Representation Theory

1 LINEAR AND UNITARY GROUPS

Representation theory

1 Linear and unitary groups

Let V denote a linear space with eld of scalars F (usually either the real or the complex numbers). The general linear group

GL(V )={A: V →V | det A 6= 0} 1 LINEAR AND UNITARY GROUPS over V consist of all invertible linear operators on V , with product the composition (multiplication) of operators.

If V has nite dimension n, then for each choice of basis B={e1,..., en} there exist (non-canonical) isomorphisms ΓB : GL(V )→GLn(F)

n X A(ei) = ΓB(A)ij ej j=1

If 0 0 0 is another basis of , then B ={e1,..., en} V

−1 ΓB0 (A) = C ΓB(A) C where 0 P (basis change). ei = j Cijej

matrices 6= linear operators 1 LINEAR AND UNITARY GROUPS

Subgroups of GL(V ) are termed linear groups. 1 LINEAR AND UNITARY GROUPS

Examples:

1. the group × Sc(V ) = {λidV | λ∈F } of scalar operators, isomorphic to the unit group F×; 2. the special linear group

SL(V ) = {A∈GL(V ) | det A=1}

of unimodular (i.e. having determinant 1) operators; 3. the stabilizer subgroup

Stab(v) = {A∈GL(V ) | Av =v}

of a vector v ∈V . 1 LINEAR AND UNITARY GROUPS

Linear groups are more amenable to study because

· linear algebra methods (e.g. spectral decomposition, determinants, etc.) are available in the characterization of their elements;

· there exist group constructions specic to linear groups (linear duality, tensor products, etc.)

· special algorithms for collections of linear operators.

Linear groups are rather the exception than the rule, but some special classes of groups (like nite, compact Lie, etc.) may be shown to have all members isomorphic to some linear group. 1 LINEAR AND UNITARY GROUPS

A subset W ⊆ V of a linear space V (with eld of scalars F) is a linear subspace if x+y ∈W and αx∈W whenever x, y ∈W and α∈F. A linear subspace W ⊆V is nontrivial if it is dierent from the zero subspace and the whole space, i.e. 0

The translates x+W ={x+y | y ∈W } of a linear subspace W ⊆V (where x runs over the elements of V ) form a linear space, the factor space V/W , with addition and multiplication by a scalar dened as

(x+W ) + (y+W ) = (x+y)+W α(x+W ) = αx+W for x, y ∈V and α∈F. 1 LINEAR AND UNITARY GROUPS

Remark. Any basis BW of a linear subspace W ⊆ V can be completed

(in many dierent ways) to a basis BV ⊇ BW of the whole space, and each element of the dierence set BV \BW corresponds to a basis vector of the factor space.

Consequence: dim (V/W )=dim V − dim W

A linear subspace W ⊆ V is an invariant subspace of the linear group

G < GL(V ) if all group elements map it onto itself, i.e. gx ∈ W for all g ∈G and x∈W .

Remark. The zero subspace and the whole space are always invariant. 1 LINEAR AND UNITARY GROUPS

Given an invariant subspace W ⊆V of G

gW : W → W x 7→ gx to W of the group elements and the factored operators

g/W : V/W → V/W x + W 7→ gx + W are well dened linear operators making up linear groups, the reduction

GW = {gW | g ∈G} < GL(W ) and the factor

G/W = {g/W | g ∈G} < GL(V/W ) of the linear group G. 1 LINEAR AND UNITARY GROUPS

Remark. Given a basis BW of W and its completion BV to one of V , the representation matrices with respect to the latter read ! ΓBW (g) T (g) ΓBV (g) = 0 ΓBV\BW (g) for some matrices T (g) that satisfy

T (gh) = ΓBW (g) T (h) + T (g) ΓBV\BW (h) for all g, h∈G, i.e. they are upper triangular block matrices.

A linear group is reducible if it has a nontrivial invariant subspace, otherwise it is called irreducible.

Schur's lemma: any operator that commutes with all elements

of an irreducible linear group is a multiple of the identity operator. 1 LINEAR AND UNITARY GROUPS

A unitary operator U on a Hilbert-space H (a complex linear space endowed with a positive denite inner product, complete with respect to the norm topology) is a linear operator preserving the inner product, i.e.

hUx, Uyi = hx, yi for all x, y ∈H.

Remark. An antiunitary operator A is one which is antilinear, i.e.

A (αx+βy) = αAx + βAy for all α, β ∈C and x, y ∈H, and satises hAx, Ayi = hx, yi

(with the bar denoting complex conjugation). 1 LINEAR AND UNITARY GROUPS

The product of unitary operators is also unitary unitary operators form a group U(H), the unitary group of H.

Remark. If H is of nite dimension n=dim H, then one usually denotes the unitary group U(H) by U(n).

Any unitary operator can be diagonalized, and its eigenvalues lie on the complex unit circle U(1)={z ∈C | zz =1}.

