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Representation Theory 2015-2016 Course Notes

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Representation Theory

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Part II — Representation Theory

Based on lectures by S. Martin

Notes taken by Dexter Chua

Lent 2016

These notes are not endorsed by the lecturers, and I have modified them (often significantly) after lectures. They are nowhere near accurate representations of what was actually lectured, and in particular, all errors are almost surely mine.

Linear Algebra and Groups, Rings and Modules are essential

Representations of finite groups Representations of groups on vector spaces, matrix representations. Equivalence of representations. Invariant subspaces and submodules. Irreducibility and Schur’s Lemma. Complete reducibility for finite groups. Irreducible representations of Abelian groups.

Character theory Determination of a representation by its character. The group algebra, conjugacy classes, and orthogonality relations. Regular representation. Permutation representations and their characters. Induced representations and the Frobenius reciprocity theorem. Mackey’s theorem. Frobenius’s Theorem. [12]

Arithmetic properties of characters Divisibility of the order of the group by the degrees of its irreducible characters. Burnside’spaqbtheorem. [2]

Tensor products Tensor products of representations and products of characters. The character ring. Tensor, symmetric and exterior algebras. [3]

Representations ofS 1 andSU 2 The groupsS 1 ,SU 2 andSO(3), their irreducible representations, complete reducibility. The Clebsch-Gordan formula. Compact groups. [4]

Further worked examples The characters of one of GL 2 (Fq), Snor the Heisenberg group. [3]

Contents II Representation Theory

0 Introduction Contents
1 Group actions
2 Basic definitions
3 Complete reducibility and Maschke’s theorem
4 Schur’s lemma
5 Character theory
6 Proof of orthogonality
7 Permutation representations
8 Normal subgroups and lifting
9 Dual spaces and tensor products of representations
- 9 Dual spaces
- 9 Tensor products
- 9 Powers of characters
- 9 Characters ofG×H
- 9 Symmetric and exterior powers
- 9 Tensor algebra
- 9 Character ring
10 Induction and restriction
11 Frobenius groups
12 Mackey theory
13 Integrality in the group algebra
14 Burnside’s theorem
15 Representations of compact groups
- 15 Representations of SU(2)
- 15 Representations of SO(3), SU(2) andU(2)
Index

1 Group actions II Representation Theory

1 Group actions

We start by reviewing some basic group theory and linear algebra.

Basic linear algebra Notation represents a field.

Usually, we takeF=C, but sometimes it can also beRorQ. These fields all have characteristic zero, and in this case, we call what we’re doingordinary representation theory. Sometimes, we will takeF=FporF ̄p, the algebraic closure ofFp. This is calledmodularrepresentation theory.

Notation writeV for a vector space overF— this will always be finite dimensional overF. We writeGL(V) for the group of invertible linear maps θ:V→V. This is a group with the operation given by composition of maps, with the identity as the identity map (and inverse by inverse).

Notation a finite-dimensional vector space overF. We writeEnd(V) for the endomorphism algebra, the set of all linear mapsV→V.

We recall a couple of facts from linear algebra: IfdimFV =n &lt;∞, we can choose a basise 1 ,···,enofV overF. So we can identifyV withFn. Then every endomorphismθ∈GL(V) corresponds to a matrixAθ= (aij)∈Mn(F) given by

θ(ej) =

∑

aijei.

In fact, we haveAθ∈GLn(F), the general linear group. It is easy to see the following:

Proposition groups,GL(V)∼=GLn(F), with the isomorphism given by θ7→Aθ.

Of course, picking a different basis ofV gives a different isomorphism to GLn(F), but we have the following fact:

Proposition 1 ,A 2 represent the same element ofGL(V) with respect to different bases if and only if they areconjugate, namely there is someX∈ GLn(F) such that A 2 =XA 1 X− 1.

