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The two nontrivial characters of $C_{3}$ form one rational representation

Example

For $C_{3} = ⟨ g ∣ g^{3} = 1 ⟩$ and $ζ = e^{2 π i / 3}$ , the characters $χ (g) = ζ$ and $\overline{χ} (g) = ζ^{2}$ are Galois conjugate. Their sum is afforded over $Q$ by the action of $g$ on $Q^{2}$ with matrix $(\begin{array}{cc} 0 & - 1 \\ 1 & - 1 \end{array}) .$ Thus the rational Galois orbit occurs with multiplicity one; in particular the Schur index of either character over its character field $Q (ζ)$ is one.

Facts & Assumptions

Given: $C_{3}$ , $g$ , and $ζ$ as displayed.

[L1]

Over a nonsplitting field, an irreducible character orbit occurs with its Schur-index multiplicity (Character formula over a nonsplitting field).

Verification

technique · computation

1.1

The displayed matrix has characteristic polynomial $x^{2} + x + 1$ , hence eigenvalues $ζ, ζ^{2}$ , and its cube is the identity. It therefore gives a rational $C_{3}$ -representation with complex character $χ + \overline{χ}$ .

L1algebra

2.1

The orbit appears once, so [L1] identifies its scalar-extension multiplicity over $Q$ as $1$ . Since the one-dimensional character itself is defined over $Q (ζ)$ , its Schur index over its character field is also $1$ .

L1step 1.1algebra∎

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Sources

Gabor Wiese, Galois Representations, Section 2.5 (standard reference, not scraped)

The two nontrivial characters of C3C3​ form one rational representation