The product of two antiunitary operators is unitary, and any two of them dier by a unitary operator antiunitary operators form a coset of U(H), hence it is enough to understand the structure of the latter. 1 LINEAR AND UNITARY GROUPS

Unitary groups in physics: quantum systems and gauge symmetries.

observable quantities ! self-adjoint operators on Hilbert-space

energy ! Hamiltonian (generator of time translations)

Wigner's theorem: the symmetries of a quantum system correspond to

(anti-)unitary operators commuting with its Hamiltonian.

Remark. Antiunitary operators from time reversal symmetry (exchanging cause and eect).

Gauge symmetries of fundamental interactions described by

(special) unitary groups in the Standard Model. 2 LINEAR REPRESENTATIONS

2 Linear representations

Problem: given an abstract group, nd a linear group isomorphic to it.

No solution for a generic group need to look for suitable (linear) homomorphic images.

A (linear) representation of the group G over the linear space V is a homomorphism D :G→GL(V ) (the representation is called faithful when its kernel is trivial).

The image of a representation is always a linear group, which is isomorphic to G precisely if the representation is faithful. 2 LINEAR REPRESENTATIONS

The eld of scalars and the dimension of D equal those of the representation space V , in particular

dim D = dim V

The theory becomes pretty complicated for generic eld of scalars best to look at representations over the complex numbers C.

Remark. For each choice of a basis B={e1,..., en} of V , the linear representation D : G → GL(V ) determines a degree n matrix representation via the rule , where DB : G→GLn(F) DB =ΓB ◦D ΓB : GL(V )→GLn(F) is the isomorphism associated to B. 2 LINEAR REPRESENTATIONS

Examples: 1. For any group G and linear space V , the map

1V : G → GL(V )

g 7→ idV is a representation, the trivial representation of G over V ;

2. For a linear group G

DG : G → GL(V ) g 7→ g is a representation of G, the dening representation; 2 LINEAR REPRESENTATIONS

3. Given a eld F, the ring F(X)={f :X →F} of F-valued functions on the set X (with pointwise addition and multiplication) is a linear space over F of dimension X . Then the assignment

D(π): F(X) → F(X) f 7→ f ◦ π−1 denes a linear operator acting on F(X), and for each subgroup G

D : G → GL(F(X)) π 7→ D(π) is a representation of G on F(X), the permutation representation associated to the action of G on X. 2 LINEAR REPRESENTATIONS

4. Given an action α: G→Aut(X, µ) of G on the measure space (X, µ)

(i.e. an action of G on X for which µ is an invariant measure, that is

µ(gS)=µ(S) for every g ∈G and every measurable set S ⊆X), consider the Hilbert-space L2(X, µ) of square integrable functions on X. Because the measure µ is invariant, the maps

L(g): L2(X, µ) → L2(X, µ)

ψ 7→ ψ ◦ αg−1 are unitary operators that satisfy L(gh) = L(g)◦L(h) for g, h ∈ G, hence they dene a (unitary) representation of G on L2(X, µ), the (left) regular representation. 2 LINEAR REPRESENTATIONS

A unitary representation of G on a Hilbert-space H is a homomorphism

U : G → U(H) into the group of unitary operators on H, i.e. a linear representation all of whose representation operators are unitary:

hU(g)x, U(g)yi = hx, yi for all x, y ∈H and g ∈G.

A unitarizable representation D : G→GL(V ) is one for which there exists a positive denite scalar product on V (making it a Hilbert-space) for which all representation operators are unitary.

All representations of a nite or compact group are unitarizable. 3 EQUIVALENCE AND REDUCIBILITY

3 Equivalence and reducibility

The representations D1 : G→GL(V1) and D2 : G→GL(V2) are (linearly) ∼ equivalent, denoted D1 = D2, if there exists an invertible linear map

A: V1 →V2 such that D2(g)A=AD1(g) for all g ∈G.

Remark. The dimension of equivalent representations is the same.

Linear equivalence is a reexive, symmetric and transitive relation.

Equivalent representations are practically the same, e.g. the representation matrices of group elements coincide (w.r.t. suitable bases).

Problem: classify all representations of a given group (up to equivalence). 3 EQUIVALENCE AND REDUCIBILITY

A representation D : G → GL(V ) is reducible if its image is a reducible linear group (i.e. it has a nontrivial invariant subspace), otherwise it is irreducible.

Remark. One dimensional representations are always irreducible.

Schur's lemma: any operator that commutes with all representation operators of an irreducible representation is a multiple of the identity.

