g-lecture.pptx

Introduction to Group Theory
With Applications to Quantum Mechanics
and Solid State Physics
Roland Winkler
rwinkler@niu.edu
August 2011
(Lecture notes version: November 3, 2015)
Please, let me know if you find misprints, errors or inaccuracies in these notes.
Thank you.
Roland Winkler, NIU, Argonne, and NCTU 2011−2015

General Literature
) J. F. Cornwell, Group Theory in Physics (Academic, 1987)
general introduction; discrete and continuous groups
) W. Ludwig and C. Falter, Symmetries in Physics
(Springer, Berlin, 1988).
general introduction; discrete and continuous groups
) W.-K. Tung, Group Theory in Physics (World Scientific, 1985).
general introduction; main focus on continuous groups
) L. M. Falicov, Group Theory and Its Physical Applications
(University of Chicago Press, Chicago, 1966).
small paperback; compact introduction
) E. P. Wigner, Group Theory (Academic, 1959).
classical textbook by the master
) Landau and Lifshitz, Quantum Mechanics, Ch. XII (Pergamon, 1977)
brief introduction into the main aspects of group theory in physics
) R. McWeeny, Symmetry (Dover, 2002)
elementary, self-contained introduction
) and many others

Specialized Literature
) G. L. Bir und G. E. Pikus, Symmetry and Strain-Induced Effects in
Semiconductors (Wiley, New York, 1974)
thorough discussion of group theory and its applications in solid state
physics by two pioneers
) C. J. Bradley and A. P. Cracknell, The Mathematical Theory of
Symmetry in Solids (Clarendon, 1972)
comprehensive discussion of group theory in solid state physics
) G. F. Koster et al., Properties of the Thirty-Two Point Groups
(MIT Press, 1963)
small, but very helpful reference book tabulating the properties of
the 32 crystallographic point groups (character tables, Clebsch-Gordan
coefficients, compatibility relations, etc.)
) A. R. Edmonds, Angular Momentum in Quantum Mechanics
(Princeton University Press, 1960)
comprehensive discussion of the (group) theory of angular momentum
in quantum mechanics
) and many others

These notes are dedicated
to
Prof. Dr. h.c. Ulrich R¨ossler
from whom I learned group theory
R.W.

Definition: Group
A set G = {a, b, c, . . .} is called a group, if there exists a group multiplication
connecting the elements in G in the following way
(1) a,b ∈ G : c = ab ∈ G
(2) a,b, c ∈ G : (ab)c = a(bc)
(3) ∃e ∈ G : ae = a ∀
a∈ G
(
4
) ∀
a∈ G ∃b ∈ G : ab = e,
(closure)
(associativity)
(identity / neutral element)
(inverse element)
i.e., b ≡ a−1
Corollaries
(a) e−1 = e
(b) a−1a = aa−1 = e ∀
a∈ G
(c) ea = ae = a ∀
a∈ G
(d) ∀a,b ∈ G : c = ab ⇔ c−1 = b−1a−1
Commutative (Abelian) Group
(5) ∀a,b ∈ G : ab = ba
Order of a Group = number of group elements
(left inverse = right inverse)
(left neutral = right neutral)
(commutatitivity)
Introduction to Group Theory and Overview

Example
s
) integer numbers Z with addition
(Abelian group, infinite order)
) rational numbers Q{0} with multiplication
(Abelian group, infinite order)
) complex numbers {exp(2πi m/n) : m = 1, . . . , n} with multiplication
(Abelian group, finite order, example of cyclic group)
) invertible (= nonsingular) n × n matrices with matrix multiplication
(nonabelian group, infinite order, later important for representation theory!)
) permutations of n objects: Pn
(nonabelian group, n! group elements)
n
) symmetry operations (rotations, reflections, etc.) of equilateral triangle
≡ P3 ≡ permutations of numbered corners of triangle – more later!
) (continuous) translations in Rn : (continuous) translation group
≡ vector addition in R
) symmetry operations of a sphere
only rotations: SO(3) = special orthogonal group in R
3
= real orthogonal 3 × 3 matrices

Group Theory in
Physics
Group theory is the natural language to describe symmetries of a physical
system
) symmetries correspond to conserved quantities
) symmetries allow us to classify quantum mechanical states
• representation theory
• degeneracies / level splittings
) evaluation of matrix elements ⇒ Wigner-Eckart theorem
e.g., selection rules: dipole matrix elements for optical transitions
) Hamiltonian Ĥ must be invariant under the symmetries
of a quantum system
⇒ construct Ĥ via symmetry arguments
) . . .

Group Theory in Physics
Classical
Mechanics
) Lagrange function L(q, q
˙
),
) Lagrange equations
d ∂L
dt ∂q̇ i
= i = 1, . . . , N
) If for one j : j
∂L
= 0 ⇒ p ≡
∂qj
∂L
∂qi
∂L
∂q̇ j
is a conserved quantity
Examples
) qj linear coordinate
• translational invariance
• linear momentum pj = const.
• translation group
) qj angular coordinate
• rotational invariance
• angular momentum pj = const.
• rotation group

Quantum Mechanics
(1) Evaluation of matrix elements
) Consider particle in potential V(x) = V(−x) even
) two possiblities for eigenfunctions ψ(x)
ψe(x) even: ψe(x) = ψe(−x)
ψo (x) odd: ψo (x) = −ψo(−x)
∫ ∗
i
) overlapp ψ (x) ψ (x) dx = δ
j ij i, j ∈ {e, o}
) expectation value ⟨i|x|i⟩ =
∫
ψi
∗(x) x ψi (x) dx = 0
well-known explanation
) product of two even / two odd functions is even
) product of one even and one odd function is odd
) integral over an odd function vanishes

Quantum Mechanics
(1) Evaluation of matrix elements (cont’d)
Group theory provides systematic generalization of these statements
) representation theory
≡ classification of how functions and operators transform
under symmetry operations
) Wigner-Eckart theorem
≡ statements on matrix elements if we know how the functions
and operators transform under the symmetries of a system

Quantum Mechanics
(2) Degeneracies of Energy
Eigenvalues
) Schrödinger equation Ĥψ = Eψ
) Let Ô with ik∂t Ô = [Ô, Ĥ] = 0
⇒ eigenvalue equations Ĥψ = Eψ
or ik∂tψ = Ĥψ
⇒ Ô is conserved quantity
and Ôψ = λOˆ ψ
can be solved simultaneously
⇒ eigenvalue λOˆ of Ô is good quantum number for ψ
Example: H atom
) ˆ
2 2
k ∂ 2 ∂ L̂ 2 e2
H = + + − ⇒ group SO(3)
2m ∂r 2 r ∂r 2mr2 r
⇒ [L̂2, Ĥ] = [L̂z, Ĥ] = [L̂2, L̂z ] = 0
⇒ eigenstates ψnlm(r): index l ↔ L̂ 2 , m ↔ L̂ z
) really another example for representation theory
) degeneracy for 0 ≤ l ≤ n −1: dynamical symmetry (unique for H atom)

