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The class-function inner product $⟨ χ_{V}, χ_{W} ⟩$ equals $\dim {Hom}_{G} (W, V)$

Statement

Let $G$ be a finite group and let $V$ and $W$ be finite-dimensional complex representations of $G$ with characters $χ_{V}$ and $χ_{W}$ . Then

$⟨ χ_{V}, χ_{W} ⟩ = \dim {Hom}_{G} (W, V) .$

Facts & Assumptions

Given: Finite-dimensional complex representations $V$ and $W$ of a finite group $G$ , with characters $χ_{V}$ and $χ_{W}$ .

[F1]

The inner product of class functions is $⟨ φ, ψ ⟩ = \frac{1}{∣ G ∣} \sum_{g \in G} φ (g) \overline{ψ (g)}$ (The standard inner product on $c f (G)$ ).

[F2]

For a finite-dimensional complex representation $U$ , the averaging operator $P_{U} = \frac{1}{∣ G ∣} \sum_{g} ρ_{U} (g)$ has image $U^{G}$ and trace $\dim U^{G}$ (The averaging operator projects onto the fixed subspace).

[F3]

The fixed points of the diagonal representation on $W^{*} \otimes V$ are exactly the intertwiners $W \to V$ , so $(W^{*} \otimes V)^{G} ≅ {Hom}_{G} (W, V)$ (For finite-dimensional complex $V$ , the intertwiners $V \to W$ are exactly the fixed points of $V^{*} \otimes W$ ).

[F4]

Characters add on direct sums, multiply on tensor products, and conjugate on duals (Characters add on direct sums, multiply on tensor products, and conjugate on duals).

[A1]

The character of a representation is by definition the trace of its action operator, $χ_{U} (g) = tr ρ_{U} (g)$ .

Proof

technique · direct

1.1

By [F3], $\dim {Hom}_{G} (W, V) = \dim (W^{*} \otimes V)^{G}$ . By [F2] applied to $U = W^{*} \otimes V$ , this equals $tr P_{U}$ for the averaging operator $P_{U} = \frac{1}{∣ G ∣} \sum_{g} ρ_{W^{*} \otimes V} (g)$ .

F2F3given

1.2

Trace is linear, so $tr P_{U} = \frac{1}{∣ G ∣} \sum_{g} tr ρ_{W^{*} \otimes V} (g) = \frac{1}{∣ G ∣} \sum_{g} χ_{W^{*} \otimes V} (g)$ , the second equality by [A1].

A1algebragiven

1.3

By [F4], the tensor-product and dual clauses give $χ_{W^{*} \otimes V} (g) = χ_{W^{*}} (g) χ_{V} (g) = \overline{χ_{W} (g)} χ_{V} (g)$ .

F4given

2.1

Combining steps 1.1 through 1.3, $\dim {Hom}_{G} (W, V) = \frac{1}{∣ G ∣} \sum_{g} \overline{χ_{W} (g)} χ_{V} (g) = \frac{1}{∣ G ∣} \sum_{g} χ_{V} (g) \overline{χ_{W} (g)}$ , reordering the product of two complex numbers in each summand. This is $⟨ χ_{V}, χ_{W} ⟩$ by [F1].

F1step 1.1step 1.2step 1.3algebra∎

Depends on

Used by

The first orthogonality relation for irreducible complex characters Theorem

Dependency tree · two levels

17 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

Pavel Etingof et al., Introduction to Representation Theory, Theorem 3.8 (standard reference, not scraped)
Peter Webb, A Course in Finite Group Representation Theory, Section 3.2 (standard reference, not scraped)

The class-function inner product ⟨χV,χW⟩⟨χV​,χW​⟩ equals dim⁡Hom⁡G(W,V)dimHomG​(W,V)