Remark. Since, by Wigner's theorem, the symmetries of a quantum system are described by (anti-)unitary operators commuting with its Hamil- tonian, the degeneracy of energy levels can be related to symmetries via Schur's lemma. 4 DIRECT SUM OF REPRESENTATIONS

4 Direct sum of representations

The direct sum of the linear spaces V1 and V2 is the linear space V1⊕V2 whose elements are ordered pairs (x1, x2) with x1 ∈V1 and x2 ∈V2, while addition and multiplication by a scalar are dened component-wise:

(x1, x2)+(y1, y2) = (x1 +y1, x2 +y2)

α(x1, x2) = (αx1, αx2) for and ( ). α∈F xi, yi ∈Vi i=1, 2

Remark. If Bi denotes a basis of Vi, then a basis of V1 ⊕V2 is given by

B1 ]B2 = {(x1, 0) | x1 ∈B1} ∪ {(0, x2) | x2 ∈B2} hence

dim(V1 ⊕V2)=dim V1 + dim V2 4 DIRECT SUM OF REPRESENTATIONS

The direct sum of the linear maps A1 ∈GL(V1) and A2 ∈GL(V2) is

A1 ⊕A2 : V1 ⊕ V2 → V1 ⊕ V2

(x1, x2) 7→ (A1x1,A2x2)

The matrix of A1⊕A2 with respect to the basis B1]B2 is block-diagonal ΓB1 (A1) 0 ΓB1]B2 (A1 ⊕A2) = 0 ΓB2 (A2)

Given representations D1 : G→GL(V1) and D2 : G→GL(V2), the map

D1 ⊕D2 : G → GL(V1 ⊕V2)

g 7→ D1(g)⊕D2(g) is a new representation, the direct sum of D1and D2, whose equivalence class is completely determined by the classes of D1 and D2. 4 DIRECT SUM OF REPRESENTATIONS

The direct sum is commutative and associative (up to equivalence)

∼ D1 ⊕D2 =D2 ⊕D1 ∼ D1 ⊕(D2 ⊕D3)=(D1 ⊕D2)⊕D3

Remark. The subspaces

Vˆ1 = {(x1, 0) | x1 ∈V1} and Vˆ2={(0, x2) | x2 ∈ V2} are invariant subspaces of D ⊕D , with reductions (D ⊕D ) =∼ D . 1 2 1 2 Vˆi i

A representation is completely reducible if it can be decomposed into a direct sum of irreducible representations (e.g. trivial representations, which can always be decomposed into a direct sum of trivial one dimensional representations, or the irreducibles themselves). 4 DIRECT SUM OF REPRESENTATIONS

Every completely reducible representation D : G → GL(V ) has an irreducible decomposition M D = nii i∈Irr(G) where is the multiplicity of the irreducible . ni ∈Z+ i∈Irr(G)

completely reducible representations maps from into ! Irr(G) Z+

All unitary (unitarizable) representations are completely reducible Maschke's theorem: all representations (over C) of a nite group are completely reducible.

A similar result holds for the representations of compact groups (Peter-

Weyl theorem). 5 THE CONTRAGREDIENT

5 The contragredient

The dual V ∨ of a linear space V (with eld of scalars F) is the set of all linear functionals, i.e. maps φ:V →F from V into F such that φ(αx + βy) = αφ(x) + βφ(y) for all α, β ∈F and x, y ∈V . The dual is itself a linear space over F with pointwise operations

φ1 +φ2 : V → F x 7→ φ1(x) + φ2(x) and αφ1 : V → F x 7→ αφ1(x) for and ∨. α∈F φ1, φ2 ∈V 5 THE CONTRAGREDIENT

If B={b1, . . . , bn} is a basis of V , then the functionals

∨ bi : V → F bj 7→ δij form a basis of ∨, the dual basis ∨ ∨ . V B ={bi | bi ∈B} Consequence: dim V ∨ =dim V . To each v ∈V one can associate a linear functional

[ ∨ v : V → F φ 7→ φ(v) of the dual space V ∨, and the mapping

[ : V → (V ∨)∨ v 7→ v[ is an invertible linear map. 5 THE CONTRAGREDIENT

The dual of the dual can be identied naturally with the original space.

tr The transpose A of a linear map A: V1 →V2is the linear map

tr ∨ ∨ A : V2 → V1 φ 7→ φ ◦ A between dual spaces.

Given a basis B={b1, . . . , bn} of V , the matrix, with respect to the dual basis ∨ ∨ , of the transpose of reads B ={bi | bi ∈B} A∈GL(V )

tr | ΓB∨ (A ) = ΓB(A) where | for . Mip =Mpi M ∈Matn (F)

Transposition is an antihomomorphism, i.e. (AB)tr =BtrAtr . 5 THE CONTRAGREDIENT

The contragredient of a representation D:G→GL(V) is the representation

D∨ : G → GL(V ∨) g 7→ Dtrg−1 of G on the dual space V ∨. Note that

dim D∨ = dim D

The contragredient of an irreducible representation is itself irreducible The contragredient of the contragredient is the original representation

(D∨)∨ =∼ D and the contragredient of a direct sum is the sum of the contragredients

