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The second orthogonality relation for irreducible complex characters

Statement

Let $G$ be a finite group with irreducible complex characters $χ_{1}, \dots, χ_{r}$ . For $g, h \in G$ ,

$\sum_{i = 1}^{r} χ_{i} (g) \overline{χ_{i} (h)} = {\begin{cases} ∣ C_{G} (g) ∣, & h \in {Cl}_{G} (g), \\ 0, & h \notin {Cl}_{G} (g) . \end{cases}$

Facts & Assumptions

Given: A finite group $G$ , its irreducible characters $χ_{1}, \dots, χ_{r}$ , and elements $g, h \in G$ .

[F1]

The irreducible characters form an orthonormal basis of the class functions $c f (G)$ (The irreducible complex characters form an orthonormal basis of $c f (G)$ ).

[F2]

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

[A1]

The indicator function $δ_{g}$ of the conjugacy class of $g$ , defined by $δ_{g} (x) = 1$ for $x \in {Cl}_{G} (g)$ and $δ_{g} (x) = 0$ otherwise, is a class function, and its class has $∣ {Cl}_{G} (g) ∣ = ∣ G ∣ / ∣ C_{G} (g) ∣$ elements.

[A2]

If $f = \sum_{i} c_{i} χ_{i}$ in the orthonormal basis of [F1], then $c_{i} = ⟨ f, χ_{i} ⟩$ .

Proof

technique · direct

1.1

The function $δ_{g}$ of [A1] is a class function, so by [F1] it expands as $δ_{g} = \sum_{i} c_{i} χ_{i}$ with $c_{i} = ⟨ δ_{g}, χ_{i} ⟩$ by [A2].

F1A1A2given

1.2

By [F2], $⟨ δ_{g}, χ_{i} ⟩ = \frac{1}{∣ G ∣} \sum_{x} δ_{g} (x) \overline{χ_{i} (x)}$ . By [A1] the only nonzero terms are at $x \in {Cl}_{G} (g)$ , where $δ_{g} (x) = 1$ and $\overline{χ_{i} (x)} = \overline{χ_{i} (g)}$ because characters are class functions. Hence $c_{i} = \frac{∣ G ∣ / ∣ C_{G} (g) ∣}{∣ G ∣} \overline{χ_{i} (g)} = \overline{χ_{i} (g)} / ∣ C_{G} (g) ∣$ .

F2A1givenalgebra

2.1

Evaluating the expansion of step 1.1 at $h$ and substituting the coefficients of step 1.2 gives $δ_{g} (h) = \frac{1}{∣ C_{G} (g) ∣} \sum_{i} \overline{χ_{i} (g)} χ_{i} (h)$ .

step 1.1step 1.2algebra

3.1

By [A1], $δ_{g} (h) = 1$ exactly when $h \in {Cl}_{G} (g)$ , and $0$ otherwise. Multiplying the identity of step 2.1 by $∣ C_{G} (g) ∣$ and taking complex conjugates gives $\sum_{i} χ_{i} (g) \overline{χ_{i} (h)} = ∣ C_{G} (g) ∣ δ_{g} (h)$ , which is the stated formula.

A1step 2.1algebra∎

Depends on

Used by

For g∈ G, the sum of |χᵢ(g)|² over the irreducible complex characters is |C_G(g)| Corollary
The character table is square and invertible Corollary
The character table of a finite cyclic group over ℂ Example
The character table of A₄ Example
The character table of Dih(C₄) Example
The character table of Q₈ Example
The character table of S₃ Example
The character table of S₄ and the normal subgroups it reveals Example

Dependency tree · two levels

8 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, Corollary 3.4.4 (standard reference, not scraped)
Pavel Etingof et al., Introduction to Representation Theory, Theorem 3.9 (standard reference, not scraped)