Part II — Representation Theory Theorems

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 paqb theorem. [2]

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

1 Representations of S and SU2 1 The groups S , SU2 and SO(3), their irreducible representations, complete reducibility. The Clebsch-Gordan formula. *Compact groups.* [4]

Further worked examples

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

1 Contents II Representation Theory (Theorems)

Contents

0 Introduction 3

1 Group actions 4

2 Basic deﬁnitions 5

3 Complete reducibility and Maschke’s theorem 6

4 Schur’s lemma 7

5 Character theory 8

6 Proof of orthogonality 10

7 Permutation representations 11

8 Normal subgroups and lifting 12

9 Dual spaces and tensor products of representations 13 9.1 Dual spaces ...... 13 9.2 Tensor products ...... 13 9.3 Powers of characters ...... 13 9.4 Characters of G × H ...... 14 9.5 Symmetric and exterior powers ...... 14 9.6 Tensor algebra ...... 14 9.7 Character ring ...... 14

10 Induction and restriction 15

11 Frobenius groups 16

12 Mackey theory 17

13 Integrality in the group algebra 19

14 Burnside’s theorem 20

15 Representations of compact groups 21 15.1 Representations of SU(2) ...... 21 15.2 Representations of SO(3), SU(2) and U(2) ...... 23

2 0 Introduction II Representation Theory (Theorems)

0 Introduction

3 1 Group actions II Representation Theory (Theorems)

1 Group actions ∼ Proposition. As groups, GL(V ) = GLn(F), with the isomorphism given by θ 7→ Aθ.

Proposition. Matrices A1,A2 represent the same element of GL(V ) with respect to diﬀerent bases if and only if they are conjugate, namely there is some X ∈ GLn(F) such that −1 A2 = XA1X . Proposition. tr(XAX−1) = tr(A).

Proposition. Let α ∈ GL(V ), where V is a finite-dimensional vector space over C and αm = id for some positive integer m. Then α is diagonalizable. Proposition. Let V be a finite-dimensional vector space over C, and α ∈ End(V ), not necessarily invertible. Then α is diagonalizable if and only if there is a polynomial f with distinct linear factors such that f(α) = 0. Proposition. A finite family of individually diagonalizable endomorphisms of a vector space over C can be simultaneously diagonalized if and only if they commute. Lemma. Given an action of G on X, we obtain a homomorphism θ : G → Sym(X), where Sym(X) is the set of all permutations of X.

4 2 Basic deﬁnitions II Representation Theory (Theorems)

2 Basic deﬁnitions

Lemma. The relation of “being isomorphic” is an equivalence relation on the set of all linear representations of G over F. Lemma. If ρ, ρ0 are isomorphic representations, then they have the same dimension. Lemma. Let ρ : G → GL(V ) be a representation, and W be a G-subspace of V . If B = {v1, ··· , vn} is a basis containing a basis B1 = {v1, ··· , vm} of W (with 0 < m < n), then the matrix of ρ(g) with respect to B has the block upper triangular form ∗ ∗ 0 ∗ for each g ∈ G. Lemma. Let ρ : G → GL(V ) be a decomposable representation with G-invariant decomposition V = U ⊕ W . Let B1 = {u1, ··· , uk} and B2 = {w1, ··· , w`} be bases for U and W , and B = B1 ∪ B2 be the corresponding basis for V . Then with respect to B, we have [ρu(g)]B1 0 [ρ(g)]B = 0 [ρu(g)]B2

5 3 Complete reducibility and Maschke’sII theorem Representation Theory (Theorems)

3 Complete reducibility and Maschke’s theorem

Theorem. Every finite-dimensional representation V of a finite group over a ∼ field of characteristic 0 is completely reducible, namely, V = V1 ⊕ · · · ⊕ Vr is a direct sum of irreducible representations.

Theorem (Maschke’s theorem). Let G be a finite group, and ρ : G → GL(V ) a representation over a finite-dimensional vector space V over a field F with char F = 0. If W is a G-subspace of V , then there exists a G-subspace U of V such that V = W ⊕ U. Proposition. Let W be G-invariant subspace of V , and V have a G-invariant inner product. Then W ⊥ is also G-invariant. Theorem (Weyl’s unitary trick). Let ρ be a complex representation of a finite group G on the complex vector space V . Then there is a G-invariant Hermitian inner product on V .

