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characters are cyclotomic integers

Contents

Contents

Statement

Proposition

Let GG be a finite group, kk a field, and VGRep kV \in G Rep_k a finite-dimensional kk-linear representation of GG.

Then the character χ V:Gk\chi_{V} \colon G \to k of VV 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 kk 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=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 nn \in \mathbb{N}, n2n \geq 2, consider the cyclic group G=/nG = \mathbb{Z}/n. Its irreducible linear representations over the real numbers are, up to isomorphism,

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

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

  3. the 2-dimensional rotation-representation 2 k\mathbf{2}_k for 0<k<n/20 \lt k \lt n/2, generated by the matrix

    ρ 2 k(1)=(cos(θ) sin(θ) sin(θ) cos(θ))AAwithθ2πkn, \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 χ 1\chi_{\mathbf{1}} and χ 1 sgn\chi_{\mathbf{1}_{sgn}} clearly take values in {±1}\{\pm 1\} \subset \mathbb{Z} \subset \mathbb{Q} \subset \mathbb{R}.

But the characters of the 2-dimensional irreps 2 k\mathbf{2}_{k} are given by traces of 2×22 \times 2 rotation matrices by rational angles:

χ 2 k:[q]/ntr((cos(θ) sin(θ) sin(θ) cos(θ)) q)=2cos(2πqkn) \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(2πkn)AAAAcos(2πkn)12AAAAcos(2πkn){1,1/2,0,+1/2,+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,nk,n \in \mathbb{Z})

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

References

Last revised on May 9, 2019 at 12:28:38. See the history of this page for a list of all contributions to it.