∨ M ∼ M ∨ Di = Di i i 6 TENSOR PRODUCTS AND THE FUSION RING

6 Tensor products and the fusion ring

A bilinear functional on the linear spaces V1 and V2 (with common eld of scalars is a map that is linear in each of its arguments: F) b:V1×V2 →F

b(αx1 + βy1, x2) = αb(x1, x2) + βb(y1, x2)

b(x1, αx2 + βy2) = αb(x1, x2) + βb(x1, y2) for scalars and ( ). α, β ∈F xi, yi ∈Vi i=1, 2

The set B(V1,V2) of bilinear functionals is a linear space with the obvious pointwise operations

b1 + b2 : V1 ×V2 → F (v, w) 7→ b1(v, w)+b2(v, w) 6 TENSOR PRODUCTS AND THE FUSION RING and αb : V1 ×V2 → F (v, w) 7→ αb(v, w) for and . α∈F b1, b2 ∈B(V1,V2)

The dyadic product v1 ⊗v2 of v1 ∈V1 and v2 ∈V2 is the linear functional

v1 ⊗v2 : B(V1,V2) → F b 7→ b(v1, v2) i.e. evaluation of bilinear functionals at the given arguments.

Dyadic products span the dual of the space of bilinear functionals, called the tensor product V1 ⊗V2 of the linear spaces V1 and V2

∨ V1 ⊗V2 = B(V1,V2) 6 TENSOR PRODUCTS AND THE FUSION RING

Physical signicance: the state space of a composite quantum system is the tensor product of the Hilbert spaces of its components; dyadic products correspond to separable states, with no correlation between the subsytems.

Given bases Bi of Vi (i = 1, 2), the set {e⊗f | e∈B1, f ∈B2} of dyadic products is a basis (the product basis) of the tensor product V1 ⊗V2.

Consequence: dim (V1 ⊗V2)=dim V1 dim V2

Remark. While every element of V1⊗V2 is a linear combination of dyadic products, not all of them are dyadic products themselves (corresponding to inseparable states in quantum theory, leading to the phenomenon of quantum entanglement). 6 TENSOR PRODUCTS AND THE FUSION RING

The tensor product of the linear operators A1 : V1 →W1 and A2 : V2 →W2 is the linear operator A1⊗A2 : V1⊗V2 →W1⊗W2 that maps each dyadic product v1⊗v2 into A1v1⊗A2v2 (well-dened, since dyadic products span the tensor product space).

Given operators Ai ∈ GL(Vi) and bases Bi of Vi (i = 1, 2), the matrix of the tensor product A1⊗A2 ∈GL(V1 ⊗V2) with respect to the product basis is the Kronecker product of the matrices {e⊗f | e∈B1, f ∈B2} ΓB1 (A1) and , with matrix elements ΓB2 (A2)

[ΓB1 (A1) ⊗ ΓB2 (A2)] (ip)(jq) = [ΓB1 (A1)]ij [ΓB2 (A2)]pq for i, j ∈B1and p, q ∈B2. 6 TENSOR PRODUCTS AND THE FUSION RING

Remark. The trace of a tensor product is the product of the traces

Tr(A1 ⊗A2)=Tr(A1) Tr(A2)

The tensor product of D1 : G→GL(V1) and D2 : G→GL(V2) is the representation

D1 ⊗D2 : G → GL(V1 ⊗V2)

g 7→ D1(g)⊗D2(g) The tensor product is compatible with linear equivalence, i.e. the class of a product is determined by the classes of its factors. Moreover, it is commutative and associative (up to equivalence),

∼ ∼ D1 ⊗D2 =D2 ⊗D1 és D1 ⊗(D2 ⊗D3)=(D1 ⊗D2)⊗D3 and distributive with respect to direct sums

∼ D1 ⊗(D2 ⊕D3)=(D1 ⊗D2)⊕(D1 ⊗D3) 6 TENSOR PRODUCTS AND THE FUSION RING

The identity representation 1, which is (the equivalence class of) any 1D trivial representation, is an identity element for the tensor product:

1 ⊗ D =∼ D ⊗ 1 =∼ D

Remark. The tensor product of two irreducibles may contain the identity at most once, in which case they are contragredients of each other.

Equivalence classes of representations with the operations of direct sum and tensor product form the fusion ring (which is actually not a ring, since representations have no additive inverses).

Physics relevance: composition of symmetry charges (e.g. angular momentum of composite systems). 6 TENSOR PRODUCTS AND THE FUSION RING

For completely reducible representations (e.g. nite or compact groups) it is enough to know the fusion rules

M ∨ i⊗j = Nijpp p∈Irr(G) i.e. the irreducible decomposition of the tensor products of irreducibles

(where i, j∈Irr(G)). The non-negative integer multiplicities , the fusion coecients, Nijp ∈Z+ are completely symmetric in their indices,

Nijp = Njip = Npij and satisfy

1 if j=i∨; N = 1ij 0 otherwise. 6 TENSOR PRODUCTS AND THE FUSION RING

Since the tensor product of 1D representations is again of dimension 1

(because of multiplicativity of dimensions under tensor products), the

1D representations of a group G form an Abelian group Gab with the operation of tensor products (inverses being equal to contragredients), which may be shown to be isomorphic to the commutator factor group

G/G0 (the largest commutative homomorphic image of G) for nite G.