Corollary. Every finite subgroup of GLn(C) is conjugate to a subgroup of U(n). Proposition. Let ρ be an irreducible representation of the finite group G over a field of characteristic 0. Then ρ is isomorphic to a subrepresentation of ρreg.

6 4 Schur’s lemma II Representation Theory (Theorems)

4 Schur’s lemma

Theorem (Schur’s lemma).

(i) Assume V and W are irreducible G-spaces over a ﬁeld F. Then any G-homomorphism θ : V → W is either zero or an isomorphism.

(ii) If F is algebraically closed, and V is an irreducible G-space, then any G-endomorphism V → V is a scalar multiple of the identity map ιV . Corollary. If V,W are irreducible complex G-spaces, then ( 1 V,W are G-isomorphic dim HomG(V,W ) = C 0 otherwise

Corollary. If G is a ﬁnite group and has a faithful complex irreducible representation, then its center Z(G) is cyclic. Corollary. The irreducible complex representations of a ﬁnite abelian group G are all 1-dimensional.

Proposition. The ﬁnite abelian group G = Cn1 × · · · × Cnr has precisely |G| irreducible representations over C.

Lemma. Let V,V1,V2 be G-vector spaces over F. Then ∼ (i) HomG(V,V1 ⊕ V2) = HomG(V,V1) ⊕ HomG(V,V2) ∼ (ii) HomG(V1 ⊕ V2,V ) = HomG(V1,V ) ⊕ HomG(V2,V ).

Lemma. Let F be an algebraically closed ﬁeld, and V be a representation of G. Ln Suppose V = i=1 Vi is its decomposition into irreducible components. Then for each irreducible representation S of G, ∼ |{j : Vj = S}| = dim HomG(S, V ).

7 5 Character theory II Representation Theory (Theorems)

5 Character theory

Theorem.

(i) χV (1) = dim V .

(ii) χV is a class function, namely it is conjugation invariant, i.e.

−1 χV (hgh ) = χV (g)

for all g, h ∈ G. Thus χV is constant on conjugacy classes.

−1 (iii) χV (g ) = χV (g). (iv) For two representations V,W , we have

χV ⊕W = χV + χW .

Lemma. Let ρ : G → GL(V ) be a complex representation aﬀording the character χ. Then |χ(g)| ≤ χ(1), with equality if and only if ρ(g) = λI for some λ ∈ C, a root of unity. Moreover, χ(g) = χ(1) if and only if g ∈ ker ρ. Lemma. (i) If χ is a complex (irreducible) character of G, then so isχ ¯. (ii) If χ is a complex (irreducible) character of G, then so is εχ for any linear (1-dimensional) character ε. Theorem (Completeness of characters). The complex irreducible characters of G form an orthonormal basis of C(G), namely (i) If ρ : G → GL(V ) and ρ0 : G → GL(V 0) are two complex irreducible representations aﬀording characters χ, χ0 respectively, then ( 1 if ρ and ρ0 are isomorphic representations hχ, χ0i = 0 otherwise

This is the (row) orthogonality of characters. (ii) Each class function of G can be expressed as a linear combination of irreducible characters of G. Corollary. Complex representations of ﬁnite groups are characterised by their characters. Corollary (Irreducibility criterion). If ρ : G → GL(V ) is a complex representation of G aﬀording the character χ, then ρ is irreducible if and only if hχ, χi = 1.

Theorem. Let ρ1, ··· , ρk be the irreducible complex representations of G, and let their dimensions be n1, ··· , nk. Then

X 2 |G| = ni .

8 5 Character theory II Representation Theory (Theorems)

Corollary. The number of irreducible characters of G (up to equivalence) is k, the number of conjugacy classes.

Corollary. Two elements g1, g2 are conjugate if and only if χ(g1) = χ(g2) for all irreducible characters χ of G.

