# nLab characters are cyclotomic integers

Contents

### Context

#### Representation theory

representation theory

geometric representation theory

# Contents

## 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.

• 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)