Quantum Mechanics
(3) Solid State Physics
in particular: crystalline solids, periodic assembly of atoms
⇒ discrete translation invariance
(i) Electrons in periodic potential V(r)
) V(r + R) = V (r) ∀R ∈ {lattice vectors}
⇒ translation operator T̂R : T̂R f (r) = f (r + R)
[T̂R,Ĥ] = 0
ik·r
⇒ Bloch theorem ψk(r) = e uk(r) with uk(r + R) = uk(r)
⇒ wave vector k is quantum number for the discrete translation invariance,
k ∈ first Brillouin zone

Quantum Mechanics
(3) Solid State
Physics
(ii) Phonons
) Consider square lattice
rotation
by 90o
) frequencies of modes are equal
) degeneracies for particular propagation directions
(iii) Theory of Invariants
) How can we construct models for the dynamics of electrons
or phonons that are compatible with given crystal symmetries?

Quantum Mechanics
(4) Nuclear and Particle Physics
Physics at small length scales: strong interaction
Proton mp = 938.28 MeV
Neutron mn = 939.57 MeV
rest mass of nucleons almost equal
~ degeneracy
[ˆ
I, Ĥstrong] = 0
) Symmetry: isospin ˆ
I with
1 1
2 2
) SU(2): proton | ⟩, 1 1
2 2
neutron | − ⟩

Mathematical Excursion:
Groups
Basic Concepts
Group Axioms: see above
Definition: Subgroup Let G be a group. A subset U ⊆ G that is itself
agroup with the same multiplication as G is called a subgroup of G.
Group Multiplication Table: compilation of all products of group elements
⇒ complete information on mathematical structure of a (finite) group
Example: permutation group P3
e =
1 2 3
1 2 3
a =
1 2 3
2 3 1
b =
1 2 3
3 1 2
c =
1 2 3
1 3 2
d =
1 2 3
3 2 1
f =
1 2 3
2 1 3
P3 e a b c d f
e
e a b c d f
a a b e f c d
b b e a d f c
c c d f e a b
d d f c b e a
f f c d a b e
) {e}, {e, a,b}, {e, c}, {e, d}, {e, f }, G are subgroups of G

Conclusions from
Group Multiplication Table
P3 e a b c d f
e e a b c d f
a a b e f c d
b b e a d f c
c c d f e a b
d d f c b e a
f f c d a b e
) Symmetry w.r.t. main diagonal
⇒ group is Abelian
) order n of g ∈ G: smallest n > 0 with gn = e
) {g, g2, . . . , gn = e} with g ∈ G is Abelian subgroup (a cyclic group)
) in every row / column every element appears exactly once because:
Rearrangement Lemma: for any fixed g′ ∈ G, we have
G = {g ′g : g ∈ G } = {gg′ : g ∈ G}
i.e., the latter sets consist of the elements in G rearranged in order.
proof: g1 /= g2 ⇔ g′
g1 /= g′
g2 ∀g1,g2, g′
∈ f

Goal: Classify elements in a
group
(1) Conjugate Elements and Classes
) Let a ∈ G. Then b ∈ G is called conjugate to a if ∃x ∈ G with b = xax−1.
Conjugation b ∼ a is equivalence relation:
• a ∼ a
• b ∼ a ⇔ a ∼ b
• a ∼ c,
⇒ a ∼ b
reflexive
symmetric
transitive
a = xcx−1
⇒ c = x−1
ax
b ∼ c b = ycy−1
= (xy−1
)−1
a(xy−1
)
) For fixed a, the set of all conjugate elements
C = {x ax−1 : x ∈ G } is called a class.
• identity e is its own class xex−1
= e ∀x ∈ f
• Abelian groups: each element is its own class
xax−1
= axx−1
= a ∀a,x ∈ f
• Each b ∈ f belongs to one and only one class
⇒ decompose f into classes
• in broad terms: “similar” elements form a class
Example: P3
x e a b c d f
e
a
b
c
d
f
e
e
e
e
e
e
a b
a b
a. b
b. a
b a
b a
c. d f
d. f c
f c d
c f d
f d c
d c f
⇒ classes {e}, {a, b},
{c, d, f }

group (2) Subgroups and Cosets
) Let U ⊂ G be a subgroup of G and x ∈ G. The set x U ≡ {x u : u ∈ U}
(the set Ux) is called the left coset (right coset) of U.
) In general, cosets are not groups.
If x ∈
/ U, the coset x U lacks the identity element:
suppose ∃u ∈ U with xu = e ∈ x U ⇒ x−1
= u ∈ U ⇒ x = u−1
∈ U
) If x′ ∈ x U, then x′U = x U any x′
∈ x U can be used to define coset x U
) If U contains s elements, then each coset also contains s elements
(due to rearrangement lemma).
) Two left (right) cosets for a subgroup U are either equal or disjoint
(due to rearrangement lemma).
) Thus: decompose G into cosets
G = U ∪ x U ∪ y U ∪ . . . x, y, . . . ∈
/ U
) Thus Theorem 1: Let h order of G ⇒ h
Let s order of U ⊂ G s
∈ N
) Corollary: The order of a finite group is an integer multiple of the
orders of its subgroups.
) Corollary: If h prime number ⇒ {e}, G are the only subgroups
⇒ G is isomorphic to cyclic group

group
(3) Invariant Subgroups and Factor Groups
connection: classes and cosets
) A subgroup U ⊂ G containing only complete classes of G is called
invariant subgroup (aka normal subgroup).
) Let U be an invariant subgroup of G and x ∈ G
⇔ x Ux−1
= U
⇔ x U = Ux (left coset = right coset)
) Multiplication of cosets of an invariant subgroup U ⊂ G:
x, y ∈ G : (x U) (y U) = xy U = z U where z = xy
well-defined: (x U) (y U) = x (U y) U = xy U U = z U U = z U
) An invariant subgroup U ⊂ G and the distinct cosets x U
form a group, called factor group F = G/U
• group multiplication: see above
• U is identity element of factor group
• x−1
U is inverse for x U
) Every factor group F = f/U is homomorphic to f (see below).