Recall thattr(A) =

∑

iaii, whereA= (aij)∈Mn(F), is the trace ofA. A nice property of the trace is that it doesn’t notice conjugacy:

Proposition. tr(XAX− 1 ) = tr(A).

Hence we can define the trace of an operatortr(θ) =tr(Aθ), which is independent of our choice of basis. This is an important result. When we study representations, we will have matrices flying all over the place, which are scary. Instead, we often just look at the traces of these matrices. This reduces our problem of studying matrices to plain arithmetic.

1 Group actions II Representation Theory

When we have too many matrices, we get confused. So we want to put a matrix into a form as simple as possible. One of the simplest form a matrix can take is being diagonal. So we want to know something about diagonalizing matrices.

Propositionα∈GL(V), whereVis a finite-dimensional vector space over Candαm= id for some positive integerm. Thenαis diagonalizable.

Proposition. LetV be a finite-dimensional vector space overC, andα∈ End(V), not necessarily invertible. Thenαis diagonalizable if and only if there is a polynomialfwith distinct linear factors such thatf(α) = 0.

This proposition allows us to prove the previous proposition immediately, noting thatxm−1 =

∏

(x−ωj), whereω=e 2 πi/m. Instead of just one endomorphism, we can look at many endomorphisms.

Proposition finite family of individually diagonalizable endomorphisms of a vector space overCcan be simultaneously diagonalized if and only if they commute.

Basic group theory

We will not review the definition of a group. Instead, we look at some of our favorite groups, since they will be handy examples later on.

Definition(Symmetric groupSn).Thesymmetric groupSnis the set of all permutations ofX={ 1 ,···,n}, i. the set of all bijectionsX→X. We have |Sn|=n!.

Definition(Alternating groupAn). Thealternating groupAnis the set of products of an even number of transpositions (i j) inSn. We know|An|=n 2 !. So this is a subgroup of index 2 and hence normal.

Definition(Cyclic groupCm).Thecyclic groupof orderm, writtenCmis

Cm=〈x:xm= 1〉.

This also occurs naturally, asZ/mZover addition, and also the group ofnth roots of unity inC. We can view this as a subgroup ofGL 1 (C)∼=C×. Alternatively, this is the group of rotation symmetries of a regularm-gon inR 2 , and can be viewed as a subgroup of GL 2 (R).

Definition(Dihedral groupD 2 m).Thedihedral groupD 2 mof order 2mis

D 2 m=〈x,y:xm=y 2 = 1,yxy− 1 =x− 1 〉.

This is the symmetry group of a regularm-gon. Thexiare the rotations and xiyare the reflections. For example, inD 8 ,xis rotation by π 2 andyis any reflection. This group can be viewed as a subgroup ofGL 2 (R), but since it also acts on the vertices, it can be viewed as a subgroup ofSm.

2 Basic definitions II Representation Theory

2 Basic definitions

We now start doing representation theory. We boringly start by defining a representation. In fact, we will come up with several equivalent definitions of a representation. As always,Gwill be a finite group andFwill be a field, usually C.

Definition(Representation).LetV be a finite-dimensional vector space over F. A(linear) representationofGonVis a group homomorphism

ρ=ρV:G→GL(V).

We sometimes writeρgforρV(g), so for eachg∈G,ρg∈GL(V), andρgρh=ρgh andρg− 1 = (ρg)− 1 for allg,h∈G.

Definition(Dimension or degree of representation).Thedimension(ordegree) of a representationρ:G→GL(V) is dimF(V).

Recall thatkerρ⊳GandG/kerρ∼=ρ(G)≤GL(V). In the very special case where kerρis trivial, we give it a name:

Definition(Faithful representation).Afaithfulrepresentation is a representa- tionρsuch that kerρ= 1.

These are the representations where the identity is the only element that does nothing. An alternative (and of course equivalent) definition of a representationis to observe that a linear representation is “the same” as a linear action ofG.

