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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

Statement

Let $G$ be a finite group and let $k$ be an algebraically closed field with $char k ∤ ∣ G ∣$ . Then there are finitely many irreducible representations $V_{1}, \dots, V_{r}$ of $G$ over $k$ , up to equivalence, and the regular representation decomposes as

k [G] ≅ V_{1}^{\oplus \dim_{k} V_{1}} \oplus \dots \oplus V_{r}^{\oplus \dim_{k} V_{r}} .

In particular, each irreducible representation occurs in the regular representation with multiplicity equal to its degree.

Facts & Assumptions

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

[L1]

Under these hypotheses, there is a $k$ -algebra decomposition

k [G] ≅ \prod_{i = 1}^{r} M_{n_{i}} (k)

for positive integers $n_{i}$ (If $k$ is algebraically closed and $char k ∤ ∣ G ∣$ , then $k [G] ≅ \prod_{i = 1}^{r} M_{n_{i}} (k)$ ).

[L2]

For such a product ring $\prod_{i = 1}^{r} M_{n_{i}} (k)$ with $r \geq 1$ , the simple left modules are exactly the column modules $k^{n_{i}}$ , one isomorphism class for each factor (Simple modules over a product of matrix rings over division rings).

[L3]

For $M_{n_{i}} (k)$ , the left regular module is the direct sum of $n_{i}$ copies of its simple column module $k^{n_{i}}$ (Matrix rings over division rings are semisimple).

[L4]

Under the dictionary, irreducible representations are exactly simple left $k [G]$ -modules (Under the dictionary, subrepresentations are exactly submodules and irreducible representations are exactly simple modules).

[L5]

The regular representation of $G$ over $k$ is the left action on $k [G]$ (The trivial representation, the regular representation, and permutation representations from finite $G$ -sets).

Proof

technique · direct

1.1

By [L1], the left regular $k [G]$ -module is isomorphic to the left regular module of $\prod_{i = 1}^{r} M_{n_{i}} (k)$ . As a module over that product, the regular module splits as the direct sum of the factor regular modules, and [L3] decomposes factor $i$ into $n_{i}$ copies of the column module $k^{n_{i}}$ . Thus the regular representation [L5] is a direct sum of finitely many simple modules, with the factor- $i$ simple occurring exactly $n_{i}$ times.

L1L3L5givenalgebra

2.1

By [L2], those factor column modules give all simple module isomorphism classes, one for each factor. Translating with [L4], there are finitely many irreducible representations of $G$ , one for each factor, and the factor- $i$ representation has degree $n_{i}$ because its underlying vector space is $k^{n_{i}}$ . Therefore the regular representation contains each irreducible representation with multiplicity equal to its degree.

L2L4step 1.1givenalgebra∎

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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∣, there are finitely many irreducible representations, and each occurs in the regular representation with multiplicity equal to its degree