1.1. Lecture 1

1.1.1. Definition of a representation

Let $k$ be a field (we will almost always take $k=\mathbb{C}$ ), and let $G$ be a group.

Definition 1.1.1.

A representation of $G$ over $k$ is a pair $(\pi,V)$ , where

(i)

$V$ is a vector space over $k$ , and
(ii)

$\pi:G\rightarrow\operatorname{GL}(V)$ is a group homomorphism.

We also say that $\pi$ is a representation of $G$ on $V$ , or simply that $\pi$ is a representation of $G$ ; this makes sense since the vector space $V$ is part of the definition of $\pi$ .

The dimension of a representation $(\pi,V)$ is the dimension of $V$ . We will often denote this by $\operatorname{dim}\pi$ .

There is another way to think of this. Suppose that $(\rho,V)$ is a representation of $G$ . Then we may define an action of $G$ on $V$ by letting

g\cdot{\bf v}:=\rho(g){\bf v}

for all $g\in G$ and ${\bf v}\in V$ . This is an action because $\rho$ is a homomorphism, and it is linear, meaning that for every $g\in G$ the map taking ${\bf v}\mapsto\rho(g){\bf v}$ is a linear map on $V$ . Conversely, given a linear action of $G$ on $V$ , we can define a representation $(\pi,V)$ of $G$ by $\pi(g){\bf v}:=g\cdot{\bf v}$ . In other words:

A representation of $G$ is a linear action on a vector space.

Suppose that $(\pi,V)$ is a representation of a group $G$ and $\operatorname{dim}\pi=n$ . By choosing a basis for $V$ , we may identify $V$ with $k^{n}$ , and under this identification any (invertible) linear map $V\rightarrow V$ is just the same thing as an (invertible) $n\times n$ matrix.

Thus, once you choose a basis, a representation is just the same as a homomorphism $G\rightarrow\operatorname{GL}_{n}(k)$ . In particular:

A 1-dimensional representation of $G$ is a homomorphism $G\rightarrow k^{\times}$ .

1.1.2. Examples

Example 1.1.2.

If $V$ is any vector space, then we can always take $\rho:G\rightarrow\operatorname{GL}(V)$ to be the homomorphism sending every element to the identity. We call this the trivial representation of $G$ on $V$ .

Example 1.1.3.

Let $G=S_{n}$ . Recall that there is a homomorphism

\epsilon:S_{n}\rightarrow\{\pm 1\}

taking a permutation to its sign. Since $\{\pm 1\}\subseteq\mathbb{C}^{\times}$ , this gives a 1-dimensional representation of $S_{n}$ , called the sign representation.

Example 1.1.4.

Suppose that $G=({\mathbb{Z}},+)$ . Then, if $\rho$ is a representation of $G$ , it is completely determined by $V$ and the invertible linear map $\rho(1):V\rightarrow V$ (which may be any element of $\operatorname{GL}(V)$ ). This is because we then have

\rho(n)=\rho(1+\ldots+1)=\rho(1)^{n}.

Thus, a representation of ${\mathbb{Z}}$ is just a vector space $V$ together with an invertible linear map from $V$ to itself.

We can push this a bit further. Suppose that $G$ is a cyclic group of order $n$ with generator $a$ , hence

G=\left\langle a\,|\,a^{n}=e\right\rangle.

A representation $(\pi,V)$ of $G$ is once again determined by $V$ and $\rho(a)$ , which can be any linear map $T:V\rightarrow V$ such that $T^{n}=\mathrm{Id}$ .

Many interesting examples arise from geometry.

Example 1.1.5.

Let $G=D_{n}$ be the dihedral group of order $2n$ , the group of symmetries (rotations and reflections) of a regular $n$ -gon. Since each rotation/reflection is an invertible linear map from $\mathbb{R}^{2}\rightarrow\mathbb{R}^{2}$ , we get a representation $\rho$ of $G$ on $\mathbb{R}^{2}$ . Letting $r$ be counter-clockwise rotation by $2\pi/n$ radians and $s$ be reflection in the horizontal axis, recall that $D_{n}$ has the presentation

\left\langle r,s\,|\,r^{n}=s^{2}=e,sr=r^{-1}s\right\rangle.