Definition(Linear action).A groupGacts linearlyon a vector spaceV if it acts onV such that

g(v 1 +v 2 ) =gv 1 +gv 2 , g(λv 1 ) =λ(gv 1 )

for allg∈G,v 1 ,v 2 ∈Vandλ∈F. We call this alinear action.

Now ifgacts linearly onV, the mapG→GL(V) defined byg7→ρg, withρg:v7→gv, is a representation in the previous sense. Conversely, given a representationρ:G→GL(V), we have a linear action ofGonVviagv=ρ(g)v. In other words, a representation is just a linear action.

Definition(G-space/G-module).If there is a linear actionGonV, we sayV is aG-spaceorG-module.

Alternatively, we can define aG-space as a module over a (not so) cleverly picked ring.

Definition(Group algebra).Thegroup algebraFGis defined to be the algebra (i. a vector space with a bilinear multiplication operation) of formalsums

FG=







∑

g∈G

αgg:αg∈F







with the obvious addition and multiplication.

2 Basic definitions II Representation Theory

Then we can regardFGas a ring, and aG-space is just anFG-module in the sense of IB Groups, Rings and Modules.

Definition(Matrix representation).Ris amatrix representationofGof degree nifRIs a homomorphismG→GLn(F).

We can view this as a representation that acts onFn. Since all finite- dimensional vector spaces are isomorphic toFnfor somen, every representation is equivalent to some matrix representation. In particular, given a linear repre- sentationρ:G→GL(V) withdimV=n, we can get a matrix representation by fixing a basisB, and then define the matrix representationG→GLn(F) by g7→[ρ(g)]B. Conversely, given a matrix representationR, we get a linear representationρ in the obvious way —ρ:G→GL(Fn) byg7→ρgviaρg(v) =Rgv. We have defined representations in four ways — as a homomorphism to GL(V), as linear actions, asFG-modules and as matrix representations. Now let’s look at some examples.

Example(Trivial representation).Given any groupG, takeV=F(the one- dimensional space), andρ:G→GL(V) byg7→(id:F→F) for allg. This is thetrivial representationofG, and has degree 1.

Despite being called trivial, trivial representations are highly non-trivial in representation theory. The way they interact with other representations geometrically, topologically etc, and cannot be disregarded. This is a very important representation, despite looking silly.

Example=C 4 =〈x:x 4 = 1〉. Letn= 2, and work overF=C. Then we can define a representation by picking a matrixA, and then defineR:x7→A. Then the action of other elements follows directly byxj7→Aj. Of course, we cannot chooseAarbitrarily. We need to haveA 4 =I 2 , and this is easily seen to be the only restriction. So we have the following possibilities:

(i)Ais diagonal: the diagonal entries can be chosen freely from{± 1 ,±i}. Since there are two diagonal entries, we have 16 choices.

(ii) Ais not diagonal: then it will be equivalent to a diagonal matrix since A 4 =I 2. So we don’t really get anything new.

What we would like to say above is that any matrix representation in which Xis not diagonal is “equivalent” to one in whichXis. To make this notion precise, we need to define what it means for representations to be equivalent, or “isomorphic”. As usual, we will define the notion of a homomorphism of representations, and then an isomorphism is just an invertible homomorphism.

Definition(G-homomorphism/intertwine).Fix a groupGand a fieldF. Let V,V′be finite-dimensional vector spaces overFandρ :G→GL(V) and ρ′:G→GL(V′) be representations ofG. The linear mapφ:V →V′is a G-homomorphismif φ◦ρ(g) =ρ′(g)◦φ (∗)

2 Basic definitions II Representation Theory

Thus, in terms of matrix representations, the representationsR:G→GLn(F) andR′:G→GLn(F) areG-isomorphic if there exists some non-singular matrix X∈GLn(F) such that R′(g) =XR(g)X− 1 for allg. Alternatively, in terms of linearG-actions, the actions ofGonVandV′are G-isomorphic if there is some isomorphismφ:V→V′such that

gφ(v) =φ(gv).

for allg∈G,v∈V. It is an easy check that this is just a reformulation of our previous definition. Just as we have subgroups and subspaces, we have the notion of sub- representation.

