Representation Theory IV Representation Theory IV 2 Representation theory of finite groups

1 Introduction

1.1 Outline

The aim of this course is to study the representation theory of groups. If $G$ is a group, then a (finite-dimensional, complex) representation of $G$ is a homomorphism

\rho:G\rightarrow\operatorname{GL}_{n}(\mathbb{C})

for some $n\geq 0$ . We can view this as an action of $G$ on $\mathbb{C}^{n}$ . Two representations are equivalent if they are related by a change of basis. In the situations we will consider, every representation can be broken down into building blocks, called irreducible representations, and for a group $G$ , there are two basic (related) problems:

1.

classify the irreducible representations;
2.

construct representations and decompose them into irreducibles.

The first half of the course will be devoted to the representation theory of finite groups. Here we will develop character theory: the character of a representation $\rho$ is the function $g\mapsto\operatorname{tr}\rho(g)$ . It turns out that this function knows a lot about the representation — in some sense, it knows everything. We will develop methods to determine the irreducible characters of a finite group $G$ . They fit into a “character table”, which has many beautiful properties.

The second half of the course focuses on the representation theory of certain groups of matrices — called linear Lie groups — such as $\operatorname{SL}_{n}(\mathbb{R})$ . In this case, we require that the representations $\rho$ are continuous. These groups are naturally smooth manifolds and their tangent spaces at the identity have the extra structure of a Lie algebra. We will explain how to relate Lie groups to their Lie algebras via the exponential map; we may then study the representation theory of Lie groups via the representation theory of Lie algebras. In the case of $\operatorname{SL}_{2}(\mathbb{C})$ we will obtain an essentially complete understanding. In the case of $\operatorname{SL}_{3}(\mathbb{C})$ , we will make substantial progress. In both cases, the method is essentially to consider the action of the diagonal matrices, leading to the theory of weights.

1.2 References

These notes are based on those written by Jens Funke for a previous iteration of this course. Those in turn are based on multiple sources, especially [MR1153249] and [MR2553682]. Here is a brief rundown of a few references you could look at:

•

We rely a lot on “Fulton and Harris”: [MR1153249]. This book takes the point of view that examples should come before theory. It has lots of good exercises. It would be particularly useful for the $\mathfrak{sl}_{3,\mathbb{C}}$ -theory. It doesn’t discuss Lie groups (as opposed to algebras) much.
•

Kosmann–Schwarzbach’s book [MR2553682] is a relatively short source, and is good for its concrete discussion of representations of Lie groups.
•

Serre’s book [MR0450380] is unusual in that it develops the theory of finite group representation theory in three chapters at three different levels: the first part, aimed at chemistry students (!) covers character theory and is very concrete; the second takes a more abstract point of view and proves the main theoretical results, while the third deals with issues that come up when you look at representations over fields other than $\mathbb{C}$ .
•

James and Liebeck, [MR1864147], is an elementary and accessible book on finite group representation theory, with a focus on character theory.
•

I like Hall, [MR3331229], for the theory of linear Lie groups, Lie algebras and representations. It is complete yet readable.

1.3 Background review

We very rapidly recall the material from Algebra II that we will be requiring in this course. We won’t be covering this in lectures but it is worth taking the time to make sure you remember this stuff (and if there’s anything you haven’t seen before, please let me know!)

1.4 Linear algebra

1.4.1 Fields

A field is a commutative ring in which every nonzero element is invertible (colloquially, a place where we can do normal arithmetic). Linear algebra takes place over fields. We will use the following examples:

•

the field $\mathbb{Q}$ of rational numbers
•

the field $\mathbb{R}$ of real numbers
•

the field $\mathbb{C}$ of complex numbers
•

the fields $\mathbb{F}_{p}$ of integers modulo $p$ , for $p$ prime.

1.4.2 Linear maps

If $V, W$ are vector spaces over a field $k$ , then a linear map $T:V\rightarrow W$ is a function such that

T(\lambda v+\mu w)=\lambda T(v)+\mu T(w)

for all $\lambda,\mu\in k$ and all $v,w\in V$ . If bases have been chosen for $V$ and $W$ (and they are finite dimensional), then every linear map can be written as a matrix. The linear map $T$ is invertible if it is a bijection, in which case its inverse $T^{-1}$ is also a linear map. We write $\operatorname{Hom}(V,W)$ for the set of linear maps from $V$ to $W$ , which is also vector space over $k$ . If $V=W$ , we write $\operatorname{End}(V)=\operatorname{Hom}(V,V)$ . We write $\operatorname{GL}(V)$ for the group of invertible linear maps from $V$ to itself ( $\operatorname{GL}$ stands for ’general linear’). If a basis of $V$ is chosen, and $\dim V=n$ , then $\operatorname{End}(V)$ is given by the $n\times n$ matrices over $k$ while $\operatorname{GL}(V)=\operatorname{GL}_{n}(k)$ is the group of $n\times n$ invertible matrices.