Structural information from representation theory!

Remark. The tensor product of a 1D representation with an irreducible is again irreducible permutation action of on . Gab Irr(G)

Over C, all irreducible representations of an Abelian group are 1D, hence ∼ Irr(G)=Gab = G (Pontryagin duality) for nite G. 7 SYMMETRIC POWERS

7 Symmetric powers

For any linear space V , the braiding operator R : V ⊗V →V ⊗V that permutes the factors of dyadic products

R(v1 ⊗v2)=v2 ⊗v1 is an involution (its square is the identity), hence its eigenvalues are ±1. Consequence: one has the spectral decomposition

2 2 V ⊗V = ∧+V ⊕ ∧−V into eigenspaces of R (symmetric and antisymmetric subspaces). 7 SYMMETRIC POWERS

Remark. If B={b1, . . . , bn} is a basis of V , then the sets

2 ∧+B = {bi ⊗ bj + bj ⊗ bi | 1 ≤ i ≤ j ≤ n} 2 ∧−B = {bi ⊗ bj − bj ⊗ bi | 1 ≤ i < j ≤ n} are natural bases of 2 , hence ∧±V n(n±1) dim ∧2 V = ± 2

For any operator A∈GL(V ), its tensor square A⊗A commutes with the braiding operator R, since for any v1, v2 ∈V one has

{R◦(A⊗A)}(v1 ⊗ v2) = R(Av1 ⊗ Av2) = Av2 ⊗ Av1 =

= (A⊗A)(v2 ⊗ v1) = {(A⊗A) ◦ R}(v1 ⊗ v2) and dyadic products span the tensor product. 7 SYMMETRIC POWERS

Consequence: the eigenspaces 2 of are invariant subspaces of the ∧±V R operator A⊗A, and the reduction of the tensor square to these subspaces are the symmetric and antisymmetric squares 2 of . ∧±A A

Remark. Tr(A)2 ± TrA2 Tr∧2 A = ± 2

Since the representation operators of the tensor square D⊗D of any representation D : G→GL(V ) have the form D(g)⊗D(g), the maps

2 ∧±D : G → GL ∧±V

g 7→ ∧±D(g) dene representations 2 of , the symmetric and antisymmetric ∧±D G squares of D. 7 SYMMETRIC POWERS

More generally, for an n-fold tensor power V ⊗n =V ⊗...⊗V , there exist ⊗n ⊗n braiding operators Rij : V →V (for 1 ≤ i < j ≤ n) exchanging the i-th and j-th factor of a dyadic product

Rij(v1 ⊗ ... ⊗vi ⊗ ... ⊗vj ⊗ ... ⊗vn)=v1 ⊗ ... ⊗vj ⊗ ... ⊗vi ⊗ ... ⊗vn

The th symmetric power of is the subspace n of ⊗n consisting n V ∧+V V of the vectors invariant all braidings, and the nth antisymmetric power is the subspace n on which all braidings act as . ∧−V −1

Remark. ⊗n n n for . Actually, the braiding opera- V 6=∧+V ⊕ ∧−V n > 2 tors provide a linear representation of the symmetric group Sn on ⊗n, and the subspaces n correspond to the two inequivalent 1D V ∧±V representations of ; for there are more irreducibles of , and Sn n > 2 Sn correspondingly more nontrivial subspaces of the tensor power. 7 SYMMETRIC POWERS

For and a positive integer , we shall denote by n the A ∈ GL(V ) n ∧±A reduction of the operator A⊗n (acting factor-wise on dyadic products) to the subspaces n . The generating function of the traces of symmetric ∧±V powers is given by

∞ X n n ±1 Molien's formula 1 + Tr ∧∓A z = det(idV ± zA) n=1

For a representation D : G→GL(V ), the maps

n n ∧±D : G → GL ∧±V n g 7→ ∧±D(g) dene representations n of , the th (anti)symmetric powers of . ∧±D G n D 7 SYMMETRIC POWERS

Physics relevance: second quantization of composite systems made up of identical subsystems symmetric/antisymmetric powers for bosonic/fer- mionic excitations (Pauli principle).