Definition(G-subspace).Letρ:G→GL(V) be a representation ofG. We sayW≤Vis aG-subspaceif it is a subspace that isρ(G)-invariant, i.

ρg(W)≤W

for allg∈G.

Obviously,{ 0 }andVareG-subspaces. These are the trivialG-subspaces.

Definition(Irreducible/simple representation).A representationρisirreducible orsimpleif there are no proper non-zeroG-subspaces.

Example 1-dimensional representation ofGis necessarily irreducible, but the converse does not hold, or else life would be very boring. We will later see thatD 8 has a two-dimensional irreducible complex representation.

Definition(Subrepresentation).IfWis aG-subspace, then the corresponding mapG→GL(W) given byg7→ρ(g)|W gives us a new representation ofW. This is asubrepresentationofρ.

There is a nice way to characterize this in terms of matrices.

Lemmaρ:G→GL(V) be a representation, andWbe aG-subspace of V. IfB={v 1 ,···,vn}is a basis containing a basisB 1 ={v 1 ,···,vm}ofW (with 0&lt; m &lt; n), then the matrix ofρ(g) with respect toBhas the block upper triangular form ( ∗ ∗ 0 ∗

)

for eachg∈G.

This follows directly from definition. However, we do not like block triangular matrices. What we really likeis block diagonal matrices, i. we want the top-right block to vanish. There is noa priorireason why this has to be true — it is possible that we cannot find anotherG-invariant complement toW.

2 Basic definitions II Representation Theory

Definition((In)decomposable representation).A representationρ:G→GL(V) isdecomposableif there are properG-invariant subspacesU,W≤Vwith V=U⊕W.

We sayρis a direct sumρu⊕ρw. If no such decomposition exists, we say thatρisindecomposable. It is clear that irreducibility implies indecomposability. The converse is not necessarily true. However, over a field of characteristic zero, it turns out irreducibility is the same as indecomposability for finite groups, as we will see in the next chapter. Again, we can formulate this in terms of matrices. Lemmaρ:G→GL(V) be a decomposable representation withG-invariant decompositionV=U⊕W. LetB 1 ={u 1 ,···,uk}andB 2 ={w 1 ,···,wℓ}be bases forUandW, andB=B 1 ∪B 2 be the corresponding basis forV. Then with respect toB, we have

[ρ(g)]B=

(

[ρu(g)]B 1 0 0 [ρu(g)]B 2

)

Example=D 6. Then every irreducible complex representation has dimension at most 2. To show this, letρ:G→GL(V) be an irreducibleG-representation. Let r∈Gbe a (non-identity) rotation ands∈Gbe a reflection. These generateD 6. Take an eigenvectorvofρ(r). Soρ(r)v=λvfor someλ 6 = 0 (sinceρ(r) is invertible, it cannot have zero eigenvalues). Let W=〈v,ρ(s)v〉≤V be the space spanned by the two vectors. We now check this is fixed byρ. Firstly, we have ρ(s)ρ(s)v=ρ(e)v=v∈W, and ρ(r)ρ(s)v=ρ(s)ρ(r− 1 )v=λ− 1 ρ(s)v∈W.

Also, ρ(r)v=λv∈W andρ(s)v∈W. SoW isG-invariant. SinceV is irreducible, we must haveW=V. SoV has dimension at most 2. The reverse operation of decomposition is taking direct sums. Definition(Direct sum). Letρ :G→GL(V) and ρ′ :G→GL(V′) be representations ofG. Then thedirect sumofρ,ρ′is the representation ρ⊕ρ′:G→GL(V⊕V′) given by (ρ⊕ρ′)(g)(v+v′) =ρ(g)v+ρ′(g)v′. In terms of matrices, for matrix representationsR:G→GLn(F) and R′:G→GLn′(F), defineR⊕R′:G→GLn+n′(F) by

(R⊕R′)(g) =

(

R(g) 0 0 R′(g)

)

.

