3. Group Representations 3.2 Irreducible, Inequivalent Representations 3.3 Unitary Representations 3.4 Schur's Lemmas 3.5 Orthonormality and Completeness Relations of Irreducible Representation Matrices 3.6 Orthonormality and Completeness Relations of Irreducible Characters 3.7 The Regular Representation 3.8 Direct Product Representations, Clebsch-Gordan Coefficients

3.1. Representations Def: Group of Operators / Linear Transformations = Set of invertible linear operators on a linear space that is also a group Definition 3.1: Group Representations Let L be the space of linear operators on an n-D vector space V. Let U: G  L, g  U(g) be a homomorphism, i.e, U(g g') = U(g) U(g')  g, g'  G Then U(G) = { U(g) | gG } is a n-D representation (rep) of G. The rep is faithful if U is an isomorphism.

Let { ej | j = 1, 2, …, n } be a basis for the n-D vector space V. U(g) can be realized as n n matrices D(g) according to (Einstein's summation notation)   D(G) forms a matrix representation of G. Example 1: Trivial 1-D Rep for Every Group, V = C U: g  U(g) = 1  gG

Example 2: Non-Trivial 1-D Rep for Group of Matrices G = group of matrices & V = C. U(g) = det g. Example 3: 1-D Rep for Td U [ T(n) ] = e – i n { – <   ) Example 4: 2-D Rep for D2 = { e, C2, x, y } = Symm of Rectangle

Example 5: R(2) = { R(), 0   < 2 } , V = E2 

Similarly  Also { U() , 0   < 2 } is a 2-D representation of R(2) on E2 { D() , 0   < 2 } is a 2-D matrix rep of R(2) wrt basis {e1, e2 } Example 6: D3 or C3v = { E, C3, C32, σ1, σ2, σ3 } , V = E2 The 2-D matrix rep is

Example 7: Vf = { f(x,y) = a x + b y | x,y R, a,b  C } Vf is the space of complex–valued linear homogeneous functions of 2 real variables. Let G be any one of the groups in the previous examples. Let U be a rep of G on E2 so that  x = {x,y}  E2 & g G This induces a rep of G on Vf as follows where  Let g = h k, then  This mapping is indeed a homomorphism The ensuring 2-D matrix rep's are identical to those found earlier ( prove it ! ) Vf is an (scalar) example of a field. Higher dim versions  vector, spin, isospin, …

Let U be a rep of K on V so that Theorem 3.1: G/H If G has a non–trivial invariant subgroup H, then any rep of K = G/H is also a degenerate rep of G. Conversely, if U(G) is a degenerate rep of G, then G has at least one invariant subgroup H such that U(G) defines a faithful rep of G/H. Proof: Let U be a rep of K on V so that U: K  L, k  U(k) L = space of linear operators on V. : G  L, g  (g) = U(k) where g h = k  K for some hH is a homomorphism.   is a rep for G. Mapping is many–1 since H is non-trivial. Follows from Theorem 2.5 Theorem 2.5: Let : G  G' be a homomorphism and Kernel = K = { g | (g) = e' } Then K is an invariant subgroup of G and G/K  G'

Example: S3 = { e, (123), (132), (23), (13), (12) } Corollary: All rep's, except the trivial one, of simple groups are faithful. e.g., Cn with n prime Example: S3 = { e, (123), (132), (23), (13), (12) } Invariant subgroup: H = { e, (123), (321) }  C3. Factor group: S3 / H  C2 = { e, a } Non-trivial 1–D rep for C2: { e, a }  {1, –1} Induced degenerate rep for S3 : { e, (123), (132), (23), (13), (12) }  { 1,1,1, –1,–1,–1 } e (123) (132) (23) (13) (12)

3.2. Irreducible, Inequivalent Representations Definition 3.2: Equivalent Representations Two rep's of G are equivalent if they are related by a similarity transform. Definition 3.3: Characters The character of gG in rep U(G) is defined as For a matrix rep D(G), Since trace is preserved in a similarity transform, all elements in a class have the same character. Definition 3.4: Invariant Subspace Let U(G) be a rep of G on V. A subspace V1 of V is invariant wrt U(G) if U(g) | x   V1  xV1 & gG V1 is proper / minimal if it doesn't contain any non-trivial invariant subspace