Remark. In space-time dimension 2=1+1, the braiding of particles is no more involutive (because of the topology of the light cone). This leads to the possibility of anyonic excitations that obey general braid statistics (e.g. in (quasi-)2D electronic systems relevant to the quantum Hall eect, or the internal degrees of freedom of fundamental strings). Braid statistics is a signal of quantum symmetries that cannot be described via symmetry groups, but need a more general framework. 8 BRANCHING RULES

8 Branching rules

The restriction of a representation D :G→GL(V ) to a subgroup H

resH D : H → GL(V ) h 7→ D(h) of the subgroup H. One can restrict in stages: if K

resK (resH D) = resK D

Restriction is compatible with direct sums and a tensor products

resH (D1 ⊕ D2) = resH D1 ⊕ resH D2

resH (D1 ⊗ D2) = resH D1 ⊗ resH D2 8 BRANCHING RULES

It follows that, for completely reducible representations it is sucient to know the restrictions

M p resH i = Bi p p∈Irr(H) of the irreducible representations i∈Irr(G), the so-called branching rules (the p are non-negative integer multiplicities). Bi

For nite or compact groups, the branching rules characterize completely the restriction of representations.

Important in the description of symmetry breaking, when the ground state of a system breaks some of the symmetries of the dynamics. 9 FROBENIUS-SCHUR INDICATORS

9 Frobenius-Schur indicators

A representation D :G→GL(V ) over the complex numbers is called real, if there is a basis B = {b1, . . . , bn} of V such that all matrix elements (wrt. B) of all representation operators D(g) are real numbers.

Every real representation is self-conjugate (equivalent to its contragredient), but not all self-conjugate representations are real; a self-conjugate representation that is not real is called pseudo-real (aka. quaternionic).

An irreducible representation i∈Irr(G) is self-conjugate if its tensor square i⊗i contains the identity representation 1, with multiplicity

N1ii = 1 9 FROBENIUS-SCHUR INDICATORS

A self-conjugate irreducible representation i ∈ Irr(G) is real or pseudo- real depending on whether the identity representation 1 is contained in the symmetric square 2 or the antisymmetric one 2 . ∧+i ∧−i

The Frobenius-Schur indicator of an irreducible representation i∈Irr(G) is dened as   +1 if i is real;   νi = 0 if i is not self-conjugate;    −1 if i is pseudo-real.

Remark. Can be generalized to quantum symmetries. 10 REPRESENTATION CHARACTERS

10 Representation characters

Basic tasks of representation theory:

1. Classify all representations of a given group (up to equivalence).

This involves in particular deciding whether two representations

are equivalent, and (for completely reducible representations) com-

pute an irreducible decomposition.

2. Determine the fusion and branching rules, Frobenius-Schur indica-

tors, symmetric powers, etc.

3. Use of the above to get structural information on the given group. 10 REPRESENTATION CHARACTERS

To achieve the above goals, one needs a simple numerical characterization of the equivalence classes of representations.

Frobenius' observation: for a representation D :G→GL(V ) over the complex numbers, the function

χD : G → C g 7→ Tr(D(g)) that assigns to each group element the trace of its representation operator (called the character of the representation) provides such a tool.

Two representations are equivalent precisely when their characters coincide.

Especially useful for nite groups. 10 REPRESENTATION CHARACTERS

Remark. Since all elements of a nite group have nite order, all eigenvalues of the representation operators are roots of unity, hence

−1 χD g = χD (g)

|χD (g)| ≤ χD (1)=dim D for representations of nite groups.

The character of the contragredient reads

−1 χD∨(g) = χD g while those of direct sums and tensor products is given by

χ (g) = χ (g) + χ (g) D1⊕D2 D1 D2 χ (g) = χ (g) χ (g) D1⊗D2 D1 D2 10 REPRESENTATION CHARACTERS

Consequence: the character of a completely reducible representation D :G→GL(V ) reads X χD = niχi i∈Irr(G) with ni denoting the multiplicity of the irreducible i∈Irr(G) in the decomposition of D. Representation characters are class functions, i.e. they are constant on conjugacy classes of elements:

−1 χD h gh =χD (g) for all g, h∈G, because

TrDh−1gh=TrDh−1D(g)D(h)=TrDhh−1D(g)=Tr(D(g)) 10 REPRESENTATION CHARACTERS

Remark. Class functions form a linear space over C, which becomes a nite dimensional Hilbert-space in case G is nite, with scalar product

1 X hφ, ψi = φ(g) ψ(g) |G| g∈G

Orthogonality relations: the irreducible characters of a nite

group form an orthonormal basis in the space of class functions.