The direct sum was easy to define. It turns out we can also multiply two representations, known as the tensor products. However, to do this, we need to know what the tensor product of two vector spaces is. We will not do this yet.

3 Complete reducibility and Maschke’s theorem II Representation Theory

Next, we claim that forw∈W, we haveq ̄(w) =w. This follows from the fact thatqitself fixesW. SinceW isG-invariant, we haveg− 1 w∈Wfor all w∈W. So we get

q ̄(w) =

1 |G|

∑

g∈G

gq(g− 1 w) =

1 |G|

∑

g∈G

gg− 1 w=

1 |G|

∑

g∈G

w=w.

Putting these together, this tells us ̄qis a projection ontoW. Finally, we claim that forh∈G, we haveh ̄q(v) = ̄q(hv), i. it is invariant under theG-action. This follows easily from definition:

h ̄q(v) =h

1 |G|

∑

g∈G

gq(g− 1 v)

=

1 |G|

∑

g∈G

hgq(g− 1 v)

=

1 |G|

∑

g∈G

(hg)q((hg)− 1 hv)

We now putg′=hg. Sincehis invertible, summing over allgis the same as summing over allg′. So we get

1 |G|

∑

g′∈G

g′q(g′− 1 (hv))

= ̄q(hv).

We are pretty much done. We finally show thatker ̄qisG-invariant. Ifv∈ker ̄q andh∈G, then ̄q(hv) =hq ̄(v) = 0. Sohv∈ker ̄q. Thus V= im ̄q⊕ker ̄q=W⊕ker ̄q is aG-subspace decomposition.

The crux of the whole proof is the definition of ̄q. Once we have that, everything else follows easily. Yet, for the whole proof to work, we need|G 1 |to exist, which in particular meansGmust be a finite group. There is no obvious way to generalize this to infinite groups. So let’s try a different proof. The second proof uses inner products, and hence we must takeF=C. This can be generalized to infinite compact groups, as we will later see. Recall the definition of an inner product: Definition(Hermitian inner product). ForV a complex space,〈·,·〉is a Hermitian inner productif

(i)〈v,w〉=〈w,v〉 (Hermitian)

(ii) 〈v,λ 1 w 1 +λ 2 w 2 〉=λ 1 〈v,w 1 〉+λ 2 〈v,w 2 〉 (sesquilinear)

(iii)〈v,v〉&gt;0 ifv 6 = 0 (positive definite) Definition(G-invariant inner product).An inner product〈·,·〉is in addition G-invariantif 〈gv,gw〉=〈v,w〉.

3 Complete reducibility and Maschke’s theorem II Representation Theory

Proposition-invariant subspace ofV, andV have aG-invariant inner product. ThenW⊥is alsoG-invariant.

Proof prove this, we have to show that for allv∈W⊥,g∈G, we have gv∈W⊥. This is not hard. We knowv∈W⊥if and only if〈v,w〉= 0 for allw∈W. Thus, using the definition ofG-invariance, forv∈W⊥, we know

〈gv,gw〉= 0

for allg∈G,w∈W. Thus for allw′∈W, pickw=g− 1 w′∈W, and this shows〈gv,w′〉= 0. Hencegv∈W⊥.

Hence if there is aG-invariant inner product on any complexG-spaceV, then we get another proof of Maschke’s theorem.

Theorem(Weyl’s unitary trick).Letρbe a complex representation of a finite groupGon the complex vector spaceV. Then there is aG-invariant Hermitian inner product onV.

Recall that the unitary group is defined by

U(V) ={f∈GL(V) :〈f(u),f(v)〉=〈u,v〉for allu,v∈V} ={A∈GLn(C) :AA†=I} = U(n).