9 6 Proof of orthogonality II Representation Theory (Theorems)

6 Proof of orthogonality

Theorem (Row orthogonality relations). If ρ : G → GL(V ) and ρ0 : G → GL(V 0) are two complex irreducible representations aﬀording characters χ, χ0 respectively, then ( 1 if ρ and ρ0 are isomorphic representations hχ, χ0i = . 0 otherwise

Theorem (Column orthogonality relations). We have

k X χi(gj)χi(g`) = δj`|CG(g`)|. i=1 Corollary. k X 2 |G| = χi (1). i=1 Theorem. Each class function of G can be expressed as a linear combination of irreducible characters of G.

10 7 Permutation representations II Representation Theory (Theorems)

7 Permutation representations

Lemma. πX always contains the trivial character 1G (when decomposed in the basis of irreducible characters). In particular, span{ex + ··· + ex } is a trivial P 1 P n G-subspace of CX, with G-invariant complement { x axex : ax = 0}.

Lemma. hπX , 1i, which is the multiplicity of 1 in πX , is the number of orbits of G on X.

Lemma. Let G act on the sets X1,X2. Then G acts on X1 × X2 by

g(x1, x2) = (gx1, gx2).

Then the character

πX1×X2 = πX1 πX2 , and so

hπX1 , πX2 i = number of orbits of G on X1 × X2. Lemma. Let G act on X, with |X| > 2. Then

πX = 1G + χ, with χ irreducible if and only if G is 2-transitive on X.

Lemma. Let g ∈ An, n > 1. If g commutes with some odd permutation in Sn,

then CSn (g) = CAn (g). Otherwise, CSn splits into two conjugacy classes in An of equal size.

11 8 Normal subgroups and lifting II Representation Theory (Theorems)

8 Normal subgroups and lifting

Lemma. Let N C G. Let ρ˜ : G/N → GL(V ) be a representation of G/N. Then the composition ρ˜ ρ : G natural G/N GL(V )

is a representation of G, where ρ(g) =ρ ˜(gN). Moreover,

(i) ρ is irreducible if and only ifρ ˜ is irreducible. (ii) The corresponding characters satisfy χ(g) =χ ˜(gN). (iii) deg χ = degχ ˜. (iv) The lifting operationχ ˜ 7→ χ is a bijection

{irreducibles of G/N} ←→ {irreducibles of G with N in their kernel}.

We sayχ ˜ lifts to χ. Lemma. Given a group G, the derived subgroup or commutator subgroup

G0 = h[a, b]: a, b ∈ Gi, where [a, b] = aba−1b−1, is the unique minimal normal subgroup of G such that G/G0 is abelian. So if G/N is abelian, then G0 ≤ N. Moreover, G has precisely ` = |G : G0| representations of dimension 1, all with kernel containing G0, and are obtained by lifting from G/G0. In particular, by Lagrange’s theorem, ` | G. Lemma. G is not simple if and only if χ(g) = χ(1) for some irreducible character χ 6= 1G and some 1 6= g ∈ G. Any normal subgroup of G is the intersection of T the kernels of some of the irreducible characters of G, i.e. N = ker χi.

12 9 Dual spaces and tensor products of representationsII Representation Theory (Theorems)

9 Dual spaces and tensor products of representations 9.1 Dual spaces

Lemma. Let ρ : G → GL(V ) be a representation over F, and let V ∗ = ∗ HomF(V, F) be the dual space of V . Then V is a G-space under (ρ∗(g)ϕ)(v) = ϕ(ρ(g−1)v).

∗ −1 This is the dual representation to ρ. Its character is χ(ρ )(g) = χρ(g ).

9.2 Tensor products Lemma.

(i) For v ∈ V , w ∈ W and λ ∈ F, we have (λv) ⊗ w = λ(v ⊗ w) = v ⊗ (λw).

(ii) If x, x1, x2 ∈ V and y, y1, y2 ∈ W , then

(x1 + x2) ⊗ y = (x1 ⊗ y) + (x2 ⊗ y)

x ⊗ (y1 + y2) = (x ⊗ y1) + (x ⊗ y2).

Lemma. Let {e1, ··· , em} be any other basis of V , and {f1, ··· , fm} be another basis of W . Then {ei ⊗ fj : 1 ≤ i ≤ m, 1 ≤ j ≤ n} is a basis of V ⊗ W . Proposition. Let ρ : G → GL(V ) and ρ0 : G → GL(V 0). We deﬁne

ρ ⊗ ρ0 : G → GL(V ⊗ V 0)

by 0 X X 0 (ρ ⊗ ρ )(g): λijvi ⊗ wj 7→ λij(ρ(g)vi) ⊗ (ρ (g)wj). Then ρ ⊗ ρ0 is a representation of g, with character

χρ⊗ρ0 (g) = χρ(g)χρ0 (g)

for all g ∈ G.

