1 Representation theory of finite groups 1.4 Schur’s Lemma 1.6 The group ring and the regular representation

1.5 Maschke’s theorem and complete reducibility

1.5.1 Maschke’s theorem

Theorem 1.40.

Suppose that $G$ is a finite group and that $V$ is a representation of $G$ over a field of characteristic {not dividing $|G|$ }. Suppose that $W$ is a subrepresentation of $V$ . Then there is a subrepresentation $W^{\prime}$ of $V$ such that

V\cong W\oplus W^{\prime}.

Proof.

First we make a useful definition.

Definition 1.41.

If $W\subset V$ are vector spaces, then a linear map $\pi:V\rightarrow W$ is a projection if

\pi(w)=w

for all $w\in W$ .

Exercise 1.42.

If $\pi:V\rightarrow W$ is a projection, then

V=W\oplus\ker(\pi).

We will construct a projection $\pi:V\rightarrow W$ that is a $G$ -homomorphism (we could call this a $G$ -projection). Given such a $\pi,$ the proof is easy: let $W^{\prime}=\ker(\pi)$ . This is a subrepresentation as $\pi$ is a $G$ -homomorphism, and by exercise we have $V=W\oplus W^{\prime}$ as required.

It remains to construct $\pi$ . Let $\pi_{0}:V\rightarrow W$ be any linear map such that $\pi_{0}|_{W}=\mathrm{id}_{W}$ (to construct it, choose a basis for $W$ and extend it to a basis for $V$ . Then define $\pi$ to be the identity on the basis of $W$ and whatever you like on the other basis vectors). This might not be a $G$ -homomorphism, but we turn it into one using an ‘averaging trick’: define

\pi(v)=\frac{1}{|G|}\sum_{g\in G}g^{-1}\pi_{0}(gv).

Then this is a $G$ -homomorphism: for any $h\in G,$

$\displaystyle h^{-1}\pi(hv)$	$\displaystyle=\frac{1}{\|G\|}\sum_{g\in G}h^{-1}g^{-1}\pi_{0}(ghv)$
	$\displaystyle=\frac{1}{\|G\|}\sum_{k\in G}k^{-1}\pi_{0}(kv)$	writing $k=gh$
	$\displaystyle=\pi(v),$

whence $\pi(hv)=h\pi(v)$ .

Finally, if $v\in W$ then $\pi(v)=\frac{1}{|G|}\sum_{g\in G}g^{-1}gv=v,$ so $\pi$ is a projection. ∎

Maschke’s Theorem and Schur’s Lemma both hold in the situation that $k=\mathbb{C}$ and $G$ is finite. From now on, we assume that this is the case, and that all representations are finite-dimensional.

1.5.2 Complete reducibility

The following corollary of Maschke’s theorem says that any (finite-dimensional) representation of a (finite) group can be written as a direct sum of irreducible representations, in an essentially unique way. So irreducible representations are the ’prime numbers’ of representation theory.

Corollary 1.43.

Let $V$ be a representation of $G$ . Then

V\cong W_{1}\oplus W_{2}\oplus\ldots\oplus W_{r}

for some irreducible representations $W_{1},\ldots,W_{r}$ .

Moreover, the number of times each isomorphism class of irreducible representation shows up in the above decomposition is independent of the exact choice of decomposition.

Proof.

The existence of such a decomposition follows from Maschke’s theorem and induction: let $W_{1}$ be an irreducible subrepresentation, write $V=W_{1}\oplus W_{1}^{\prime}$ by Maschke, repeat starting with $W^{\prime}_{1}$ .

If $W$ is an irreducible representation and $V\cong W_{1}\oplus\ldots\oplus W_{r},$ then

\dim\operatorname{Hom}_{G}(W,V)=\sum\dim\operatorname{Hom}_{G}(W,W_{i})=\#\{i:% W\cong W_{i}\}

by Schur’s Lemma (specifically, part (3) of 1.32). This only depends on $V$ and $W,$ not on the choice of decomposition of $V$ . ∎

In the previous proof, we used an easy, but important, property of $\operatorname{Hom}$ :

Lemma 1.44.

If $V,V^{\prime},W,W^{\prime}$ are representations of $G,$ then

	$\displaystyle\operatorname{Hom}_{G}(V,W\oplus W^{\prime})$	$\displaystyle\cong\operatorname{Hom}_{G}(V,W)\oplus\operatorname{Hom}_{G}(V,W^% {\prime})$
	$\displaystyle\operatorname{Hom}_{G}(V\oplus V^{\prime},W)$	$\displaystyle\cong\operatorname{Hom}_{G}(V,W)\oplus\operatorname{Hom}_{G}(V^{% \prime},W).$

We isolate the following part of the proof of 1.43 for later use.

Lemma 1.45.

If $\rho$ is an irreducible representation of $G$ and $\sigma$ is some other representation of $V,$ then the number of times $\rho$ appears in the irreducible decomposition of $\sigma$ is exactly

\dim\operatorname{Hom}_{G}(\rho,\sigma).