Then we have an easy corollary:

Corollary finite subgroup ofGLn(C) is conjugate to a subgroup of U(n).

Proof start by defining an arbitrary inner product onV: take a basis e 1 ,···,en. Define (ei,ej) =δij, and extend it sesquilinearly. Define a new inner product 〈v,w〉=

1 |G|

∑

g∈G

(gv,gw).

We now check this is sesquilinear, positive-definite andG-invariant. Sesquilin- earity and positive-definiteness are easy. So we just checkG-invariance: we have

〈hv,hw〉=

1 |G|

∑

g∈G

((gh)v,(gh)w)

=

1 |G|

∑

g′∈G

(g′v,g′w)

=〈v,w〉.

Note that this trick also works for real representations. Again, we had to take the inverse|G 1 |. To generalize this to compact groups,

we will later replace the sum by an integral, and|G 1 |by a volume element. This is

4 Schur’s lemma II Representation Theory

4 Schur’s lemma

The topic of this chapter isSchur’s lemma, an easy yet extremely useful lemma in representation theory.

Theorem(Schur’s lemma).

(i)AssumeV andW are irreducibleG-spaces over a fieldF. Then any G-homomorphismθ:V→Wis either zero or an isomorphism.

(ii) IfFis algebraically closed, andV is an irreducibleG-space, then any G-endomorphismV→V is a scalar multiple of the identity mapιV.

Proof.

(i)Letθ:V→Wbe aG-homomorphism between irreducibles. Thenkerθis aG-subspace ofV, and sinceVis irreducible, eitherkerθ= 0 orkerθ=V. Similarly,imθis aG-subspace ofW, and asWis irreducible, we must haveimθ= 0 orimθ=W. Hence eitherkerθ=V, in which caseθ= 0, or kerθ= 0 and imθ=W, i.θis a bijection.

(ii) SinceFis algebraically closed,θhas an eigenvalueλ. Thenθ−λιVis a singularG-endomorphism ofV. So by (i), it must be the zero map. So θ=λιV.

Recall that theF-spaceHomG(V,W) is the space of allG-homomorphisms V→W. IfV=W, we write EndG(V) for theG-endomorphisms ofV.

Corollary,Ware irreducible complexG-spaces, then

dimCHomG(V,W) =

{

1 V,WareG-isomorphic 0 otherwise

Proof andW are not isomorphic, then the only possible map between them is the zero map by Schur’s lemma. Otherwise, supposeV ∼=W and letθ 1 ,θ 2 ∈HomG(V,W) be both non- zero. By Schur’s lemma, they are isomorphisms, and hence invertible. So θ 2 − 1 θ 1 ∈EndG(V). Thusθ− 21 θ 1 =λιVfor someλ∈C. Thusθ 1 =λθ 2.

We have another less obvious corollary.

Corollary a finite group and has a faithful complex irreducible repre- sentation, then its centerZ(G) is cyclic.

This is a useful result — it allows us transfer our representation-theoretic knowledge (the existence of a faithful complex irreducible representation) to group theoretic knowledge (center of group being cyclic). This willbecome increasingly common in the future, and is a good thing since representations are easy and groups are hard. The converse, however, is not true. For this, see example sheet 1, question 10.

4 Schur’s lemma II Representation Theory

Proofρ:G→GL(V) be a faithful irreducible complex representation. Let z∈Z(G). Sozg=gzfor allg∈G. Henceφz:v7→zvis aG-endomorphism onV. Hence by Schur’s lemma, it is multiplication by a scalarμz, say. Thus zv=μzvfor allv∈V. Then the map

σ:Z(G)→C× z7→μg

is a representation ofZ(G). Sinceρis faithful, so isσ. SoZ(G) ={μz:z∈ Z(G)}is isomorphic to a finite subgroup ofC×, hence cyclic.