As an explicit homomorphism $\rho:D_{n}\rightarrow\operatorname{GL}_{2}(\mathbb{R})$ , we have

\rho(r)=\left(\begin{matrix}\cos(\theta_{n})&-\sin(\theta_{n})\\ \sin(\theta_{n})&\cos(\theta_{n})\end{matrix}\right),\qquad\rho(s)=\left(% \begin{matrix}1&0\\ 0&-1\end{matrix}\right),

where $\theta_{n}=2\pi/n$ . Since $\operatorname{GL}_{2}({\mathbb{R}})\subseteq\operatorname{GL}_{2}({\mathbb{C}})$ , we may also view this as a representation of $D_{n}$ on ${\mathbb{C}}^{2}$ ; we call $(\rho,{\mathbb{C}}^{2})$ the defining representation of $D_{n}$ .

Example 1.1.6.

Let $G=S_{4}$ . You might remember that this is isomorphic to the group of symmetries (rotations/reflections) of the regular tetrahedron in $\mathbb{R}^{3}$ . We therefore get a representation

\rho:S_{4}\rightarrow\operatorname{GL}_{3}(\mathbb{R}).

It would be a slightly unpleasant exercise to work the matrices out explicitly.

Note that $S_{4}$ is also isomorphic to the group of rotations of the cube, giving another (different!) 3-dimensional representation.

Another source of representations comes from actions of groups on (usually finite) sets.

Example 1.1.7.

Define a representation $(\pi,k^{n})$ of $S_{n}$ via

\pi(\sigma)\left(x_{1}{\bf e}_{1}+\ldots+x_{n}{\bf e}_{n}\right)=x_{1}{\bf e}_% {\sigma(1)}+\ldots+x_{n}{\bf e}_{\sigma(n)},

where ${\bf e}_{1},\ldots,{\bf e}_{n}$ is the standard basis. This is called the permutation representation of $S_{n}$ on $k^{n}$ .

Important: If we write elements of $k^{n}$ as $(x_{1},\ldots,x_{n})\,^{\mathrm{t}}\!$ (as we commonly do), then it is not the case that

\pi(\sigma)\left(\begin{matrix}x_{1}\\ \vdots\\ x_{n}\end{matrix}\right)=\left(\begin{matrix}x_{\sigma(1)}\\ \vdots\\ x_{\sigma(n)}\end{matrix}\right).

This actually would define a right action, not a left action. The correct formula is

\pi(\sigma)\left(\begin{matrix}x_{1}\\ \vdots\\ x_{n}\end{matrix}\right)=\left(\begin{matrix}x_{\sigma^{-1}(1)}\\ \vdots\\ x_{\sigma^{-1}(n)}\end{matrix}\right).

This may be generalised whenever we have a group action on a set. Firstly, we associate to any set $X$ an $k$ -vector space:

Definition 1.1.8.

Let $X$ be a set. The free vector space $k(X)$ over $k$ generated by $X$ is the vector space consisting of all formal sums

\sum_{x\in X}z_{x}x,

where $z_{x}\in k$ and $z_{x}=0$ for all but finitely many $x\in X$ . The vector space operations are as follows: vector addition is given by

\left(\sum_{x\in X}z_{x}x\right)+\left(\sum_{x\in X}y_{x}x\right):=\left(\sum_% {x\in X}(z_{x}+y_{x})x\right),

and scalar multiplication

\lambda\left(\sum_{x\in X}z_{x}x\right):=\left(\sum_{x\in X}\lambda z_{x}x\right)

for all $\lambda\in k$ .

Observe that a basis for $k(X)$ is $\{x\}_{x\in X}$ (here $x=1x$ , i.e. the formal sum $\sum_{y\in X}z_{y}y$ , with $z_{y}=1$ if $y=x$ and $z_{y}=0$ otherwise)

Definition 1.1.9.

Given a group $G$ acting on a set $X$ , we define the permutation representation $\big{(}\pi,k(X)\big{)}$ for this action by

\pi(g)\left(\sum_{x\in X}z_{x}x\right):=\left(\sum_{x\in X}z_{x}(g\cdot x)% \right)=\left(\sum_{x\in X}z_{g^{-1}\cdot x}x\right).

1.1.3. Exercises

Remark 1.1.10.

We elaborate on two ways of checking whether a map $\pi:G\rightarrow\mathrm{GL}(V)$ is actually a representation. This point was discussed a bit during the lecture, but since it will be used frequently throughout the course, I would like to emphasise this a bit further:

(i) Assume that a rule or formula is given for $\pi(g)$ for every $g\in G$ . In order to verify that $(\pi,V)$ really is a representation, one needs to either check or show that $\pi(gh)=\pi(g)\pi(h)$ for all $g,h\in G$ .