Definition 3.5: Irreducible Representations (IR) A rep U(G) of G on V is irreducible if there is no non-trivial invariant subspace in V wrt U(G). Otherwise, U(G) is reducible. A reducible rep is decomposable if the orthogonal complement of the invariant subspace is also invariant. Example 1: D2  C2v = { e, C2, x, y } V = E2 c.f. Eg 4, §3.1 The x-axis, or span( e1), is a minimal invariant subspace wrt D2. Ditto y-axis = span(e2).  2-D rep in Eg 4, §3.1, is decomposable.

Example 2: R(2) V = E2 c.f. Eg 5, §3.1 1-D invariant subspaces wrt R(2) are spanned by  Matrix rep wrt { e+, e– } is Example 3: D3 = { E, C3, C32, C2, C2', C2'' } V = E2 E2 is minimal wrt D3  2-D rep in Problem 3.1 is an IR

Let V1 be an n1-D invariant subspace wrt a reducible U(G). Choose basis { ej | j = 1,…,n } so that 1st n1 vectors are in V1.   where D1(g) is n1 n1 Since D(g g') = D(g) D(g') is also upper triangular If V2 = span { ej | j = n1+1,…,n } is also invariant, then U(G) is decomposable & all D'(g) = O Restricting U(G) to an invariant subspace results in a rep of lower dim.

3.3. Unitary Representations Definition 3.6: Unitary Representation A rep U(G) of G on V is unitary if V is an inner product space U(g) are unitary  g G Unitary operators preserve inner products & thereby, lengths & angles

 Theorem 3.2: Unitarity: Reducibility  Decomposability If a unitary representation is reducible, then it is also decomposable. Proof: Let U(G) be the unitary reducible rep of G on inner prod space V = span{ ei | i = 1, …, n } Let V1 = span{ ej | j = 1, …, n1 } be the relevant invariant subspace & V2 = span{ ek | k = n1+1, …, n } its orthogonal complement. V1 is invariant  Orthogonal complement  U(g) is unitary   QED

Theorem 3.3: Every rep D(G) of a finite group on an inner prod space is equiv to a unitary rep. Proof : We shall show that S D(g) S–1 = U(g) is unitary  gG if S satisfies The existence of S is established by showing that ( , ) is an inner product so that S is just the relevant basis transformation. ( Proof of this is left as an exercise: see Prob 3.4 ) Proof of unitarity of U(G) is as follows:

Comments: Theorem can be extended to infinite groups for which an invariant measure can be defined, e.g., compact & semi-simple Lie groups. All reducible rep's of a finite group are decomposable. The inner product space is a direct sum of the invariant subspaces. Definition 3.7: Direct Sum Representation Let U(G) be decomposable on V. Then V = V1  V2  U(G) = U1(G)  U2(G) where U is an IR that occurs n times. D(G) will be block diagonal for a properly chosen basis

3.4. Schur's Lemmas Schur's Lemma 1: Let U(G)  L be an IR of G on V, and A  L. Then AU(g) = U(g)A  gG  A =  E C Proof: Let U(G) be unitary, else replaced by its unitary equivalence S U(G) S–1. Let A be hermitian, else replaced by one of its hermitian components A+ = ( A + A+) / 2 or A– = ( A – A+) / 2i Since A is hermitian, it is diagonalizable & its eigenvalues are all real. Let |  j  be the eigenstate corresponding to the eigenvalue j of A : where { |  j  } can be chosen to be orthonormal:

Label  is introduced to account for possible degeneracies, viz., A U(g) = U(g) A  is also an eigenvector belonging to j Let n be the degree of degeneracies. is an invariant subspace wrt U(G). U(G) is IR  V has no non-trivial invariant subspaces  Vj = V & j = 1 only A = 1 E where D(E) is n n If A itself is hermitian, we have A =  E, where  = 1 is real. Otherwise, A = ( + + i – ) E, where  is the (real) eigenvalue of A