Corollary irreducible complex representations of a finite abelian groupG are all 1-dimensional. Proof can use the fact that commuting diagonalizable matrices are simulta- neously diagonalizable. Thus for every irreducibleV, we can pick somev∈V that is an eigenvector for eachg∈G. Thus〈v〉is aG-subspace. AsV is irreducible, we must haveV=〈v〉. Alternatively, we can prove this in a representation-theoretic way. LetVbe an irreducible complex representation. For eachg∈G, the map

θg:V→V v7→gv

is aG-endomorphism ofV, since it commutes with the other group elements. SinceVis irreducible,θg=λgιVfor someλg∈C. Thus

gv=λgv

for anyg. AsVis irreducible, we must haveV=〈v〉.

Note that this result fails overR. For example,C 3 has a two irreducible real representations, one of dimension 1 and one of dimension 2. We can do something else. Recall that every finite abelian groupGisomorphic to a product of abelian groups. In fact, it can be written as a product ofCpα for various primespandα≥1, and the factors are uniquely determined up to order. This you already know from IB Groups Rings and Modules. You might be born knowing it — it’s such a fundamental fact of nature.

Proposition finite abelian groupG=Cn 1 ×···×Cnrhas precisely|G| irreducible representations overC.

This is not a coincidence. We will later show that the number of irreducible representations is the number of conjugacy classes of the group. In abelian groups, each conjugacy class is just a singleton, and hence this result.

Proof G=〈x 1 〉×···×〈xr〉,

where|xj|=nj. Any irreducible representationρmust be one-dimensional. So we have ρ:G→C×.

4 Schur’s lemma II Representation Theory

where V(λ) ={v∈V:α(v) =λv}.

This is canonical in that it depends onαalone, and nothing else. IfVis moreover aG-representation, how does this tie in to the decomposition ofVinto the irreducible representations? Let’s do an example.

Example=D 6 ∼=S 3 =〈r,s:r 3 =s 2 = 1,rs=sr− 1 〉. We have previously seen that each irreducible representation has dimension at most 2. We spot at least three irreducible representations:

1 triad r7→ 1 s7→ 1 S sign r7→ 1 s7→− 1 W 2-dimensional

The last representation is the action ofD 6 onR 2 in the natural way, i. the rotations and reflections of the plane that corresponds to the symmetries of the triangle. It is helpful to view this as a complex representation in order to make the matrix look nice. The 2-dimensional representation (ρ,W) is defined by W=C 2 , whererandsact onWas

ρ(r) =

(

ω 0 0 ω 2

)

, ρ(s) =

(

0 1

1 0

)

,

andω=e 2 πi/ 3 is a third root of unity. We will now show that these are indeed all the irreducible representations, by decomposing any representation into sum of these. So let’s decompose an arbitrary representation. Let (ρ′,V) be any complex representation ofG. Sinceρ′(r) has order 3, it is diagonalizable has eigenvalues 1 ,ω,ω 2. We diagonalizeρ′(r) and thenV splits as a vector space into the eigenspaces V=V(1)⊕V(ω)⊕V(ω 2 ). Sincesrs− 1 =r− 1 , we knowρ′(s) preservesV(1) and interchangesV(ω) and V(ω 2 ). Now we decomposeV(1) intoρ′(s) eigenspaces, with eigenvalues±1. Sincer has to act trivially on these eigenspaces,V(1) splits into sums of copies of the irreducible representations 1 and S. For the remaining mess, choose a basisv 1 ,···,vnofV(ω), and letv′j=

ρ′(s)vj. Thenρ′(s) acts on the two-dimensional space〈vj,v′j〉as

(

0 1

1 0

)

, while

ρ′(r) acts as

(

ω 0 0 ω 2

)

. This meansV(ω)⊕V(ω 2 ) decomposes into many copies of W.