If $T:V\rightarrow W$ is a linear map, then its kernel and image are

\ker(T)=\{v\in V:Tv=0\}

and

\operatorname{im}(T)=\{w\in W:w=Tv\text{ for some $v\in V$}\}.

1.4.3 Subspaces, quotients, sums

A subspace of $V$ is a subset $W$ closed under addition and scalar multiplication.

If $W$ is a subspace of $V$ , then the quotient space $V/W$ is the set of cosets (for addition) of $W$ in $V$ . We denote its elements by

\overline{v}=v+W=\{v+w:w\in W\}.

In this situation, the map $V\rightarrow V/W$ sending $v$ to $\overline{v}$ is a surjective linear map whose kernel is $W$ . If $T:V\rightarrow U$ is a linear map, then the map $T:V/\ker(T)\rightarrow U$ taking $\overline{v}$ to $T(v)$ gives a well-defined isomorphism from $V/\ker(T)$ to $\operatorname{im}(T)$ . Compare the first isomorphism theorem in group theory, and also the rank-nullity theorem

\dim\ker(T)+\dim\operatorname{im}(T)=\dim V.

If $V$ and $W$ are two vector spaces, then their (external) direct sum is

V\oplus W=\{(v,w):v\in V,w\in W\}

with componentwise addition and scalar multiplication. If $V$ and $W$ are both subspaces of some common space $U$ , we say that $U$ is the internal direct sum of $V$ and $W$ if every element of $U$ can be written uniquely as $u=v+w$ for $v\in V$ , $w\in W$ . This is equivalent to requiring $U=V+W$ and $V\cap W=\{0\}$ , or to requiring that the map

V\oplus W\rightarrow U

sending

(v,w)\mapsto v+w

is an isomorphism. Often in this situation we will simply say that $U$ is the direct sum of $V$ and $W$ .

We can generalise this to more than one subspace. If $V_{1},\ldots,V_{n}$ are subspaces of $V$ , then we say that $V$ is their internal direct sum if every element of $V$ can be written uniquely as $v_{1}+\ldots+v_{n}$ with $v_{i}\in V_{i}$ for all $i$ . Equivalently, if the map

	$\displaystyle V_{1}\oplus\ldots\oplus V_{n}$	$\displaystyle\rightarrow V$
	$\displaystyle(v_{1},\ldots,v_{n})$	$\displaystyle\mapsto v_{1}+\ldots+v_{n}$

is an isomorphism.

1.4.4 Eigenspaces

If $T:V\rightarrow V$ is a linear map from a vector space to itself, then an eigenvector of $T$ with eigenvalue $\lambda$ is a non-zero vector $v\in V$ such that $Tv=\lambda v$ .

The linear map $T$ is diagonalizable if there is a basis of $V$ consisting of eigenvectors of $T$ . This is equivalent to there being a basis for which the matrix of $T$ is diagonal.

For later use, we record the following theorem from linear algebra: if $T_{1},\ldots,T_{n}$ are linear maps $V\rightarrow V$ that commute with each other and that are diagonalizable, then there is a basis of $V$ consisting of simultaneous eigenvectors of the $T_{i}$ . Equivalently, a basis for which the matrices of the $T_{i}$ are all diagonal.

1.5 Group theory

1.5.1 Subgroups, cosets, quotients

If $G$ is a group then a subgroup $H$ is a subset containing the identity, closed under the group law and taking inverses. If $H$ is a subgroup, then a left coset of $H$ in $G$ is a subset $gH=\{gh:h\in H\}$ . The left cosets partition $G$ , and we write $G/H$ for the set of left cosets (not a group!). Similarly we define right cosets and $H\backslash G$ .

A subgroup $H$ is normal if, for every $g\in G$ , $gHg^{-1}=H$ . Equivalently, $gH=Hg$ for all $g\in G$ . In this case, the rule

(gH)(g^{\prime}H)=(gg^{\prime})H

defines a group law on $G/H=H\backslash G$ (the quotient group).

1.5.2 Homomorphisms

If $G$ and $G^{\prime}$ are groups, a homomorphism $f:G\rightarrow G^{\prime}$ is a function such that $f(gh)=f(g)f(h)$ . The kernel of $f$ is $\ker(f)=\{g\in G:f(g)=e\}$ and the image is $\operatorname{im}(f)=\{f(g):g\in G\}$ and these are subgroups of $G$ and $H$ respectively. The subgroup $\ker(f)$ is normal — in fact, normal subgroups are precisely those that are the kernel of some homomorphism.

A homomorphism is injective if and only if its kernel is trivial. A bijective homomorphism is called an isomorphism, in which case the inverse is also an isomorphism and we say that the groups are isomorphic.

