TheoremStatement: Literature-sourcedProof: AI-adaptedprecheck passjudge pass (gpt-5.6-terra)audited 2026-08-29

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The character of a permutation representation counts fixed points

Statement

Let $X$ be a finite left $G$ -set and let $C^{(X)}$ be the permutation representation over $C$ . Then the character of $C^{(X)}$ is

$χ_{C^{(X)}} (g) = ∣ {x \in X : g \cdot x = x} ∣,$

the number of fixed points of $g$ acting on $X$ .

Facts & Assumptions

Given: A finite group $G$ , a finite left $G$ -set $X$ , and $g \in G$ .

[F1]

The character is $χ (g) = tr ρ (g)$ (The character $χ_{V} (g) = tr (ρ_{V} (g))$ of a finite-dimensional complex representation).

[F2]

The permutation representation has basis $(e_{x})_{x \in X}$ and action $g \cdot e_{x} = e_{g \cdot x}$ (The trivial representation, the regular representation, and permutation representations from finite $G$ -sets).

Proof

technique · direct

1.1F2given

In the basis $(e_{x})_{x \in X}$ of [F2], the matrix of $ρ (g)$ is the permutation matrix with a $1$ in row $g \cdot x$ , column $x$ : its $x$ -th diagonal entry is $1$ exactly when $g \cdot x = x$ , and $0$ otherwise.

2.1F1step 1.1algebra∎

The trace of that matrix is the sum of its diagonal entries, counting one for each $x \in X$ with $g \cdot x = x$ . By [F1] this is $χ (g)$ , which equals the stated number of fixed points.

Depends on

Used by

Dependency tree · two levels

9 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Peter Webb, A Course in Finite Group Representation Theory, Example 4.3.4 (standard reference, not scraped)