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The regular character gives a second proof of the sum-of-squares formula

Statement

Let $G$ be a finite group and let $χ_{1}, \dots, χ_{r}$ be its irreducible complex characters, of degrees $n_{i} = χ_{i} (1)$ . Then

$\sum_{i = 1}^{r} n_{i}^{2} = ∣ G ∣ .$

Facts & Assumptions

Given: A finite group $G$ with irreducible characters $χ_{1}, \dots, χ_{r}$ , the regular representation $C [G]$ , and the regular character $χ_{r e g}$ .

[F1]

The regular character is $∣ G ∣$ at $1$ and $0$ elsewhere (The regular character is $∣ G ∣$ at $1$ and $0$ away from $1$ ).

[F2]

The multiplicity of an irreducible summand is an inner product: $m_{i} = ⟨ χ_{r e g}, χ_{i} ⟩$ (The multiplicity of an irreducible summand is a character inner product).

[F3]

Characters add on direct sums: $χ_{V \oplus W} = χ_{V} + χ_{W}$ (Characters add on direct sums, multiply on tensor products, and conjugate on duals).

[A1]

The degree of a character is its value at $1$ , $χ_{i} (1) = n_{i}$ .

Proof

technique · direct

1.1

By [F2], the regular representation decomposes as $C [G] ≅ ⨁_{i} m_{i} V_{i}$ with $m_{i} = ⟨ χ_{r e g}, χ_{i} ⟩$ . By [F1] and the definition of the inner product this is $\frac{1}{∣ G ∣} \cdot ∣ G ∣ \cdot χ_{i} (1) = χ_{i} (1) = n_{i}$ , because the only nonzero term is at $g = 1$ .

F1F2A1given

2.1

Characters add on direct sums by [F3], so evaluating both sides of the decomposition of step 1.1 at $1$ gives $χ_{r e g} (1) = \sum_{i} m_{i} n_{i}$ . By [F1] the left side is $∣ G ∣$ , and step 1.1 gives $m_{i} = n_{i}$ , so $∣ G ∣ = \sum_{i} n_{i}^{2}$ .

F1F3step 1.1algebra∎

Depends on

Used by

Dependency tree · two levels

13 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 3.3.7 (standard reference, not scraped)