Theorem 3.4: IR of any abelian group must be 1-D. Proof: Let U(G) be an IR of the Abelian G. For a given pG, U(p) U(g) = U(g) U(p)  gG Schur‘s 1st lemma  U(p) = p E  pG  U(G) is equivalent to the 1-D rep { p  p  C }

Schur's Lemma 2: Let U(G) & U'(G) be IRs of G on V & V' , resp. Let A: V'  V be linear & satisfies A U' (g) = U(g) A  gG Then, either 1) A = O , or 2) V  V' (A is isomorphism) & U(G)  U'(G) (equivalent) Proof: Let R = Range A = { x  V | x = A x' , x'  V' }  R is an invariant subspace of V wrt U(G) U(G) is IR  Either R = { 0 } & hence A = O or R = V so that map A is onto Proof is done if for 2), A is also 1-1

Let N' = Null ( V' ) = { x'  V' | A x' = 0 } i.e., N' is an invariant subspace of V' wrt U’(G) U' (G) is IR  Either N' = V' & hence A = O or N' = { 0 } so that map A is 1–1 since Thus, A  O is an isomorphism & A U' (g) = U(g) A  U(g) = A U' (g) A–1

3.5.Orthonormality & Completeness Relations of IR Matrices Notations: nG order of the group G , labels for inequivalent IRs of G n dimension of the -rep D(g) matrix version of gG in the -rep wrt an orthonormal basis j character of elements of class  j in the -rep nj number of elements in the class  j nC number of classes in G

Theorem 3.5a : Orthonormality of IR Matrices dual is an nG–D orthonormal vector for given ( , k, m ) Proof to be given after some examples

Theorem 3.5: Orthonormality of Unitary IR Matrices dual is an nG–D orthonormal vector for given ( , k, m )

For Abelian groups, all IRs are 1–D: Example 1: C2 = { e, a } nG = 2 e a C2 e a 1 1 2 –1 Identity rep: 1-D rep orthonormal to d1 : Example 2: D2 = { e, a = a–1, b = b–1, c = a b } nG = 4 e a b c Identity rep: D2 e a b c 1 1 2 –1 3 4 Invariant subgroup { e, a }  C2 Factor group D2 / C2  C2 Example 3: Td = { T(n) | n  Z } nG =  c.f. Chap 1 Td Abelian  All IRs are 1–D Orthonormality:

Example 4: C3v = { E, C3, C32, σ1, σ2, σ3 } nG = 6

Proof of Theorem 3.5a: Let X be any nn matrix and Schur's lemmas: Either    & MX = 0 or  =  & MX = cX E Let   Take trace on both sides   QED

Corollary 1: Proof: is an nG–D orthonormal vector for given ( , k, m ) Since for a given rep , there are n2 choices of ( k, m ).  is the number of orthonormal vectors in this nG–D space QED Comments: Corollary allows meaningful search for all inequivalent IRs for finite / compact / semi-simple groups. Next theorem shows that

Theorem 3.6: Completeness of IR Matrices 1) 2) Proof of 1) is deferred to § 3.7. Given 1) , 2) is just the completeness relation of nG orthonormal vectors in an nG–D vector space. Comments: If G is Abelian, n = 1    nG inequivalent IRs. Application of Thm 3.5-6 to infinite groups requires existence of an invariant measure to replace group sum with an integral Thm 3.5-6 are basis-dependent; character versions of them are not.

Example : C3v = { E, C3, C32, σ1, σ2, σ3 } nG = 6

3.6. Orthonormality and Completeness Relations of Irreducible Characters Lemma: Sum over a Class Let U(G) be an IR of G . Then Proof: Let then Schur's 1st lemma: Take trace :  QED

Theorem 3.7: Orthonormality & Completeness of  Proof:    QED

Define the normalized character as  Orthonormality Completeness Since j = 1, …, nC , { j } is an nCnC matrix ( character table ). Corollary: ( for finite groups ) Number of inequiv IRs = Number of classes = nC Note: Tables of characters are independent of basis. Example: Abelian groups Each group element forms a class by itself and all IRs are 1–D, i.e., D(gj) = j  Tables of D(G) are also tables of characters