1 X hχ , χ i = χ (g) χ (g) = δ i j |G| i j ij g∈G

Remark. A similar result holds for compact groups, with summation replaced by the Haar-measure in the denition of the scalar product. 10 REPRESENTATION CHARACTERS

Consequences: 1. # of inequivalent irreducible representations = # of conjugacy classes 2. X 2 X 2 Burnside's theorem (dim i) = χi (1) = |G| i∈Irr(G) i

Remark. Burnside's theorem is a special case of the second orthogonality relations ( X |CG(g)| if g and h are conjugates; χ (g) χ (h) = i i 0 otherwise. i∈Irr(G) 3. 1 X n = hχ , χ i = χ (g) χ (g) i D i |G| D i g∈G is the multiplicity of an irreducible i∈Irr(G) in the decomposition of D. 10 REPRESENTATION CHARACTERS

In particular, the fusion coecients are given by

D E 1 X N = χ , χ = χ (g) χ (g) χ (g) ijp i⊗j p∨ |G| i j p g∈G the branching rules from G to H

1 X Bp = χ (h) χ (h) i H i p h∈H with i∈Irr(G) and p∈Irr(H), and the Frobenius-Schur indicators by

1 X ν = χ g2 i |G| i g∈G since the character of symmetric squares read

2 2 χD (g) ± χD g χ 2 (g) = ∧±D 2 10 REPRESENTATION CHARACTERS

Generalized orthogonality relations

1 X χ (h) χ (g) χ g−1h = δ j |G| i j ij χ (1) g∈G j

Consequence: for i∈Irr(G), the linear combinations

χ (1) X P = i χ (g)D(g) i |G| i g∈G of the representation operators are orthogonal to each other, i.e.

PiPj = δijPi and project onto the isotypical components of D :G→GL(V ), meaning that the image PiV ⊆V is an invariant subspace such that the reduction of D to PiV does not contain any irreducible dierent from i. 10 REPRESENTATION CHARACTERS

Character table of a nite group

C1 ={1} ··· Cj ···

1 χ1 (1)=1 ··· 1 ··· ......

i χi (1)=dim i ··· χi (Cj) ··· ......

Rows indexed by irreducible characters, and columns by conjugacy classes (with the rst row corresponding to the identity representation, and the rst column to the trivial class). 10 REPRESENTATION CHARACTERS

Character table of D3

−1 C1 ={1} C2 = C,C C3 ={σ1, σ2, σ3} 1 1 1 1

1∗ 1 1 −1

2 2 −1 0

Fusion rules of D3

1 1∗ 2

1 1 1∗ 2

1∗ 1∗ 1 2

2 2 2 1⊕1∗ ⊕2 10 REPRESENTATION CHARACTERS

Example: the character of the symmetric squares 2 read ∧±2

−1 C1 ={1} C2 = C,C C3 ={σ1, σ2, σ3} 2 ∧+2 3 0 1 2 ∧−2 1 1 −1

Clearly, 2 ∗ by inspection, while ∧−2=1 1 ∧2 2, 1 = (1 · 3 · 1 + 2 · 0 · 1 + 3 · 1 · 1) = 1 + 6 1 ∧2 2, 1∗ = (1 · 3 · 1 + 2 · 0 · 1 + 3 · 1 · (−1)) = 0 + 6 1 ∧2 2, 2 = (1 · 3 · 2 + 2 · 0 · (−1) + 3 · 1 · 0) = 1 + 6 hence 2 and ∗ . ∧+2=1⊕2 2⊗2=1⊕1 ⊕2

Remark. The above argument also shows that 2 is real, i.e. ν2 =1. 11 PROJECTIVE REPRESENTATIONS

11 Projective representations

Because (pure) states of a quantum system correspond to the one dimensional subspaces of its Hilbertspace, the symmetry operators are determined only up to scalar factors.

A mapping D:G→GL(V ) for which D(1)=idV is called a projective representation (of G over V ) if there exists a function α:G×G→C× (called the cocycle of D) such that

D(g) D(h) = α(g, h) D(gh) for all g, h∈G. 11 PROJECTIVE REPRESENTATIONS

Since the product of operators is associative

D(g1)(D(g2)D(g3))= α(g2, g3) D(g1)D(g2g3)= α(g1, g2g3)α(g2, g3) D(g1g2g3) || ||

(D(g1)D(g2))D(g3)= α(g1, g2) D(g1g2)D(g3)= α(g1, g2)α(g1g2, g3) D(g1g2g3) for all g1, g2, g3 ∈G, hence

α(g1, g2)α(g1g2, g3) = α(g1, g2g3)α(g2, g3) cocycle equation

Cocycles are normalized,

α(g, 1)=α(1, g)=1 because of D(1)=idV . 11 PROJECTIVE REPRESENTATIONS

The pointwise product of two cocycles is again a cocycle, hence cocycles form an Abelian group Z2(G), whose identity element is the trivial cocycle (with all values equal to 1).

2 The cocycles α1, α2 ∈Z (G) are cohomologous, denoted α1 'α2, if there exists a function λ:G→C× such that α (g, h) λ(g) λ(h) 2 = α1(g, h) λ(gh) for all g, h∈G.

