1 Representation theory of finite groups 1 Representation theory of finite groups 1.2 Formalities

1.1 Definitions and first examples

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

Definition 1.1.

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

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$V$ is a vector space over $k,$ and
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$\rho:G\rightarrow\operatorname{GL}(V)$ is a group homomorphism.

The dimension of the representation is the dimension of $V$ . We will very often say that $V$ is a representation of $G$ , or that $\rho$ is a representation of $G$ , without mentioning the other part of the definition.

There is another way to think of this. Suppose that $(\rho,V)$ is a representation of $G$ . Then we define an action of $G$ on $V$ by $gv=\rho(g)v$ . This is an action because $\rho$ is a homomorphism, and it is linear, meaning that for every $g$ the map taking $v$ to $g v$ is a linear map. Conversely, given a linear action of $G$ on $V$ , we can define $\rho$ by $\rho(g)v=gv$ . In other words:

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

We will often use $\rho(g)v$ and $g v$ interchangeably.

If a basis is given for $V$ , then an (invertible) linear map $V\rightarrow V$ is just the same thing as an (invertible) $n\times n$ matrix, where $n=\dim V$ . So, once you choose a basis, a representation is just the same as a homomorphism $G\rightarrow\operatorname{GL}_{n}(k)$ . In particular:

A one-dimensional representation of $G$ is the same as a homomorphism $G\rightarrow k^{\times}$ .

Example 1.2.

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

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

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

Example 1.3.

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 on $V$ .

Example 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 can be anything). 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 $g,$ so

G=\{e,g,g^{2},\ldots,g^{n-1}\}

and $g^{n}=e$ . Then a representation of $G$ is again determined by $V$ and $\rho(g),$ which can be any linear map $T:V\rightarrow V$ such that $T^{n}=I$ .

One source of more interesting examples is geometry.

Example 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 — see Figure 1. 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 rotation by $2\pi/n$ and $s$ be reflection in the vertical axis, $D_{n}$ 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 (with $\theta=2\pi/n$ )

	$\displaystyle\rho(r)$	$\displaystyle=\begin{pmatrix}\hidden@noalign{}\hfil\textstyle\cos(\theta)&-% \sin(\theta)\\ \hidden@noalign{}\hfil\textstyle\sin(\theta)&\cos(\theta)\end{pmatrix}$
	$\displaystyle\rho(s)$	$\displaystyle=\begin{pmatrix}\hidden@noalign{}\hfil\textstyle-1&0\\ \hidden@noalign{}\hfil\textstyle 0&1\end{pmatrix}.$

Figure 1: The action of the dihedral group on a polygon.

Example 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!) three-dimensional representation.

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

Example 1.7.

Let $G=S_{n},$ and let $V=k^{n}$ . Define a representation of $S_{n}$ on $V$ via

\sigma\cdot\left(x_{1}e_{1}+\ldots+x_{n}e_{n}\right)=x_{1}e_{\sigma(1)}+\ldots% +x_{n}e_{\sigma(n)}

where $e_{1},\ldots,e_{n}$ is the standard basis. This is called the permutation representation (over $k$ ).

There is a warning here! If you write elements of $V$ as $(x_{1},\ldots,x_{n})$ , as well you might, then it is not the case that $g(x_{1},\ldots,x_{n})=(x_{g(1)},\ldots,x_{g(n)})$ . This actually would define a right action, not a left action. The correct formula is

g(x_{1},\ldots,x_{n})=(x_{g^{-1}(1)},\ldots,x_{g^{-1}(n)}).

Definition 1.8.

If $G$ is a group acting on a set $X,$ we consider a $k$ -vector space $V$ with basis $\{e_{x}:x\in X\}$ . It has a representation of $G$ given by $ge_{x}=e_{gx},$ called the permutation representation associated to $X$ .

Remark 1.9.

There is another point of view on this, the functional point of view. We didn’t cover this in lectures; as things stand this is optional. For simplicity, let $X$ be finite, and define

k^{X}=\{f:X\rightarrow k\}.

We give this an action of $G$ by the formula

(gf)(x)=f(g^{-1}(x))

for $g\in G$ , $f\in k^{X}$ , $x\in X$ .

Question 1.10.

Why don’t we define $(gf)(x)=f(gx)$ ?

Let $\delta_{x}\in k^{X}$ be the function sending $x$ to $1$ and everything else to $0$ . Then the $\delta_{x}$ for $x\in X$ are a basis for $k^{X}$ , and for $g\in G$ you can check that

g\delta_{x}=\delta_{gx}.

In other words, the $\delta_{x}$ behave exactly like the $e_{x}$ in the permutation representation; when we have a bit more language, we can say that $k^{X}$ is isomorphic to the permutation representation.

The next example combines geometry and combinatorics.

Example 1.11.

Let $G$ be the group of rotations of the cube and let $X$ be the set of faces. We can think of an element of the permutation representation as being a way of writing a complex number on each face.

This term is all about finite groups, but next term we will consider representations of Lie groups such as the group $\mathrm{SO}(3)$ of rotations of $\mathbb{R}^{3}$ . These have many applications in physics.

Example 1.12.

In a spherically symmetric situation we might be interested in (smooth) solutions $f:\mathbb{R}^{3}\rightarrow\mathbb{C}$ to Laplace’s equation

\nabla f=0

with radial behaviour $f(rx)=r^{l}f(x)$ for scalars $r,$ and some integer $l\geq 0$ . Here $\nabla$ is the Laplacian

\frac{\partial^{2}}{\partial x^{2}}+\frac{\partial^{2}}{\partial y^{2}}+\frac{% \partial^{2}}{\partial z^{2}}.

These form a representation of $\mathrm{SO}(3)$ of dimension $2l+1,$ and in fact the irreducible finite dimensional representations of $\mathrm{SO}(3)$ are exactly those obtained in this way for integers $l\geq 0$ .

The solutions are called spherical harmonics, and representation theory can be used to find particularly nice bases of these spaces!

Here is another example:

Example 1.13.

Let $G=\operatorname{GL}_{2}(\mathbb{C})$ and let $V=\left\langle X^{2},XY,Y^{2}\right\rangle\subset\mathbb{C}[X,Y]$ . Then $V$ is a representation of $G$ via

g\cdot F(X,Y)=F(aX+cY,bX+dY).