Example: Permutation Group P3
e a b c d f
e e a b c d f
a
b
c
d
f
a
b
b
e
e
a
f c d
d f c
c d f e a b
d f c b e a
f c d a b e
invariant subgroup U = {e, a,b}
U cU
cU U
⇒ one coset cU = dU = f U = {c, d, f }
factor group P3/U = {U , cU}
U cU
U
cU
) We can think of factor groups G/U as coarse-grained versions of G.
) Often, factor groups G/U are a helpful intermediate step when
working out the structure of more complicated groups G.
) Thus: invariant subgroups are “more useful” subgroups than other
subgroups.

Mappings of
Groups
) Let G and G′ be two groups. A mapping φ : G → G′ assigns to each
g ∈ G an element g′ = φ(g) ∈ G′, with every g′ ∈ G′ being the image
of at least one g ∈ G.
) If φ(g1) φ(g2) = φ(g1 g2) ∀g1,g2 ∈ G,
then φ is a homomorphic mapping of G on G′.
• A homomorphic mapping is consistent with the group structures
• A homomorphic mapping f → f ′
is always n-to-one (n ≥ 1):
The preimage of the unit element of f ′
is an invariant subgroup U of f.
f ′
is isomorphic to the factor group f/U.
) If the mapping φ is one-to-one, then it is an isomomorphic mapping
of G on G′.
• Short-hand: f isomorphic to f ′
⇒ f ' f ′
• Isomorphic groups have the same group structure.
) Examples:
•trivial homomorphism f = P3 and f ′
= {e}
•isomorphism between permutation group P3 and symmetry group C3v of
equilateral triangle Roland Winkler, NIU, Argonne, and NCTU 2011−2015

Products of
Groups
) Given two groups G1 = {ai } and G2 = {bk }, their outer direct product
(ai, ai) ∈ G ⊗G
is the group G1 × G2 with elements (ai, bk) and multiplication
(ai, bk) · (aj, bl ) = (aiaj, bkbl ) ∈ G1 × G2
•Check that the group axioms are satisfied for f1 × f2.
•Order of fn is hn (n = 1,2) ⇒ order of f1 × f2 is h1h2
•If f = f1 × f2 , then both f1 and f2 are invariant subgroups of f.
Then we have isomorphisms f2 ' f/f1 and f1 ' f/f2.
• Application: built more complex groups out of simpler groups
) If G1 = G2 = G = {ai }, the elements
define a group G
˜≡ G ⊗G called the inner product of G.
• The inner product f ⊗ f is isomorphic to f (⇒ same order as f)
• Compare: product representations (discussed below)

Matrix Representations of a
Group
Motivation
) Consider symmetry group Ci = {e, i}
e = identity
i = inversion
Ci e i
e e i
i i e
) two “types” of basis functions: even and odd
) more abstract: reducible and irreducible representations
matrix representation (based on 1 × 1 and 2 × 2 matrices)
Γ1 = {De = 1, Di = 1}
e i
Γ2 = {D = 1, D = −1}
Γ3 = De = 1 0
, Di = 1 0
0 1 0 −1
}
consistent with g
r
o
u
p
multiplication table
where Γ1 : even function fe(x) = fe(−x)
Γ2 : odd functions fo (x) = −fo(−x)
irreducible
representations
Γ3 : reducible representation:
decompose any f (x) into even and odd parts
f (x) = fe(x)+fo (x) with
e
1
2
f (x) = f (x) + f (−x)
o
1
2
f (x) = f (x) −f (−x)
How to generalize these ideas for arbitrary groups?

Matrix Representations of a
Group
) Let group G = {gi : i = 1, . . . , h}
) Associate with each gi ∈ G a nonsingular square matrix D(gi ).
If the resulting set {D(gi ) : i = 1, . . . , h} is homomorphic to G
it is called a matrix representation of G.
• gi gj = gk ⇒ D(gi ) D(gj ) = D(gk)
• D(e) = 1 (identity matrix)
i
−1 −1
i
• D(g ) = D (g )
) dimension of representation = dimension of representation matrices
Example (1): G = C∞ = rotations around a fixed axis (angle φ)
) ∞
C is isomorphic to group of orthogonal 2 × 2 matrices SO(2)
D2(φ) =
cosφ −sin φ
sin φ cosφ
⇒ two-dimensional (2D) representation
) C∞ is homomorphic to group {D1(φ) = 1} ⇒ trivial 1D representation
) C∞ is isomorphic to group
1 0
0 D2(φ)
⇒ higher-dimensional
representation
) Generally: given matrix representations of dimensions n1 and n2,
we can construct (n1 + n2) dimensional representations

Matrix Representations of a Group (cont’d)
Example (2): Symmetry group C3v of equilateral triangle
(isomorphic to permutation group P3)
y
x
identity
rotation
φ
=
120
◦
rotation
φ
=
240
◦
reflecti
on
y
↔
−y
roto-
reflectio
n
φ
=
120
◦
roto-
reflection
φ
=
240
◦
P3
Koster
e
E
a
C3
b = a2
C2
3
c = ec
σv
d = ac
σv
f = bc
σv
Γ1 (1) (1)
(1)
,
− 1 −
√
3
,
√ 2 2
3 1
2 − 2
,
(1)
(1)
— 1
√
3
,
√2 2
— 3 − 1
2 2
(1)
(−1)
1 0
0 −1
,
(1)
(−1)
— 1 −
√
3
,
√2 2
3 1
— 2 2
(1)
(−1)
,
− 1
√
3
,
√ 2 2
3 1
2 2
Γ2 (1)
Γ3 1 0
0 1
multipli-
cation
table
P3 e a b c d f
e e a b c d f
a a b e f c d
b b e a d f c
c c d f e a b
d d f c b e a
f f c d a b e
) mapping f → {D(gi )} homomorphic,
but in general not isomorphic (not faithful)
) consistent with group multiplication table
) Goal: characterize matrix representations of f
) Will see: f fully characterized by its “distinct”
matrix representations (only three for f = C3v!)

Goal: Identify and Classify
Representations
) Theorem 2: If U is an invariant subgroup of G, then every representation
of the factor group F = G/U is likewise a representation of G.
Proof: f is homomorphic to F , which is homomorphic to the representations of F.
Thus: To identify the representations of f it helps to identify the representations of F.
) Definition: Equivalent Representations
Let {D(gi )} be a matrix representation for G with dimension n.
Let X be a n-dimensional nonsingular matrix.
The set {D′(gi ) = X D(gi ) X−1} forms a matrix representation
called equivalent to {D(gi )}.
Convince yourself: {D′
(gi )} is, indeed, another matrix representation.
Matrix representations are most convenient if matrices { D } are unitary. Thus
) Theorem 3: Every matrix representation {D(gi )} is equivalent to a unitary
representation {D′(gi )} where D′ †(gi ) = D′ −1(gi )
) In the following, it is always assumed that matrix representations are unitary.