(ii) Often, we will define $\pi$ by where it sends the generators $g_{1},\ldots,g_{k}$ of a group $G$ . Since every element of $G$ can be written as a word in $g_{1},\ldots,g_{k}$ , if $\pi$ is required to be a homomorphism, then there is no choice in the values of $\pi(g)$ for the other group elements $g$ : $\pi(g)$ must be the product of the $\pi(g_{i})$ s corresponding to the decomposition of $g$ into a product of $g_{i}$ s. In order for this procedure to give a well-defined homomorphism $\pi$ on all of $G$ , it is both necessary and sufficient that the $\pi(g_{i})$ s satisfy the same relations as the $g_{i}$ s. (This is a general fact about any group homomorphism, not just ones into $\mathrm{GL}(V)$ ).

Compare the phrasing of problems 1.1.10 and 1.1.10 below!

Problem 12. Verify that $(\pi,\mathbb{C}^{3})$ is a representation of $S_{3}$ , where $\pi(\sigma)\in\mathrm{GL}_{3}(\mathbb{C})$ is defined via

\pi(\sigma)\left(\begin{matrix}z_{1}\\ z_{2}\\ z_{3}\end{matrix}\right)=\left(\begin{matrix}z_{\sigma^{-1}(1)}\\ z_{\sigma^{-1}(2)}\\ z_{\sigma^{-1}(3)}\end{matrix}\right)\qquad\forall\left(\begin{matrix}z_{1}\\ z_{2}\\ z_{3}\end{matrix}\right)\in\mathbb{C}^{3}.

Problem 13. Writing $D_{3}=\langle r,s\,|\,r^{3}=s^{2}=rsrs=e\rangle$ , let $\rho(r),\rho(s)\in\mathrm{GL}_{2}(\mathbb{C})$ be given by

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

Show that this defines a representation $(\rho,\mathbb{C}^{2})$ of $D_{3}$ .

Problem 14.

(a)

Find the matrices of all the elements of $S_{3}$ for the permutation representation $(\pi,{\mathbb{C}}^{3})$ , with respect to the basis ${\bf e}_{1},{\bf e}_{2},{\bf e}_{3}$ .
(b)

Find another basis such that the matrices all take the form

$\begin{pmatrix}1&0&0\\ 0&?&?\\ 0&?&?\end{pmatrix},$

and determine the unknown entries for your basis.

Problem 15. Let $(\pi,V)$ and $(\rho,W)$ be two representations of a group $G$ . Show that $(\pi\oplus\rho,V\oplus W)$ is a representation of $G$ , where

\pi\oplus\rho(g)(v,w):=(\pi(g)v,\rho(g)w)\qquad\forall g\in G,\,v\in V,\,w\in W.

Remark: this is called the (direct) sum of two representations.

Problem 16. Let $(\pi,V)$ be a representation of a group $G$ . Given a subgroup $H\leq G$ , show that $(\pi|_{H},V)$ is a representation of $H$ .

Problem 17. Given a group action $G\circlearrowright X$ , let $V=\{f:X\rightarrow\mathbb{C}\}$ .

a) Show that $(\pi,V)$ is a representation of $G$ , where for every $g\in G$ and $f\in V$ , $\pi(g)f\in V$ is defined via the formula

\big{(}\pi(g)f\big{)}(x)=f(g^{-1}\cdot x)\qquad\forall x\in X.

b) Show that $(\widetilde{\pi},V)$ is not a representation of $G$ , where for every $g\in G$ and $f\in V$ , $\widetilde{\pi}(g)f\in V$ is defined via the formula

\big{(}\widetilde{\pi}(g)f\big{)}(x)=f(g\cdot x)\qquad\forall x\in X.

Problem 18. Prove the following:

Proposition 1.1.11.

Let $G$ be a finite group and $(\pi,{\mathbb{C}})$ a representation of $G$ . Show that for every $g\in G$ , there exists $n_{g}\in\{0,1,2,\ldots|G|-1\}$ such that

\pi(g)=e^{2\pi in_{g}/|G|}.

Problem 19. Find all 1-dimensional representations of $D_{4}$ .