Example 1.46.

We work out the projections constructed in the proof of Maschke’s theorem when $V$ is the permutation representation of $S_{3}$ on $\mathbb{C}^{3}$ . Then $V$ has a subspace $V_{0}=\{(x,y,z):x=y=z\}$ . The map

	$\displaystyle V$	$\displaystyle\rightarrow V_{0}$
	$\displaystyle(x,y,z)$	$\displaystyle\mapsto\frac{1}{3}(x+y+z)(e_{1}+e_{2}+e_{3})$

is a $G$ -equivariant projection. As in the proof of Maschke’s theorem, its kernel $V_{1}=\{(x,y,z):x+y+z=0\}$ is a complement:

V=V_{0}\oplus V_{1}.

The representation $V_{1}$ is irreducible (under the isomorphism $S_{3}\cong D_{3},$ it is $\rho_{1}$ ). We can also write down a $G$ -equivariant projection $V\rightarrow V_{1}$ :

(x,y,z)\mapsto\frac{1}{3}(2x-y-z,2y-x-z,2z-x-y).

The kernel of this projection is $V_{0}$ .

Remark 1.47.

The decomposition of Corollary 1.43 is not unique: if $G$ is the trivial group, then $\mathbb{C}^{2}$ can be written as $\mathbb{C}\oplus\mathbb{C}$ in infinitely many ways, simply by choosing any two distinct lines. However, if we have an irreducible representation $(\rho,W)$ of $G,$ then for any representation $V$ the subspace $V(\rho)$ spanned by all the subrepresentations of $V$ isomorphic to $\rho$ is uniquely determined: it is called the $\rho$ -isotypic component of $V$ . The key example to keep in mind is, if $G\cong C_{n}$ is generated by $g,$ and $\chi$ is a character with $\chi(g)=\omega,$ then $V(\omega)$ is just the $\omega$ -eigenspace of $g$ acting on $V$ .

1.5.3 Unitarizability

We didn’t cover this section, but it is recommended reading.

Recall that a (Hermitian) inner product on a complex vector space $V$ is a map $(v,w):V\times V\rightarrow\mathbb{C}$ such that

	$\displaystyle(v+v^{\prime},w)$	$\displaystyle=(v,w)+(v^{\prime},w),$
	$\displaystyle(v,w+w^{\prime})$	$\displaystyle=(v,w)+(v,w^{\prime}),$
	$\displaystyle(\lambda v,\mu w)$	$\displaystyle=\overline{\lambda}\mu(v,w),\qquad\text{and}$
	$\displaystyle(w,v)$	$\displaystyle=\overline{(v,w)}$

for all $v,v^{\prime},w,w^{\prime}\in V$ , $\lambda,\mu\in\mathbb{C}$ , and that is positive definite, meaning that $(v,v)>0$ for all nonzero $v\in V$ . The standard example is

((w_{1},\ldots,w_{n}),(z_{1},\ldots,z_{n}))=\overline{w}_{1}z_{1}+\ldots+% \overline{w}_{n}z_{n}

on $V=\mathbb{C}^{n}$ . A (Hermitian) inner product space is a vector space together with a chosen Hermitian inner product.

Remark 1.48.

If $(,)$ is a Hermitian inner product on $\mathbb{C}^{n},$ and $\mathbf{w}$ and $\mathbf{z}\in\mathbb{C}^{n}$ are written as column vectors, then we can write

(\mathbf{w},\mathbf{z})=\mathbf{w}^{\dagger}H\mathbf{z}

where $\mathbf{w}^{\dagger}$ is the complex conjugate of the transpose of $\mathbf{w}$ . The matrix $H$ will satisfy

H^{\dagger}=H

and be diagonalizable with positive real eigenvalues.

If $(,)$ is a Hermitian inner product on $V$ and $W$ is a subspace, then the orthogonal complement is

W^{\perp}=\{v\in V:(v,w)=0\text{ for all $w\in W$}\}.

As vector spaces, we have $V=W\oplus W^{\perp}$ .

Definition 1.49.

A representation $V$ of $G$ is unitarizable if there is a $G$ -invariant inner product on $V$ ; that is, a Hermitian inner product such that

(gv,gw)=(v,w)

for all $g\in G$ .

If such an inner product is chosen, then the representation $V$ is said to be unitary.

Exercise 1.50.

Suppose that $V$ is a unitary representation of $G$ and that $W$ is a subrepresentation. Then $W^{\perp}$ is a subrepresentation of $V$ .

Theorem 1.51.

If $V$ is a complex representation of $G,$ then $V$ is unitarizable.

Proof.

Start with any Hermitian inner product, $(,),$ not necessarily $G$ -invariant. Then average it over $G$ . ∎

Exercise 1.52.

Fill in the details of the proof of Theorem 1.51, and use it together with exercise 1.50 to give a second proof of Maschke’s theorem.