Proof of Theorem 3 (cf. Falicov)
Challenge: Matrix X has to be choosen such that it makes all matrices
D′(gi ) unitary simultaneously.
) Let {D(gi ) ≡ Di : i = 1, . . . , h} be a matrix representation for G
(dimension h).
)
h
Σ
i
†
i
Define H = D D (Hermitean)
i=1
) Thus H can be diagonalized by means of a unitary matrix U.
i
−1 −1
d ≡ U HU = U Di i
Σ Σ
i
i
−1 −1 −1 −1
i
D U = U D U U D U
` ˛¸ x ` ˛¸ x
=D˜i
†
Σ
i
~ ˜†
i
= D D with dµν µ µν
= d δ diagonal=D˜i
i
) Diagonal entries dµ are positive:
d =
Σ Σ
~ ˜†
i
(D ) (D )
µ i µλ λµ
Σ
i λ iλ
˜
= (D ) ˜∗
i
(D )
i µλ µλ
Σ
iλ
˜
= |(D )
i µλ
2
| > 0
˜
) Take diagonal matrix d̃ with elements (d )
±1/2
≡ d δ
± ± µν µ µν
Σ
i
†
i
~ ˜ ˜ ˜
— − − i −
) Thus 1 = d̃ d d̃ = d D D d (identity matrix)

Proof of Theorem 3 (cont’d)
) Assertion: Di
′ = d̃— i + −
−1
= d U Di
D̃ d̃ ˜ ˜+
U d are unitary matrices
equivalent to Di
˜−
• equivalent by construction: X = d U−1
• unitarity:
Di
′
Di
′ †
=
¸ x` ˛
= d D d (d
Σ
k
— i + − k
˜ ˜ ˜ ˜ ˜ ˜
D D†
k
†
i
˜ ˜ ˜ ˜
d ) d D d
− + −
= d˜ Σ
k
— i k
˜ ˜ ˜ ˜
† †
k i
` ˛¸ x` ˛¸ x
= D˜j = D˜j
†
(rearrangement lemma)
˜
D D D D d−
`
=
˛¸
d
x
= 1
qed

Reducible and Irreducible
Representations
(RRs and IRs)
) If for a given representation {D(gi ) : i = 1, . . . , h}, an equivalent
representation {D′(gi ) : i = 1, . . . , h} can be found that is block
diagonal
′
i
D (g ) =
′
1 i
D (g ) 0
′
2 i
0 D (g )
i
∀g ∈ G
then {D(gi ) : i = 1, . . . , h} is called reducible, otherwise irreducible.
) Crucial: the same block diagonal form is obtained
for all representation matrices D(gi ) simultaneously.
) Block-diagonal matrices do not mix, i.e., if D′(g1) and D′(g2) are
block diagonal, then D′(g3) = D′(g1) D′(g2) is likewise block diagonal.
⇒ Decomposition of RRs into IRs decomposes the problem
into the smallest subproblems possible.
) Goal of Representation Theory
Identify and characterize the IRs of a group.
) We will show
The number of inequivalent IRs equals the number of classes.

Schur’s First
Lemma
Schur’s First Lemma: Suppose a matrix M commutes with
all matrices Ð(gi ) of an irreducible representation of 5
Ð(gi ) M = M Ð(gi ) ∀gi ∈ 5
then M is a multiple of the identity matrix M = c1, c ∈ C.
Corollaries
(♠)
) If (♠) holds with M /= c1, c ∈ C, then {Ð(gi )} is reducible.
) All IRs of Abelian groups are one-dimensional
Proof: Take gj ∈ f arbitrary, but fixed.
f Abelian ⇒ D(gi ) D(gj ) = D(gj ) D(gi ) ∀gi ∈ f
Lemma ⇒ D(gj) = cj 1 with cj ∈ C , i.e., {D(gj ) = cj} is an IR.

Proof of Schur’s First Lemma (cf. Bir &
Pikus)
) Take Hermitean conjugate of (♠): M† Ð†(gi ) = Ð†(gi ) M†
Multiply with Ð†(gi ) = Ð−1(gi ): Ð(gi ) M† = M† Ð(gi )
) Thus: (♠) holds for M and M†, and also the Hermitean matrices
2 2
′′ i †
M′ = 1 (M + M†) M = (M −M )
) It exists a unitary matrix U that diagonalizes M′ (similar for M′′)
d = U−1 M′ U with dµν = dµδµν
) Thus (♠) implies Ð′(gi ) d = d Ð′(gi ), where Ð′(gi ) = U−1 Ð(gi ) U
more explicitly: Ðµ
′
ν(gi ) (dµ −dν) = 0 ∀i,µ, ν

Proof of Schur’s First Lemma (cont’d)
Two possibilities:
Thus: Ð′
κλ (gi ) = 0
) All dµ are equal, i.e, d = c1.
So M′ = UdU−1 and M′′ are likewise proportional to 1, and so is
M = M′ −iM′′.
) Some dµ are different:
Say {dκ : κ = 1, . . . , r} are different from {dλ : λ = r + 1, . . . , h}.
∀κ= 1,. . . , r;
∀λ = r + 1,. . . , h
Thus {Ð′(gi ) : i = 1, . . . , h} is block-diagonal, contrary to the
assumption that {Ð(gi )} is irreducible qed

Schur’s Second
Lemma
Schur’s Second Lemma: Suppose we have two
IRs
{Ð1(gi ), dimension n1} and {Ð2(gi ), dimension n2},
as well as a n1 × n2 matrix M such that
Ð1(gi ) M = M Ð2(gi ) ∀gi ∈ 5 (★)
(1) If {Ð1(gi )} and {Ð2(gi )} are inequivalent, then M = 0.
(2) If M /= 0 then {Ð1(gi )} and {Ð2(gi )} are equivalent.

Proof of Schur’s Second Lemma (cf. Bir &
Pikus)
) † −1
i i
−1
i
Take Hermitean conjugate of (★); use Ð (g ) = Ð (g ) = Ð(g ),
so
1 2
i i
M Ð (g ) = Ð (g )M
† −1 −1 †
) 2
−1
i 1
−1
i
Multiply by M on the left; Eq. (★) implies M Ð (g ) = Ð (g ) M,
so
1 1
MM Ð (g ) = Ð (g )MM
† −1 −1 † −1
i i i
∀g ∈ 5
) Schur’s first lemma implies that MM† is square matrix with
MM† = c1 with c ∈ C
) Case a: n1 = n2
• If c /= 0 then det M /= 0 because of (*), i.e., M is invertible.
(*)
So (★) implies −1
1 i 2 i
M Ð (g ) M = Ð (g ) i
∀g ∈ f
thus {Ð1(gi )} and {Ð2(gi )} are equivalent.
• If c = 0 then MM†
= 0, i.e.,
µν νµ
ν ν
M M = Mµν
† ∗
µν
Σ Σ Σ
ν
µν
2
M = |M | = 0 ∀µ
so that M = 0.

