1.2. Lecture 2

1.2.1. Subrepresentations and irreducible representations

Definition 1.2.1.

A subrepresentation of a representation $(\rho,V)$ of $G$ is a pair $(\rho|_{W},W)$ consisting of a subspace $W\subseteq V$ such that $\rho(g){\bf w}\in W$ for all ${\bf w}\in W$ together with the group homomorphism

\rho|_{W}:G\longrightarrow\operatorname{GL}(W)\;;\;\rho|_{W}(g){\bf w}:=\rho(g% ){\bf w},

for all $w\in W$ .

Definition 1.2.2.

A representation $(\rho,V)$ of $G$ is irreducible if it is non-zero and has no subrepresentations except for $\{0\}$ and $V$ itself.

Example 1.2.3.

Since 1-dimensional vector spaces have no non-trivial subspaces, every 1-dimensional representation is irreducible.

Example 1.2.4.

Consider the permutation representation $(\pi,{\mathbb{C}}^{3})$ of $S_{3}$ . This is the representation which acts on via $\pi(\sigma){\bf e}_{i}={\bf e}_{\sigma(i)}$ , where ${\bf e}_{1},{\bf e}_{2},{\bf e}_{3}$ are the standard basis vectors. It is not irreducible!

Let ${\bf v}_{1}=(1,1,1)\,^{\mathrm{t}}\!$ . Then $\pi(\sigma){\bf v}_{1}={\bf v}_{1}$ for every $\sigma\in S_{3}$ , so $(\pi,{\mathbb{C}}{\bf v}_{1})$ is a subrepresentation of $(\pi,{\mathbb{C}}^{3})$ . Being 1-dimensional, it is in fact an irreducible subrepresentation.

Denote by $W_{0}$ the subspace of ${\mathbb{C}}^{3}$ consisting of all vectors $(z_{1},z_{2},z_{3})\,^{\mathrm{t}}\!$ such that $z_{1}+z_{2}+z_{3}=0$ . Then since $S_{3}$ permutes the coordinates of a vector, it doesn’t change their sum; so $\pi(\sigma)W_{0}\subseteq W_{0}$ , and $W_{0}$ is a subrepresentation (of dimension 2).

We claim that $W_{0}$ is irreducible. Indeed, suppose that $U\subseteq W_{0}$ is a non-zero subrepresentation; we have to show that $U=W_{0}$ . Let $(x,y,z)\,^{\mathrm{t}}\!\in U$ be non-zero. As $x=y=z$ can’t happen, we can apply an element of $S_{3}$ to permute the coordinates so that $x\neq y$ . Then applying $\pi(12)$ , we have $(y,x,z)\,^{\mathrm{t}}\!\in U$ . Taking the difference, $(x-y,y-x,0)\,^{\mathrm{t}}\!\in U$ ; scaling, $(1,-1,0)\,^{\mathrm{t}}\!\in U$ . Applying $\pi(23)$ , we have $(1,0,-1)\,^{\mathrm{t}}\!\in U$ . But these vectors span $W_{0}$ (e.g. because they are linearly independent and $\operatorname{dim}W_{0}=2$ ), so $U=W_{0}$ as required.

Example 1.2.5.

If $(\rho,V)$ is a finite-dimensional representation of $G={\mathbb{Z}}$ , with $T=\rho(1)$ , then $T$ has an eigenvector ${\bf v}$ which spans a 1-dimensional subrepresentation of $V$ . Thus $V$ is not irreducible unless $\operatorname{dim}V=1$ . The irreducible subrepresentations of $V$ are exactly the lines spanned by $T$ -eigenvectors.

1.2.2. Homomorphisms and isomorphisms

Definition 1.2.6.

