CorollaryStatement: Literature-sourcedProof: AI-adaptedSession-authored (Fable 5 assisted)precheck passjudge pass (gpt-5.6-terra)audited 2026-08-28

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If $k$ is algebraically closed and $char k ∤ ∣ G ∣$ , then $\sum_{i} (\dim_{k} V_{i})^{2} = ∣ G ∣$

Statement

Let $G$ be a finite group and let $k$ be an algebraically closed field with $char k ∤ ∣ G ∣$ . If $V_{1}, \dots, V_{r}$ is a complete list of the irreducible representations of $G$ over $k$ , up to equivalence, then

\sum_{i = 1}^{r} (\dim_{k} V_{i})^{2} = ∣ G ∣ .

Facts & Assumptions

Given: A finite group $G$ and an algebraically closed field $k$ with $char k ∤ ∣ G ∣$ .

[L1]

The regular representation decomposes as $k [G] ≅ V_{1}^{\oplus \dim_{k} V_{1}} \oplus \dots \oplus V_{r}^{\oplus \dim_{k} V_{r}},$ where $V_{1}, \dots, V_{r}$ are the irreducible representations of $G$ (If $k$ is algebraically closed and $char k ∤ ∣ G ∣$ , there are finitely many irreducible representations, and each occurs in the regular representation with multiplicity equal to its degree).

[L2]

For a finite group, $\dim_{k} k [G] = ∣ G ∣$ (If $G$ is finite then $\dim_{k} k [G] = ∣ G ∣$ ).

Proof

technique · direct

1.1

By [L1], the regular representation is a direct sum of $\dim_{k} V_{i}$ copies of each $V_{i}$ , and [L2] says its total dimension is $∣ G ∣$ .

L1L2given

2.1

Taking dimensions in step 1.1 gives $∣ G ∣ = \dim_{k} k [G] = \sum_{i = 1}^{r} \dim_{k} (V_{i}^{\oplus \dim_{k} V_{i}}) = \sum_{i = 1}^{r} (\dim_{k} V_{i})^{2} .$ This is the required identity.

step 1.1givenalgebra∎

Depends on

Used by

ℂ[Q₈] and ℂ[Dih(C₄)] both decompose as ℂ⁴× M₂(ℂ) Example
ℂ[S₃]≅ℂ×ℂ× M₂(ℂ) Example
ℂ[ℤ/3ℤ]≅ℂ×ℂ×ℂ Example
The regular representation of ℤ/4ℤ over ℂ splits into its four characters Example
False statement: the sum of the irreducible degrees equals |G| False statement

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, Corollary 2.1.5 (standard reference, not scraped)
Pavel Etingof et al., Introduction to Representation Theory, Theorem 3.1(ii) (standard reference, not scraped)

If kk is algebraically closed and char⁡k∤∣G∣chark∤∣G∣, then ∑i(dim⁡kVi)2=∣G∣∑i​(dimk​Vi​)2=∣G∣