Proof of Schur’s Second Lemma
(cont’d)
) Case b: n1 /= n2 (n1 < n2 to be specific)
•Fill up M with n2 − n1 rows to get matrix M̃ with detM̃ = 0.
• However M̃M̃†
= MM†
, so that
det(MM†
) = det(M̃ M̃ †
) = (det M̃ ) (det M̃ †
) = 0
• So c = 0, i.e., MM†
= 0, and as before M = 0.
qed

Orthogonality Relations for IRs
Notation:
) Irreducible Representations (IR): ΓI = {ÐI (gi ) : gi ∈ f }
) nI = dimensionality of IR ΓI
) h = order of group f
Theorem 4: Orthogonality Relations for Irreducible Representations
(1) two inequivalent IRs ΓI /= ΓJ
h
Σ
i=1
Ð (g )∗
I i µ' ν' J i µν
Ð (g ) = 0
∀µ′, ν′ = 1, . . . , nI
∀ µ, ν =1, . . . , nJ
(2) representation matrices of one IR ΓI
I
h
n Σ
h
Remarks
i=1
I i
Ð (g )∗
µ' ν' Ð (g ) = δ δ
I i µν µ' µ ν'ν
′ ′
∀µ , ν , µ, ν = 1, . . . , nI
) [ÐI (gi )µν : i = 1, . . . , h] form vectors in a h-dim. vector space
)
√
I
vectors are normalized to h/n (because ΓI assumed to be unitary)
) vectors for different I, µν are orthogonal
) 2
Σ Σ 2
I I I I
in total, we have n such vectors; therefore n ≤ h
Corollary: For finite groups the number of inequivalent IRs is finite.

Proof of Theorem 4: Orthogonality Relations for IRs
(1) two inequivalent IRs ΓI /= ΓJ
) Take arbitrary nJ × nI matrix X /= 0 (i.e., at least one Xµν =
/ 0)
)
Σ
J i I
−1
i
Let M ≡ Ð (g ) X Ð (g )
Σ
i
J k J k J i I
i =M
¸ x`
−1 −1
i I
` ˛¸ x ` ˛¸ x
k I k
⇒ Ð (g ) M = Ð (g ) Ð (g ) X Ð (g ) Ð (g ) Ð (g )
=1
˛¸ x` ˛
Σ
i `˛¸x
=gj
−1
I k i
`˛¸x
=gj
= ÐJ (gk gi) X Ð (g g ) ÐI (gk )
Σ
j
J j I
−1
j I k
i Roland Winkler, NIU, Argonne, and NCTU 2011−2015
= Ð (g ) X Ð (g ) Ð (g )
=
`
M
˛¸ x
ÐI (gk )
⇒ (Schur’s Second Lemma)
0 = Mµµ' ∀µ,µ′
Σ Σ
i κ,λ
= Ð (g ) X Ð
J i µκ κλ I
−1
i
(g )λµ'
in particular correct for
Xκλ = δνκ δλν'
Σ
i
J i µν I
−1
i
= Ð (g ) Ð (g )ν' µ'
=
Σ
ÐI (gi )∗
µ'ν' ÐJ (gi )µν qed

Proof of Theorem 4: Orthogonality Relations for IRs (cont’d)
(2) representation matrices of one IR ΓI
First steps similar to case (1):
Σ
I i I
−1
i
) Let M ≡ Ð (g ) X Ð (g ) with nI × nI matrix X /= 0
i
⇒ ÐI (gk) M = M ÐI (gk )
⇒ (Schur’s First Lemma): M = c 1, c ∈ C
) µµ'
Thus c δ =
Σ Σ
i κ,λ
I i µκ κλ I
−1
i
Ð (g ) X Ð (g )λµ' choose Xκλ = δνκ δλν'
Σ
i
I i µν I
= Ð (g ) Ð (g−1
i )ν' µ' = Mµµ'
) c =
1
nI
Σ
µ
µµ
M =
nI
1 Σ Σ
i µ
I i µν I
−1
i
Ð (g ) Ð (g )ν µ
` ˛
¸ x
I
−1
i
Ð (g g = e)
i ν'ν νν
= δ '
h
' =
n
δ
I
νν ' qed

Goal: Characterize different irreducible representations of a
group
Characters
) The traces of the representation matrices are called characters
i i
Σ
i
χ(g ) ≡ tr Ð(g ) = Ð(g )
i µµ
) Equivalent IRs are related via a similarity transformation
Ð′(gi ) = X Ð(gi )X −1 with X nonsingular
This transformation leaves the trace invariant: tr Ð′(gi ) = tr Ð(gi )
⇒ Equivalent representations have the same characters.
) Theorem 5: If gi, gj ∈ 5 belong to the same class Ck of 5, then for
every representation ΓI of 5 we have χI (gi ) = χI (gj )
Proof:
• gi, gj ∈ C ⇒ ∃x ∈ f with gi = x gj x−1
• Thus ÐI (gi ) = ÐI (x) ÐI (gj ) ÐI (x−1
)
• χI (gi ) = tr ÐI (x) ÐI (gj ) ÐI (x−1
) (trace invariant under
cyclic permutation)
`
=
˛¸ x
= tr ÐI (x−1
) ÐI (x) ÐI (gj ) = χI (gk )

Characters (cont’d)
Notation
) χI (Ck) denotes the character of group elements in class Ck
) The array [χI (Ck)] with I = 1, . . . , N
k = 1, . . . , Ñ
(N = number of IRs)
(Ñ = number of classes)
is called character table.
Remark: For Abelian groups the character table is the table
of the 1 × 1 representation matrices
Theorem 6: Orthogonality relations for characters
Let {ÐI (gi)} and {ÐJ (gi )} be two IRs of 5. Let hk be the number of
elements in class Ck and Ñ the number of classes. Then
N˜
Σ h
k=1
k
h
∗
χ (C ) χ (C ) = δ
I k J k IJ ∀I,J = 1,...,N Proof: Use orthogonality
relation for IRs
) Interpretation: rows [χI (Ck) : k = 1, . . . Ñ] of character table
are like N orthonormal vectors in a Ñ-dimensional vector space
⇒ N ≤ Ñ.
) If two IRs ΓI and ΓJ have the same characters, this is necessary and
sufficient for ΓI and ΓJ to be equivalent.