9.3 Powers of characters Lemma. For any G-space V , S2V and Λ2V are G-subspaces of V ⊗2, and

V ⊗2 = S2V ⊕ Λ2V.

The space S2V has basis

{vivj = vi ⊗ vj + vj ⊗ vi : 1 ≤ i ≤ j ≤ n},

13 9 Dual spaces and tensor products of representationsII Representation Theory (Theorems) while Λ2V has basis

{vi ∧ vj = vi ⊗ vj − vj ⊗ vi : 1 ≤ i < j ≤ n}.

Note that we have a strict inequality for i < j, since vi ⊗ vj − vj ⊗ vi = 0 if i = j. Hence 1 1 dim S2V = n(n + 1), dim Λ2V = n(n − 1). 2 2 Lemma. Let ρ : G → GL(V ) be a representation aﬀording the character χ. 2 2 Then χ = χS +χΛ where χS = S χ is the character of G in the subrepresentation 2 2 2 on S V , and χΛ = Λ χ the character of G in the subrepresentation on Λ V . Moreover, for g ∈ G, 1 1 χ (g) = (χ2(g) + χ(g2)), χ (g) = (χ2(g) − χ(g2)). S 2 Λ 2 9.4 Characters of G × H Proposition. Let G and H be two ﬁnite groups with irreducible characters χ1, ··· , χk and ψ1, ··· , ψr respectively. Then the irreducible characters of the direct product G × H are precisely

{χiψj : 1 ≤ i ≤ k, 1 ≤ j ≤ r}, where (χiψj)(g, h) = χi(g)ψj(h).

9.5 Symmetric and exterior powers 9.6 Tensor algebra 9.7 Character ring Lemma. Suppose α is a generalized character and hα, αi = 1 and α(1) > 0. Then α is actually a character of an irreducible representation of G.

14 10 Induction and restriction II Representation Theory (Theorems)

10 Induction and restriction

Lemma. Let H ≤ G. If ψ is any non-zero irreducible character of H, then there G exists an irreducible character χ of G such that ψ is a constituent of ResH χ, i.e.

G hResH χ, ψi= 6 0.

Lemma. Let χ be an irreducible character of G, and let

G X ResH χ = ciχi, i with χi irreducible characters of H, and ci non-negative integers. Then

X 2 ci ≤ |G : H|, with equality iﬀ χ(g) = 0 for all g ∈ G \ H.

G G Lemma. Let ψ ∈ CH . Then IndH ψ ∈ C(G), and IndH ψ(1) = |G : H|ψ(1).

Lemma. Given a (left) transversal t1, ··· , tn of H, we have

n G X ˚ −1 IndH ψ(g) = ψ(ti gti). i=1

Theorem (Frobenius reciprocity). Let ψ ∈ C(H) and ϕ ∈ C(G). Then

G G hResH ϕ, ψiH = hϕ, IndH ψiG.

G Corollary. Let ψ be a character of H. Then IndH ψ is a character of G. Proposition. Let ψ be a character of H ≤ G, and let g ∈ G. Let

m [ CG(g) ∩ H = CH (xi), i=1 where the xi are the representatives of the H conjugacy classes of elements of H G conjugate to g. If m = 0, then IndH ψ(g) = 0. Otherwise,

m X ψ(xi) IndG ψ(g) = |C (g)| . H G |C (x )| i=1 H i

G Lemma. Let ψ = 1H , the trivial character of H. Then IndH 1H = πX , the permutation character of G on the set X, where X = G/H is the set of left cosets of H.

15 11 Frobenius groups II Representation Theory (Theorems)

11 Frobenius groups

Theorem (Frobenius’ theorem (1891)). Let G be a transitive permutation group on a finite set X, with |X| = n. Assume that each non-identity element of G fixes at most one element of X. Then the set of fixed point-free elements (“derangements”)

K = {1} ∪ {g ∈ G : gα 6= α for all α ∈ X}

is a normal subgroup of G with order n. Proposition. The left action of any ﬁnite Frobenius group on the cosets of the Frobenius complement satisﬁes the hypothesis of Frobenius’ theorem.