Suppose that $(\rho,V)$ and $(\sigma,W)$ are representations of $G$ . Then a $G$ -homomorphism (or homomorphism of representations of $G$ , or map of representations of $G$ , or, if we are being lazy, just a homomorphism) $V\rightarrow W$ is a linear map $\phi:V\rightarrow W$ such that

\phi(\rho(g){\bf v})=\sigma(g)\phi({\bf v})

for all ${\bf v}\in V$ , $g\in G$ .

In other words, $\phi$ “commutes” with the action of $G$ . We write $\operatorname{Hom}_{G}(V,W)$ for the (vector space) of $G$ -homomorphisms from $V$ to $W$ .

There is another word that is sometimes used for $G$ -homomorphism: intertwiner, or $G$ -intertwiner.

Definition 1.2.7.

A $G$ -isomorphism (or just an isomorphism) is a bijective $G$ -homomorphism.

If $(\pi,V)$ and $(\rho,W)$ are two representations of a group $G$ , and there exists a $G$ -isomorphism $V\rightarrow W$ then we write $(\pi,V)\cong(\rho,W)$ , and say that they are isomorphic. The collection of all representations isomorphic to a representation $(\pi,V)$ is called the isomorphism class of $(\pi,V)$ .

Lemma 1.2.8.

Suppose that $V$ and $W$ are representations of $G$ . If $T\in\operatorname{Hom}_{G}(V,W)$ is an isomorphism, then $T^{-1}\in\operatorname{Hom}_{G}(W,V)$ .

Proof.

Exercise. ∎

Isomorphic representations are “different pictures of the same object”; if $(\pi,V)$ and $(\rho,W)$ are isomorphic, with $T:V\rightarrow W$ begin a $G$ -isomorphism, then we have

\rho(g)=T\pi(g)T^{-1}

for all $g\in G$ , i.e. $\pi$ and $T$ tells us everything about $\rho$ .

Lemma 1.2.9.

Let $(\pi,V)$ and $(\rho,W)$ be two representations of a group $G$ . If $\phi\in\operatorname{Hom}_{G}(V,W)$ , then $(\pi,\ker(\phi))$ and $(\rho,{\mathrm{im}}(\phi))$ are subrepresentations of $V$ and $W$ , respectively.

Proof.

We know from linear algebra that $\ker(\phi)$ and ${\mathrm{im}}(\phi)$ are subspaces, so we just have to show that they are preserved by the $G$ -actions.

For the kernel: suppose that ${\bf v}\in\ker(\phi)$ . Then $\phi({\bf v})={\bf 0}$ , and so for any $g\in G$ , we have

\phi(\pi(g){\bf v})=\rho(g)\phi({\bf v})=\rho(g){\bf 0}={\bf 0},

so $\pi(g){\bf v}\in\ker(\phi)$ . Thus $\ker(\phi)$ is $G$ -invariant, as required.

For the image: suppose that ${\bf w}\in{\mathrm{im}}(\phi)$ . Then ${\bf w}=\phi({\bf v})$ for some ${\bf v}\in V$ . Then for any $g\in G$ ,

\rho(g){\bf w}=\rho(g)\phi({\bf v})=\phi(\pi(g){\bf v})\in{\mathrm{im}}(\phi),

so the image of $\phi$ is also $G$ -invariant. ∎

Example 1.2.10.

Let $G=D_{3}\cong S_{3}$ where the isomorphism takes $r\mapsto(123)$ and $s\mapsto(23)$ . Let $(\pi,{\mathbb{C}}^{3})$ be the permutation representation of $G$ , so in the basis ${\bf e}_{1},{\bf e}_{2},{\bf e}_{3}$ we have

\pi(r)=\begin{pmatrix}0&0&1\\ 1&0&0\\ 0&1&0\end{pmatrix},\qquad\pi(s)=\begin{pmatrix}1&0&0\\ 0&0&1\\ 0&1&0\end{pmatrix}.

Let $(\rho,{\mathbb{C}}^{2})$ denote the defining representation of $G$ ; thus

\rho(r)=\left(\begin{matrix}-1/2&-\sqrt{3}/2\\ \sqrt{3}/2&-1/2\end{matrix}\right),\qquad\rho(s)=\left(\begin{matrix}1&0\\ 0&-1\end{matrix}\right).