Example: Symmetry group C3v of equilateral triangle
y
x
identity
rotation
φ
=
120
◦
rotation
φ
=
240
◦
reflecti
on
y
↔
−y
roto-
reflectio
n
φ
=
120
◦
roto-
reflection
φ
=
240
◦
P3
Koster
e
E
a
C3
b = a2
C2
3
c = ec
σv
d = ac
σv
f = bc
σv
Γ1 (1) (1)
(1)
,
− 1 −
√
3
,
√ 2 2
3 1
2 − 2
,
(1)
(1)
— 1
√
3
,
√2 2
— 3 − 1
2 2
(1)
(−1)
1 0
0 −1
,
(1)
(−1)
— 1 −
√
3
,
√2 2
3 1
— 2 2
(1)
(−1)
,
− 1
√
3
,
√ 2 2
3 1
2 2
Γ2 (1)
Γ3 1 0
0 1
multipli-
cation
table
P3 e a b c d f
e e a b c d f
a a b e f c d
b b e a d f c
c c d f e a b
d d f c b e a
f f c d a b e
Character table
P3
C3v
e a,b c, d, f
E 2C3 3σv
1
Γ1
Γ2
Γ3
1 1
1 1 −1
2 −1 0

Interpretation: Character Tables
) A character table is the uniquely defined signature of a group and
its IRs ΓI [independent of, e.g., phase conventions for representation
matrices ÐI (gi) that are quite arbitrary].
) Isomorphic groups have the same character tables.
) Yet: the labeling of IRs ΓI is a matter of convention. – Customary:
•Γ1 = identity representation: all characters are 1
• IRs are often numbered such that low-dimensional IRs come first;
higher-dimensional IRs come later
• If f contains the inversion, a superscript ± is added to ΓI indicating
the behavior of ΓI
±
under inversion (even or odd)
• other labeling schemes are inspired by compatibility relations
(more later)
) Different authors use different conventions to label IRs. To compare
such notations we need to compare the uniquely defined characters
for each class of an IR.
(See, e.g., Table 2.7 in Yu and Cardona: Fundamentals of Semiconductors;
here we follow Koster et al.)

Decomposing Reducible Representations
(RRs) Into Irreducible Representations
(IRs)
Given an arbitrary RR {Ð(gi )} the representation matrices {Ð(gi )} can
be brought into block-diagonal form by a suitable unitary transformation
′
i
Ð(gi ) → Ð (g ) =
1 i
Ð (g ) 0 )
..
Ð1(gi )
...
ÐN (gi )
...
ÐN (gi )
0
. a1 times
.
.
)
aN times
Theorem 7: Let aI be the multiplicity, with which the IR ΓI ≡ {ÐI (gi )}
is contained in the representation {Ð(gi )}. Then
N
Σ
I =1
(1) χ(gi ) = aI χI (gi )
I
(2) a =
1
h
h
Σ ∗
I i i
χ (g ) χ(g ) =
i=1 k=1
N˜
Σ hk
h
∗
I k k
χ (C ) χ(C )
We say: {Ð(gi )} contains the IR ΓI aI times.

Proof: Theorem 7
(1) due to invariance of trace under similarity transformations
N
(2) we have aJ χJ (gi ) = χ(gi )
Σ
J=1
1
h
h
Σ
i=1
χ∗
I (gi) ×
N
Σ
⇒ a
J=1
1
J
h
h
Σ
i=1
∗
I i J i
χ (g ) χ (g ) =
`
=
˛
δ
¸
IJ
x
1
h
h
Σ
i=1
∗
I i i
χ (g ) χ(g ) qed
Applications of Theorem 7:
) i
Corollary: The representation {Ð(g )} is irreducible if and only if
h
Σ
i=1
i
2
|χ(g )| = h
I
Proof: Use Theorem 7 with a =
1
0
for one I
otherwise
) Decomposition of Product Representations (see later)

Where Are We?
We have discussed the orthogonality relations for
) irreducible representations
) characters
These can be complemented by matching completeness relations.
Proving those is a bit more cumbersome. It requires the
introduction of the regular representation.

The Regular
Representation
Finding the IRs of a group can be tricky. Yet for finite groups we can derive
the regular representation which contains all IRs of the group.
) Interpret group elements gν as basis vectors {|gν⟩ : ν = 1, . . . h}
for a h-dim. representation
⇒ Regular representation:
νth column vector of ÐR (gi ) gives image |gµ⟩ = gi|gν⟩ ≡ |gi gν⟩
of basis vector |gν⟩
⇒ ÐR (gi )µν =
) Strategy:
1 if gµgν
−1 = gi
0 otherwise
• Re-arrange the group multiplication table
as shown on the right
1 2
g−1
g−1
g−1
3 . . .
. . .
e
g1
g2
g3
e
.
. e
.
. e
• For each gi ∈ f we have ÐR (gi )µν = 1,
if the entry (µ, ν) in the re-arranged
group multiplication table equals gi ,
otherwise ÐR (gi )µν = 0.

Properties of the Regular Representation {ÐR (gi )}
(1) {ÐR (gi)} is, indeed, a representation for the group 5
(2) It is a faithful representation, i.e., {ÐR (gi )} is isomorphic to 5 = {gi }.
R i
(3) χ (g ) =
h if gi = e
0 otherwise
Proof:
(1) Matrices {ÐR (gi )} are nonsingular, as every row / every column
contains “1” exactly once.
Show: if gi gj = gk, then ÐR(gi )ÐR (gj ) = ÐR(gk)
Take i, j, µ, ν arbitrary, but fixed
R i µλ
Ð (g ) = 1 only for gµ g−1
= gi ⇔ gλ = g−1
gµ
ÐR (gj )λν = 1
λ i
only for gλ gν
−1
= gj ⇔ gλ = gj gν
Σ
λ
⇔ Ð (g )
R i µ λ R j λν i
Ð (g ) = 1 only for g−1
gµ = gj gν
−1
⇔ gµ gν = gi gj = gk [definition of ÐR(gk )µν ]
(2) immediate consequence of definition of ÐR(gi )
(3) ÐR (gi )µµ =
1 if gi = gµ gµ
−1
= e
0 otherwise
R i
Σ
µ
⇒ χ (g ) = ÐR i µµ
(g ) =
h if gi = e
0 otherwise

Example: Regular Representation for
P3
e e a b c d f
a a b e f c d
b b e a d f c
c c d f e a b c c f d e a b
d d f c b e a d d c f b e a
f
Thus
f c d a b e f f d c a b e
⇒
g e a b c d f g−1 e b a c d f
e e b a c d f
a a e b f c d
b b a e d f c
e =
0
1 0
0 1
0 1 0
0 0 1 0
0 1 0 1 0 0 0 0 1
0 1 0 1 0 0
1 0 0 a = 0 1 0 b =
0 1 0 0 0 1
0 0 1 1 0 0
0 0 1
1 0 0
0 1 0
c =
1 0 0
0 1 0
0 0 1
1 0 0
0 1 0
0 0 1
d =
0 0 1
1 0 0
0 1 0
0 1 0
0 0 1
1 0 0
f =
0 1 0
0 0 1
1 0 0
0 0 1
1 0 0
0 1 0