We did this forD 6 by brute force. How do we generalize this? We first have the following lemma:

Lemma,V 1 ,V 2 beG-vector spaces overF. Then

(i) HomG(V,V 1 ⊕V 2 )∼=HomG(V,V 1 )⊕HomG(V,V 2 )

(ii) HomG(V 1 ⊕V 2 ,V)∼=HomG(V 1 ,V)⊕HomG(V 2 ,V).

4 Schur’s lemma II Representation Theory

Proof proof is to write down the obvious homomorphisms and inverses. Define the projection map

πi:V 1 ⊕V 2 →Vi,

which is theG-linear projection ontoVi. Then we can define theG-homomorphism

HomG(V,V 1 ⊕V 2 )7→HomG(V,V 1 )⊕HomG(V,V 2 ) φ7→(π 1 φ,π 2 φ).

Then the map (ψ 1 ,ψ 2 )7→ψ 1 +ψ 2 is an inverse. For the second part, we have the homomorphismφ7→(φ|V 1 ,φ|V 2 ) with inverse (ψ 1 ,ψ 2 )7→ψ 1 π 1 +ψ 2 π 2.

Lemma an algebraically closed field, andVbe a representation ofG. SupposeV=

⊕n i=1Viis its decomposition into irreducible components. Then for each irreducible representationSofG,

|{j:Vj∼=S}|= dim HomG(S,V).

This tells us we can count the multiplicity ofS inV by looking at the homomorphisms.

Proof induct onn. Ifn= 0, then this is a trivial space. Ifn= 1, thenV itself is irreducible, and by Schur’s lemma,dim HomG(S,V) = 1 ifV =S, 0 otherwise. Otherwise, forn &gt;1, we have

V=

(n− 1 ⊕

)

⊕Vn.

By the previous lemma, we know

dim homG

(

S,

(n− 1 ⊕

)

⊕Vn

)

= dim HomG

(

S,

n⊕− 1

)

dim homG(S,Vn).

The result then follows by induction.

Definition(Canonical decomposition/decomposition into isotypical compo- nents).A decomposition ofV as

⊕

Wj, where eachWjis (isomorphic to)nj copies of the irreducibleSj(withSj 6 ∼=Sifori 6 =j) is thecanonical decomposition ordecomposition into isotypical components.

For an algebraically closed fieldF, we know we must have

nj= dim HomG(Sj,V),

and hence this decomposition is well-defined. We’ve finally all the introductory stuff. The course now begins.

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Representation Theory 2015-2016 Course Notes

Module: Representation Theory

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Part II — Representation Theory

Based on lectures by S. Martin

Notes taken by Dexter Chua

Lent 2016

These notes are not endorsed by the lecturers, and I have modified them (often

significantly) after lectures. They are nowhere near accurate representations of what

was actually lectured, and in particular, all errors are almost surely mine.

Linear Algebra and Groups, Rings and Modules are essential

Representations of finite groups

Representations of groups on vector spaces, matrix representations. Equivalence of

representations. Invariant subspaces and submodules. Irreducibility and Schur’s

Lemma. Complete reducibility for finite groups. Irreducible representations of Abelian

groups.

Character theory

Determination of a representation by its character. The group algebra, conjugacy classes,

and orthogonality relations. Regular representation. Permutation representations and

their characters. Induced representations and the Frobenius reciprocity theorem.

Mackey’s theorem. Frobenius’s Theorem. [12]

Arithmetic properties of characters

Divisibility of the order of the group by the degrees of its irreducible characters.

Burnside’s paqbtheorem. [2]

Tensor products

Tensor products of representations and products of characters. The character ring.

Tensor, symmetric and exterior algebras. [3]

Representations of S1and SU2

The groups

SU2

and

(3), their irreducible representations, complete reducibility.

The Clebsch-Gordan formula. *Compact groups.* [4]

Further worked examples

The characters of one of GL2(Fq), Snor the Heisenberg group. [3]