16 12 Mackey theory II Representation Theory (Theorems)

12 Mackey theory

Proposition. Let G be a ﬁnite group and H,K ≤ G. Let g1, ··· , gk be the representatives of the double cosets K\G/H. Then

k G G ∼ M K ResK IndH 1H = Ind −1 1. giHgi ∩K i=1

Theorem (Mackey’s restriction formula). In general, for K,H ≤ G, we let S = {1, g1, ··· , gr} be a set of double coset representatives, so that [ G = KgiH.

−1 We write Hg = gHg ∩ K ≤ G. We let (ρ, W ) be a representation of H. For each g ∈ G, we deﬁne (ρg,Wg) to be a representation of Hg, with the same underlying vector space W , but now the action of Hg is

−1 ρg(x) = ρ(g xg), where h = g−1xg ∈ H by construction. This is clearly well-deﬁned. Since Hg ≤ K, we obtain an induced representation IndK W . Hg g Let G be ﬁnite, H,K ≤ G, and W be a H-space. Then M ResG IndG W = IndK W . K H Hg g g∈S

Corollary. Let ψ be a character of a representation of H. Then X ResG IndG ψ = IndK ψ , K H Hg g g∈S where ψg is the class function (and a character) on Hg given by

−1 ψg(x) = ψ(g xg).

Corollary (Mackey’s irreducibility criterion). Let H ≤ G and W be a H-space. G Then V = IndH W is irreducible if and only if (i) W is irreducible; and (ii) For each g ∈ S \H, the two H spaces W and ResH W have no irreducible g g Hg −1 constituents in common, where Hg = gHg ∩ H.

Corollary. Let H C G, and suppose ψ is an irreducible character of H. Then G IndH ψ is irreducible if and only if ψ is distinct from all its conjugates ψg for −1 g ∈ G \ H (where ψg(h) = ψ(g hg) as before). Theorem (Mackey’s restriction formula). In general, for K,H ≤ G, we let S = {1, g1, ··· , gr} be a set of double coset representatives, so that [ G = KgiH.

17 12 Mackey theory II Representation Theory (Theorems)

18 13 Integrality in the group algebra II Representation Theory (Theorems)

13 Integrality in the group algebra

Proposition.

(i) The algebraic integers form a subring of C.

(ii) If a ∈ C is both an algebraic integer and rational, then a is in fact an integer.

(iii) Any subring of C which is a ﬁnitely generated Z-module consists of algebraic integers.

Proposition. If χ is a character of G and g ∈ G, then χ(g) is an algebraic integer.

Proposition. The class sums C1, ··· ,Ck form a basis of Z(CG). There exists non-negative integers aij` (with 1 ≤ i, j, ` ≤ k) with

k X CiCj = aij`C`. `=1 Lemma. The values of χ(g) ω (C ) = |C | χ i χ(1) i are algebraic integers.

Theorem. The degree of any irreducible character of G divides |G|, i.e.

χj(1) | |G|

for each irreducible χj.

19 14 Burnside’s theorem II Representation Theory (Theorems)

14 Burnside’s theorem

Theorem (Burside’s paqb theorem). Let p, q be primes, and let |G| = paqb, where a, b ∈ Z≥0, with a + b ≥ 2. Then G is not simple. Lemma. Suppose m 1 X α = λ , m j j=1 n is an algebraic integer, where λj = 1 for all j and some n. Then either α = 0 or |α| = 1. Lemma. Suppose χ is an irreducible character of G, and C is a conjugacy class in G such that χ(1) and |C| are coprime. Then for g ∈ C, we have

|χ(g)| = χ(1) or 0.

Proposition. If in a ﬁnite group, the number of elements in a conjugacy class C is of (non-trivial) prime power order, then G is not non-abelian simple. Theorem (Burside’s paqb theorem). Let p, q be primes, and let |G| = paqb, where a, b ∈ Z≥0, with a + b ≥ 2. Then G is not simple.