Completeness of Irreducible
Representations
Lemma: The regular representation contains every IR nI times,
where nI = dimensionality of IR ΓI .
Σ
i
R i I I i
Proof: Use Theorem 7: χ (g ) = a χ (g ) where
I h
Σ
i
I i R i
1 ∗ 1
h
∗
` ˛¸ x ` ˛¸ x
=nI =h
a = χ (g ) χ (g ) = χ (e) χR (e) = n
I I
Corollary (Burnside’s Theorem): For a group 5 of order h,
the dimensionalities nI of the IRs ΓI obey
Σ
I
2
I
n = h R I I
Σ Σ
I I
Proof: h = χ (e) = a χ (e) = n
2
I
serious constraint for dimensionalities of IRs
Theorem 8: The representation matrices ÐI (gi ) of a group 5 of order h
obey the completeness relation
Σ Σ
I µ,ν
nI
h
∗
I
Ð (g )
i µν I j µν ij ∀i, j = 1,...,h
Ð (g ) = δ (*)
Proof:
) Theorem 4: Interpret [ÐI (gi )µν : i = 1,.. . , h] as orthonormal row vectors
of a matrix M
) Corollary: M has h columns
⇒ M is square matrix: unitary
⇒ column vectors also orthonormal
= completeness (*)

Completeness Relation for
Characters
Theorem 9: Completeness Relation for Characters
If χI (Ck) is the character for class Ck and irreducible representation I,
hk
h
then
Σ
I
∗
I k I
χ (C )χ (Ck' ) = δkk' ∀k, k′ = 1,...,N˜
) Interpretation: columns
χ1(Ck )
.
.
χN (Ck)
are like Ñ orthonormal vectors in a N-dimensional vector space
˜
of character table [k = 1, . . . , N]
) Thus Ñ ≤ N.
) Also N ≤ Ñ
(from com-
pleteness)
(from ortho-
gonality)
)
Number N of irreducible representations
= Number Ñ of classes
) Character table
• square table
• rows and column form orthogonal vectors

Proof of Theorem 9: Completeness Relation for
Characters
Lemma: Let {ÐI (gi)} be an nI -dimensional IR of 5. Let Ck be a class of
5 with hk elements. Then
Σ
i∈Ck
hk
nI
I i I k
Ð (g ) = χ (C ) 1
The sum over all representation matrices in a class of an IR is proportional to
the identity matrix.
Proof of Lemma:
) j
For arbitrary g ∈ 5
Σ
i∈Ck
ÐI (gj ) Ð (g ) Ð
I i I
−1
j
(g ) =
k
I j I i I
−1
j
i∈C ` ˛¸
= D I (gi' ) with i'∈Ck
Ð (g ) Ð (g ) Ð (g ) =
x
Σ Σ
because gj maps gi1 =
/ gi2 onto gi1
' /= gi
2
'
i'
x ∈C k
I
Ð (gi' )
Σ
i∈Ck
I i k
⇒ (Schur’s First Lemma): Ð (g ) = c 1
) ck
nI i∈Ck
I i
1 Σ hk
nI
I k
= tr Ð (g ) = χ (C ) qed

Proof of Theorem 9: Completeness Relation for
Characters
) Use Theorem 8 (Completeness Relations for Irreducible Representations)
N
I
h
∗
Ð (g ) Ð (g ) = δ
I i µν I j µν ij
Σ Σ n Σ Σ
I =1 µ,ν i∈Ck j∈Ck'
⇒
I
h
Σ n Σ
I µ,ν
Σ ∗
Ð (g )
µν
k
nI
∗
χ (C ) δ
I k µν
Σ
i∈Ck j∈Ck'
µν
` ˛¸ x ` ˛¸ x
h hk'
nI
'
χ (C ) δ
I k µν (Lemma)
` ˛¸ x
k
h hk'
n2
I
∗
I k I k
Σ
µ,ν
χ (C ) χ (C ' ) δµν
`
=
˛
n
¸
I
x
I i I j k kk
Ð (g ) = h δ '
qed

Summary: Orthogonality and Completeness
Relations
nI
h
Theorem 4: Orthogonality Relations for Irreducible
Representations
h
Σ
i=1
I
Ð (g )∗
i µ' ν' Ð (g ) = δ δ ' δ
J i µν IJ µµ νν '
I,J = 1,...,N
µ′, ν′ = 1,...,nI
µ, ν = 1,...,nJ
Theorem 8: Completeness Relations for Irreducible
Representations
N
Σ Σ
I =1 µ,ν
nI
h
∗
I
Ð (g ) Ð (g ) = δ
i µν I j µν ij ∀i, j = 1,...,h
Theorem 6: Orthogonality Relations for Characters
N˜
Σ
k=1
hk
h
∗
I
χ (C ) χ (C ) = δ
k J k IJ ∀I,J = 1,...,N
Theorem 9: Completeness Relation for Characters
hk
h
N
Σ
I =1
∗
I
χ (C )χ (C
k I k' ) = δkk ' ∀k, k′ = 1,...,N˜

h h h h h h h
Unreducible
More on Irreducibh
lh
e Problems

Group Theory in Quantum Mechanics
Topics:
) Behavior of quantum mechanical states and operators
under symmetry operations
) Relation between irreducible representations and invariant subspaces
of the Hilbert space
) Connection between eigenvalue spectrum of quantum mechanical
operators and irreducible representations
) Selection rules: symmetry-induced vanishing of matrix elements
and Wigner-Eckart theorem
Note:
Operator formalism of QM convenient to discuss group theory.
Yet: many results also applicable in other areas of physics.