Example: S3 = { e, (123), (132), (23), (13), (12) } 3 classes  3 inequiv IRs Identity rep: 1 = ( 1, 1, 1 ) 1–D rep 2 = ( 1, a, b ). Orthonormality   2 = ( 1, 1, –1 ) Dim d of 3 : 3 = ( 2, a, b ). Orthonormality   3 = ( 2, –1, 0 ) S3 e 2 (123) 3 (23) 1 1 2 –1 3 2

Theorem 3.8:  Proof: Take trace of   C2 e a 1 1 2 –1  2 Example: C2 = { e = a2, a } D(G):  = ( 2, 0 ) D(G)  D'(G): Prob 3.6

Example: Vibration of NH3 C3v e 2C3 3 v A1 1 z A2 –1 Rz E 2 (x,y), (Rx, Ry) Nxy Nz Hxy 6 A1+ A2 + 2E Hz 3 A1 + E NH3 12 3A1 + A2 + 4E Translation: A1+ E Rotation: A2 + E Vibration: 2A1+ 2E

Example: Electronic states of NH3 N: 1s2 2s22p3 H: 1s C3v e 2C3 3 v A1 1 z A2 –1 Rz E 2 (x,y), (Rx, Ry) Ns Np 3 A1 + E Hs NH3 7 3A1 + 2E Bonds

Theorem 3.9: Condition for Irreducibility U(G) is IR  i.e., Proof: Let If U(G) = U(G), then  Conversely  so that for some  i.e., U(G) = U(G) Character tables of all crystallographic point-groups are given in most texts

3.7. The Regular Representation More details concerning group algebra are in Appendix III. See also Chap 5 Notations for group multiplications in finite G = { gj ; j = 1, …, nG } :   Theorem 3.10: Regular Representation is an nG–D matrix rep of G Proof: Let a b = c , where a, b, c  G a b = c  QED

Comments: Theorem 3.10 is just a version of the Cayley's theorem ( Thm 2.1 ). with Cayley: Reg rep: with where Pa is the nn matrix rep of pa. Alternative proof for Theorem 3.10:  where

Theorem 3.11: Decomposition of the regular rep R 1. 2. Proof 1:  Let Thm 3.8: Proof 2: From 1: Take trace 

C2 e a 1 1 2 –1 Example: C2 = { e, a } e a R 2  R can be diagonalized by a similarity transform with so that

Calculation of DR   DR(a) can be obtained from the multiplication table gi vs gj1 by setting all entries that equal to a to 1, and all others to 0.

Example : C3v = { E, C3, C32, σ1, σ2, σ3 } A1+A2+2E e C32 C3 1 2 3 –1 E 2 R 6 A1+A2+2E

Comments: For any finite group G, its DR is equivalent to the block diagonal form See App III

3.8. Direct Product Rep, Clebsch- Gordan Coefficients Definition 3.8: Direct Product Vector Space U V Let U & V be inner product spaces with orthonormal bases & respectively The direct product space of U & V is defined as the inner product space with Thus

Example: 2-particle system 1-particle states: 2-particle system:

Def: Direct Product Operators Let LU & LV be the operator spaces on U & V, resp. Then the operator space on UV is the set with  Definition 3.9: Direct Product Representation Let D(G) & D(G) be reps of G on U & V, resp. Then is a rep of G on U  V

Let D(G) be IRs of G. Then with Note: D(G) can be reducible even if D(G) & D(G) are IRs.

Example: S3 S3 e 2 (123) 3 (23) 1 1 2 –1 3 2 11 1 1 12 –1 2 13 2 3 22 23 33 4 1

Comments:  are invariant subspaces For a properly chosen basis, D is block diagonal. E.g. with the order of varying indices being: k     ( sum over i, j ) [ wi j = uivj complete ] Definition 3.10: Clebsch-Gordan Coefficients ( CGC ) ( sum over i, j )

Theorem 3.12: Orthonormality & Completeness of CGCs sum: ,,k sum: i, j where Proof: Follows directly from the orthonormality & completeness of wi j since

Comments:  Set  

Theorem 3.13: Reduction of Product Representation Applications: Broken symmetry Addition of angular momenta ~ Wigner-Eckert theorem ( Chap 7 )

Short-cut to Theorem 3.13: