1}_{sgn}}$ clearly take values in $\{\pm 1\} \subset \mathbb{Z} \subset \mathbb{Q} \subset \mathbb{R}$.

But the characters of the 2-dimensional irreps $\mathbf{2}_{k}$ are given by traces of $2 \times 2$ rotation matrices by rational angles:

$$\chi_{\mathbf{2}_k} \;\colon\; \underset{ \in \mathbb{Z}/n }{ \underbrace{ [q] } } \;\mapsto\; \mathrm{tr} \left( \left( \array{ cos(\theta) & -sin(\theta) \\ sin(\theta) & \phantom{-}cos(\theta) } \right)^q \right) \;=\; 2 \, \cos \left( 2 \pi \tfrac{q k}{n} \right)$$

hence are two times the cosine of a rational angle.

This means that for cyclic groups the statement of Prop. says equivalently that if the cosine of a rational angle is a rational number, then it is in fact a half-integer

(1)$$cos \big( 2 \pi \tfrac{k}{n} \big) \;\in\; \mathbb{Q} \phantom{AA} \Leftrightarrow \phantom{AA} cos \big( 2 \pi \tfrac{k}{n} \big) \;\in\; \tfrac{1}{2}\mathbb{Z} \phantom{AA} \Leftrightarrow \phantom{AA} cos \big( 2 \pi \tfrac{k}{n} \big) \;\in\; \big\{ -1, -1/2, 0, +1/2, +1 \big\}$$

(remember that $k,n \in \mathbb{Z}$)

A direct proof of this fact, using identities of trigonometric functions, is given in Jahnel, Sec. 3.

References

• V. Naik, Characters are cyclotomic integers

• Kay Yang, Rational valued characters, (pdf, pdf)

• Jörg Jahnel, When is the (co)sine of a rational angle equal to a rational number? (pdf, pdf)