Being cohomologous is a congruence relation of the group Z2(G) of cocycles (i.e. an equivalence compatible with the group structure), hence its equivalence classes also form a group, the Schur multiplier (aka. second cohomology group) H2(G). 11 PROJECTIVE REPRESENTATIONS

The Schur multiplier of a nite or compact group (and of most groups of practical interest) is nite, but can be innite in some important applications (e.g. for the 2D conformal group, a fact of extreme importance in string theory).

Remark. Higher cohomology groups can be dened analogously, with

2-cocycles replaced by n-cocycles for some positive integer n (i.e. scalar valued functions on Gn that satisfy a suitable cocycle equation), and a corresponding notion of cocycles being cohomologous. They form the basis of the cohomology theory of groups, an important branch of ho- mological algebra, and nd interesting physics applications, e.g. in the study of gauge and gravitational anomalies, Chern-Simons dynamics,etc. 11 PROJECTIVE REPRESENTATIONS

Remark. Ordinary representations are the projective representations with trivial cocycle.

The projective representations D1 : G→GL(V1) and D2 : G→GL(V2) are equivalent if there exists a bijective linear map A:V1 →V2 and a function λ:G→C× (the scale factor) such that for all g ∈G

D2(g)A = λ(g) AD1(g)

Cocycles of equivalent projective representations are cohomologous.

Indeed

α2(g,h)D2(gh)A=D2(g)D2(h)A=λ(h)D2(g)AD1(h)=λ(g)λ(h)AD1(g)D1(h) λ(g)λ(h) = λ(g)λ(h)α (g,h)AD (gh) = α (g, h) D (gh)A 1 1 λ(gh) 1 2 11 PROJECTIVE REPRESENTATIONS

For each cohomology class, the theory is analogous to that of ordinary representations, with corresponding notions of invariant subspace, reducible and irreducible representations, direct sums, contragredients, etc., but

1. direct sums of projectives are only dened when their cocycles are

equal('superselection rule');

2. the cocycle of a tensor product is the product of the respective

cocycles;

3. the cocycle of the nth symmetric power is the nth power of the representation's cocycle. 11 PROJECTIVE REPRESENTATIONS

Gˆ is called a covering group of G if it has a central subgroup A

A =∼ H2(G) G/Aˆ =∼ G Schur: every nite group has a covering group, but there might be several non-isomorphic ones!

Isomorphism class of covering group uniquely determined if G0 =G.

ˆ ordinary representations of G ! projective representations of G (irreducibles corresponding to irreducibles) 11 PROJECTIVE REPRESENTATIONS

Examples:

1. ˆ ∼ , the binary tetrahedral group; T = SL(2, Z3)

2. ˆ ∼ , the binary icosahedral group; I = SL(2, Z5)

3. ∼ , due to the exceptional outer automorphism; Ac6 = 3·SL(2, Z9)

4. for Lie-groups, the covering group of G is the unique simply con-

nected group Gˆ locally isomorphic to G (the universal cover).

In particular, since SU(2) is simply connected and locally isomorphic to SO(3) =∼ SU(2)/Z(SU(2)), one has Z(SU(2)) =∼ H2(SO(3)), leading to two classes of projective representations for the rotation group: tensorial

(ordinary) and spinorial ones. 12 REPRESENTATIONS OF LIE-GROUPS

12 Representations of Lie-groups

In case of a (continuous) representation D :G→GL(V ) of a Lie-group G, the representation operators D(g) are dierentiable (operator-valued) functions of the parameters, and the derivatives (at the identity) satisfy ∂D ∂D ∂D ∂D X ij ∂D − = ck ∂αi ~ ∂αj ∂αj ∂αi ~ ∂αk ~ α~=0 α~=~0 α~=~0 α~=0 k α~=0 with ij denoting the structure constants of , i.e. form a representation ck G of the Lie-algebra Lie(G). A representation of a Lie-algebra g on the linear space V is a linear map d: g → gl(V ) that satises [d(a) , d(b)] = d(a) d(b) − d(b) d(a) for all a, b∈g). 12 REPRESENTATIONS OF LIE-GROUPS

representations of the Lie-algebra Lie(G)

representations of the universal cover Gˆ

projective representations of G

The Casimiroperators of Lie(G) commute with all representation operators, hence they act as scalars in any irreducible representation according to Schur's lemma.

Example: the Poincaré-group has two independent Casimirs, whose values in an irreducible are related to the mass and the spin of the corresponding particle. 12 REPRESENTATIONS OF LIE-GROUPS

The irreducible representation of SU(2) are the dierent symmetric powers of the dening representation (of dimension 2) there is precisely one irreducible for any positive integer dimension n, and the Casimir- operator J 2 acts in it as n2 − 1 J 2 = 4

One often labels these irreducibles by the quantity n−1 , the spin of j = 2 the representation, which can take on any integer or half-integer value.

Remark. The corresponding representation of the rotation group SO(3) is tensorial (ordinary) or spinorial according to whether j is integer or half-integer (i.e. whether the dimension is odd or even).