The first (and best) isomorphism theorem states that the map

	$\displaystyle G/\ker(f)$	$\displaystyle\rightarrow\operatorname{im}(f)$
	$\displaystyle g\ker(f)$	$\displaystyle\mapsto f(g)$

is an isomorphism.

1.5.3 Symmetric groups

The symmetric group $S_{n}$ is the group of permutations of $\{1,\ldots,n\}$ . We use cycle notation, so that (e.g.) $(1253)$ is the permutation taking $1$ to $2$ , $2$ to $5$ , $5$ to $3$ , and $3$ to $1$ . Every permutation can be written uniquely (up to changing the order of the factors) as a product of disjoint cycles. We don’t bother writing cycles of length one, for example

(18)(2475)\in S_{8}

fixes the elements $3$ and $6$ . The sequence of lengths of the cycles appearing (including those of length one!), written in decreasing order, is called the cycle type of the permutation. The elements of given cycle type make up a single conjugacy class of $S_{n}$ (see below for conjugacy classes).

There is a homomorphism $\epsilon:S_{n}\rightarrow\pm{1}$ uniquely determined by the property that it takes transpositions $(ij)$ to $--1$ . It is called the sign homomorphism. Its kernel is the alternating group $A_{n}$ . The sign of an $n$ –cycle is $(-1)^{n+1}$ .

If $X$ is a set, we sometimes write $S_{X}$ for the group of permutations of $X$ , so $S_{n}=S_{X}$ with $X=\{1,\ldots,n\}$ . If $|X|=n$ then $S_{X}\cong S_{n}$ , but the exact isomorphism depends on how we label the elements of $X$ by the numbers $1$ to $n$ .

1.5.4 Actions

A (left) action of a group $G$ on a set $X$ is a way of transforming an element $x$ by elements $g$ to produce $gx\in X$ , such that $(gh)x=g(hx)$ .

Exercise 1.1.

If $G$ acts on a set $X,$ show that the map

\rho:G\rightarrow S_{X}

defined by $\rho(g)(x)=gx$ is a group homomorphism.

In this case, if $x\in X$ then its stabiliser $\operatorname{stab}(x)=\{g\in G:gx=x\}$ is a subgroup. We also have the orbit $Gx=\operatorname{orb}{x}=\{gx:g\in G\}\subset X$ . One form of the orbit-stabiliser theorem states that the map

	$\displaystyle G/\operatorname{stab}(x)$	$\displaystyle\rightarrow Gx$
	$\displaystyle g\operatorname{stab}(x)\mapsto gx$

is a bijection. This implies (if $G$ is finite) that

|\operatorname{orb}{x}||\operatorname{stab}(x)|=|G|.

If $g\in G$ then the set of fixed points of $g$ is

X^{g}=\mathrm{Fix}(g)=\{x\in X:gx=x\}.

We write

X^{G}=\{x\in X:gx=x\text{ for all $g\in G$}\},

the set of fixed points of $G$ .

1.5.5 Zoo

We will need a few examples of groups:

•

the integers $\mathbb{Z}$
•

cyclic groups $C_{n}\cong\mathbb{Z}/n\mathbb{Z}$
•

symmetric groups $S_{n}$
•

alternating groups $A_{n}$
•

dihedral groups $D_{n}$ (symmetries of regular $n$ –gon)
•

the quaternion group $Q_{8}$ :

$\{\pm 1,\pm i,\pm j,\pm k\}$

with $i^{2}=j^{2}=k^{2}=ijk=-1$ .
•

the general linear group $\operatorname{GL}_{n}(k)$ (for $k$ a field) of invertible $n\times n$ matrices over $k$ [note that if $k=\mathbb{F}_{p}$ then this is finite!]
•

the special linear group $\operatorname{SL}_{n}(k)$ of matrices with determinant 1
•

the orthogonal group

$\operatorname{O}(n)=\{A\in\operatorname{GL}_{n}(\mathbb{R}):AA^{T}=I\},$

which is also the group of isometries of $\mathbb{R}^{n}$ (with its standard inner product) fixing the origin, and the subgroup $\mathrm{SO}(n)$ of elements of $\operatorname{O}(n)$ whose determinant is one (i.e. the group of rotations of $\mathbb{R}^{n}$ fixing the origin.)
•

the unitary and special unitary groups

$\mathrm{U}(n)=\{A\in\operatorname{GL}_{n}(\mathbb{C}):AA^{\dagger}=I\}$

where $A^{\dagger}=\overline{A}^{T}$ , with $\overline{\cdot}$ being complex conjugation, and

$\mathrm{SU}(n)=\{A\in\mathrm{U}(n):\det(A)=1\}.$

The group $\mathrm{U}(n)$ is the group of transformations of $\mathbb{C}^{n}$ fixing the standard Hermitian inner product.