20 15 Representations of compact groups II Representation Theory (Theorems)

15 Representations of compact groups

Theorem. Every one-dimensional (continuous) representation S1 is of the form

ρ : z 7→ zn

for some n ∈ Z. Lemma. If ψ :(R, +) → (R, +) is a continuous group homomorphism, then there exists a c ∈ R such that ψ(x) = cx

for all x ∈ R. Lemma. Continuous homomorphisms ϕ :(R, +) → S1 are of the form ϕ(x) = eicx

for some c ∈ R. Theorem. Every one-dimensional (continuous) representation S1 is of the form

ρ : z 7→ zn

for some n ∈ Z. Theorem. Let G be a compact Hausdorﬀ topological group. Then there exists a unique Haar measure on G. Corollary (Weyl’s unitary trick). Let G be a compact group. Then every representation (ρ, V ) has a G-invariant Hermitian inner product. Theorem (Maschke’s theoerm). Let G be compact group. Then every representation of G is completely reducible. Theorem (Orthogonality). Let G be a compact group, and V and W be irreducible representations of G. Then ( 1 V =∼ W hχV , χW i = . 0 V =6∼ W

15.1 Representations of SU(2) Lemma (SU(2)-conjugacy classes). (i) Let t ∈ T . Then sts−1 = t−1. (ii) s2 = −I ∈ Z(SU(2)). (iii) The normalizer

a 0 0 a N (T ) = T ∪ sT = , : a ∈ , |a| = 1 . G 0a ¯ −a¯ 0 C

(iv) Every conjugacy class C of SU(2) contains an element of T , i.e. C ∩ T 6= ∅.

21 15 Representations of compact groups II Representation Theory (Theorems)

(v) In fact, C ∩ T = {t, t−1} for some t ∈ T , and t = t−1 if and only if t = ±I, in which case C = {t}. (vi) There is a bijection

{conjugacy classes in SU(2)} ↔ [−1, 1],

given by 1 A 7→ tr A. 2 We can see that if λ 0 A = , 0 λ¯ then 1 1 tr A = (λ + λ¯) = Re(λ). 2 2 ∼ 2 Proposition. For t ∈ (−1, 1), the class Ct = S as topological spaces.

Lemma. A continuous class function f : G → C is determined by its restriction to T , and F |T is even, i.e.

λ 0 λ−1 0 f = f . 0 λ−1 0 λ

Lemma. If χ is a character of a representation of SU(2), then its restriction χ|T is a Laurent polynomial, i.e. a ﬁnite N-linear combination of functions λ 0 7→ λn 0 λ−1

for n ∈ Z.

Theorem. The representations ρn : SU(2) → GL(Vn) of dimension n + 1 are irreducible for n ∈ Z≥0. Theorem. Every ﬁnite-dimensional continuous irreducible representation of G is one of the ρn : G → GL(Vn) as deﬁned above. Proposition. Let G = SU(2) or G = S1, and V,W are representations of G. Then χV ⊗W = χV χW .

Proposition (Clebsch-Gordon rule). For n, m ∈ N, we have ∼ Vn ⊗ Vm = Vn+m ⊕ Vn+m−2 ⊕ · · · ⊕ V|n−m|+2 ⊕ V|n−m|.

22 15 Representations of compact groups II Representation Theory (Theorems)

15.2 Representations of SO(3), SU(2) and U(2) Proposition. There are isomorphisms of topological groups:

(i) SO(3) =∼ SU(2)/{±I} = PSU(2) (ii) SO(4) =∼ SU(2) × SU(2)/{±(I,I)} (iii) U(2) =∼ U(1) × SU(2)/{±(I,I)} All maps are group isomorphisms, but in fact also homeomorphisms. To show this, we can use the fact that a continuous bijection from a Hausdorﬀ space to a compact space is automatically a homeomorphism. Corollary. Every irreducible representation of SO(3) has the following form:

ρ2m : SO(3) → GL(V2m),

for some m ≥ 0, where Vn are the irreducible representations of SU(2). Proposition. SO(3) =∼ SU(2)/{±I}.

Proposition. The complete list of irreducible representations of SO(4) is ρm×ρn, where m, n > 0 and m ≡ n (mod 2). Proposition. The complete list of irreducible representations of U(2) is

⊗m det ⊗ ρn, where m, n ∈ Z and n ≥ 0, and det is the obvious one-dimensional representation.