Let $T:{\mathbb{C}}^{3}\rightarrow{\mathbb{C}}^{2}$ be the linear map defined through

T({\bf e}_{1})=\left(\begin{matrix}1\\ 0\end{matrix}\right),\qquad T({\bf e}_{2})=\left(\begin{matrix}-1/2\\ \sqrt{3}/2\end{matrix}\right),\qquad T({\bf e}_{2})=\left(\begin{matrix}-1/2\\ -\sqrt{3}/2\end{matrix}\right),

(i.e. $T$ maps the basis vectors ${\bf e}_{i}$ to the vertices of the equilateral triangle around the origin with a vertex at $(1,0)\,^{\mathrm{t}}\!$ ). We want to show that $T$ is a $G$ -homomorphism.

To do this we first consider the matrix of $T$ is

\begin{pmatrix}1&-1/2&-1/2\\ 0&\sqrt{3}/2&-\sqrt{3}/2\end{pmatrix}.

We need to show that $T\pi(g)=\rho T(g)$ for all $g\in D_{3}$ . As $D_{3}$ is generated by just $r$ and $s$ we only need to check two cases. We need to check that

\begin{pmatrix}1&-1/2&-1/2\\ 0&-\sqrt{3}/2&\sqrt{3}/2\end{pmatrix}\begin{pmatrix}0&0&1\\ 1&0&0\\ 0&1&0\end{pmatrix}=\left(\begin{matrix}-1/2&\sqrt{3}/2\\ -\sqrt{3}/2&-1/2\end{matrix}\right)\begin{pmatrix}1&-1/2&-1/2\\ 0&\sqrt{3}/2&-\sqrt{3}/2\end{pmatrix},

which is true, and a similar equation coming from $s$ .

The kernel of the homomorphism $T$ is the subspace ${\mathbb{C}}({\bf e}_{1}+{\bf e}_{2}+{\bf e}_{3})$ and in fact $T$ defines an isomorphism from the subrepresentation

W_{0}=\{(a,b,c)\,^{\mathrm{t}}\!\in{\mathbb{C}}^{3}:a+b+c=0\}\subseteq{\mathbb% {C}}^{3}

to ${\mathbb{C}}^{2}$ .

1.2.3. Exercises

Problem 20. Let $(\pi,V)$ be a representation of a group $G$ . Show that if $(\pi,W_{1})$ , $(\pi,W_{2})$ are two irreducible subrepresentations of $(\pi,V)$ , then either $W_{1}\cap W_{2}=0$ or $W_{1}=W_{2}$ .

Problem 21. Prove Lemma 1.2.8.

Problem 22. Let $(\pi,V)$ be an $n$ -dimensional representation of a group $G$ . Show that there exists a representation $(\widetilde{\pi},{\mathbb{C}}^{n})$ that is isomorphic to $(\pi,V)$ .

Problem 23. Generalise Example 1.2.4 to the permutation representation $(\pi,{\mathbb{C}}^{n})$ of $S_{n}$ .

Problem 24. Let

C_{0}(X)=\{f:X\rightarrow{\mathbb{C}}\,|\,f(x)=0\mathrm{\;for\;all\;but\;% finitely\;many}\;x\in X\}.

The quasi-regular representation of $G$ on $C_{0}(X)$ is denoted $(\lambda,C_{0}(X))$ and is defined via the formula

\big{(}\lambda(g)f\big{)}(x):=f(g^{-1}\cdot x)

for all $g\in G$ , $x\in X$ and $f\in C_{0}(X)$ . This is called the functional point of view. Let $G$ be a group acting on a set $X$ . Show that $(\pi,{\mathbb{C}}(X))$ , the representation arising from this group action, is isomorphic to $(\lambda,C_{0}(X))$ .