Symmetry Operations in Quantum Mechanics (QM)
) Let 5 = {gi } be a group of symmetry operations of a qm system
e.g., translations, rotations, permutation of particles
) Translated into the language of group theory:
In the Hilbert space of the qm system we have a group of unitary
operators 5′ = {P̂(gi )} such that 5′ is isomorphic to 5.
Examples
) translations Ta
→ unitary operator P̂(Ta) = exp(ip̂ · a/k) (p̂ = momentum)
ˆ a 2
1 2
P(T ) ψ(r) = 1 + ∇ · a + (∇ · a) + . . . ψ(r) = ψ(r + a)
) rotations Rφ (L̂ = angular momentum
→ unitary operator P̂(n, φ) = exp iL̂ · nφ/k φ = angle of rotation
n = axis of rotation)

Transformation of QM States
) Let {|ν⟩} be an orthonormal basis
) Let P̂(gi ) be the symmetry operator for the symmetry transformation gi
with symmetry group 5 = {gi }.
) ˆ i
1 =
Σ
µ |µ⟩(µ|
Σ
↑ µ
ˆ i
Then P(g ) |ν⟩ = |µ⟩ ⟨µ| P(g ) |ν⟩
` ˛¸ x
o(gi )µν
ˆ
Σ
µ
i i µν
) So P(g ) |ν⟩ = Ð(g ) |µ⟩ where Ð(gi )µν =
matrix of a unitary
representation of f
because P̂(gi ) unitary
) Note: bras and kets transform according to complex conjugate
ˆ i
representations Σ
µ
⟨ν|P(g ) = ⟨µ| Ð(g )
† ∗
i µν
) Let gi, gj ∈ 5 with gi gj = gk ∈ 5. Then (consistent with
matrix multiplication)
ˆ ˆ
i j
P(g ) P(g ) |ν⟩ =
,
,
Σ
κµ
o(gi )κµ o(gi )µν
¸ x` ˛ ¸ x` ˛
ˆ ˆ
Σ
κµ
i i i κµ i µν
|κ⟩⟨κ| P(g ) |µ⟩ ⟨µ| P(g ) |ν⟩ = Ð(g ) Ð(g ) |κ⟩
Σ
κ
P̂(gk) |ν⟩ = Ð(gk )κν |κ⟩

Transformation of (Wave) Functions ψ(r)
) i i
ˆ ˆ i
′ ′ −1
i
P(g )|r⟩ = |r =g r⟩ ⇔ ⟨r|P(g ) = ⟨r =g r| b/c P̂(g)†
= P̂(g−1
)
) Let ψ(r) ≡ ⟨r|ψ⟩
ˆ ˆ
i i
−1
i i
⇒ P(g ) ψ(r) ≡ ⟨r|P(g )|ψ⟩ = ψ(g r) ≡ ψ (r)
) In general, the functions V = {ψi (r) : i = 1, . . . , h} are linear dependent
⇒ Choose instead linear independent functions ψν(r) ≡ ⟨r|ν⟩ spanning V
⇒ Expand images P̂(gi ) ψν(r) in terms of {ψν (r)}:
ˆ ˆ
i i
µ
ˆ
Σ Σ
µ
i i µν µ
P(g ) ψ(r) = ⟨r|P(g )|ψ⟩ = ⟨r|µ⟩⟨µ|P(g )|ν⟩ = Ð(g ) ψ (r)
ˆ
i ν ν
−1
i
Σ
µ
i µν µ
⇒ P(g ) ψ (r) = ψ (g r) = Ð(g ) ψ (r)
) Thus: every function ψ(r) induces a matrix representation Γ = {Ð(gi )}
) Also: every representation Γ = {Ð(gi )} is completely characterized by a
(nonunique) set of basis functions {ψν (r)} transforming according to Γ.
) Dirac bra-ket notation convenient for formulating group theory of functions.
Yet: results applicable in many areas of physics beyond QM.

Important Representations in Physics (usually
reducible)
(1) Representations for polar and axial (cartesian)
vectors
) generally: two types of point group symmetry operations
• proper rotations gpr = (n, θ) about axis n, angle θ
Ð[gpr = (n, θ)] = Rodrigues’ rotation formula
2
x
n (1 − cos θ) + cos θ
2
y x z y y z x
n n (1 − cos θ) + n sin θ n (1 − cos θ) + cos θ n n (1 − cos θ) − n sin θ
nznx (1 − cos θ) − ny sin θ nzny (1 − cos θ) + nx sin θ 2
z
n (1 − cos θ) + cosθ
nx ny (1 − cos θ) − nz sin θ nx nz(1 − cos θ) + ny sin θ
) det Ð(gpr) = +1
) χ(gpr) = tr Ð(gpr) = 1 + 2cosθ independent of n
• improper rotations gim ≡ i gpr = gpri where i = inversion
) polar vectors
•proper rotations gpr:
pol pr
) det Ð (g ) = +1
) tr Ðpol(gpr) = 1 + 2cosθ
• inversion i: Ðpol(i) = −13×3
• improper rotations gim = i gpr:
) Dpol(gim) = −Dpol(gpr)
) det Dpol(gim) = −1
) tr Dpol(gim) = −(1 + 2 cos θ)
• Γpol = {Ðpol(g)} ⊆ O(3) always a faithful representation (i.e., isomorphic to f)
• examples: position r, linear momentum p, electric field E

Important Representations in
Physics
Γpol
Γax
= {Ðpol(gi ) : i = 1, . . . , h}
= {Ðax(gi ) : i = 1, . . . , h}
(1) Representations for polar and axial (cartesian) vectors (cont’d)
) axial vectors
•proper rotations gpr:
) Ðax(gpr) = Ðpol(gpr)
) det Ðax(gpr) = +1
) tr Ðax(gpr) = 1 + 2cosθ
•inversion i: Ðax(i) = +13×3
• improper rotations gim = i gpr:
) Ðax(gim) = Ðax(gpr) = −Ðpol(gpr)
) det Ðax(gim) = +1
) tr Ðax(gim) = 1 + 2cosθ
•Γax = {Ðax(g)} ⊆ SO(3)
• examples: angular momentum L, magnetic field B
) systems with discrete symmetry group 5 = {gi : i = 1, . . . , h}:
We have a “universal recipe” to
construct the 3 × 3 matrices Dpol(g)
and Dax(g) for each group element
gpr = (n,θ) and gim = i (n, θ)

Important Representations in Physics (cont’d)
(2) Equivalence Representations Γeq
) Consider symmetric object (symmetry group 5)
• vertices, edges, and faces of platonic solids are equivalent by symmetry
• atoms / atomic orbitals |µ⟩ in a molecule may be equivalent by symmetry
) Equivalence representation Γeq describes mapping of equivalent objects
) ˆ
Σ
eq νµ
Generally: P(g) |µ⟩ = Ð (g) |ν⟩
ν
) Example: orbitals of equivalent H atoms in NH3 molecule (group C3v )
Equivalent to: permutations of corners of triangle (group P3)

Example: Symmetry group C3v of equilateral triangle
z
y
3 2
1
x
identity
rotation
φ
=
120
◦
rotation
φ
=
240
◦
reflection
y
↔
−y
roto-
reflection
φ
=
120
◦
roto-
reflection
φ
=
240
◦
P3
Koster
e
E
a
C3
b = a2
C2
3
c = ec d = ac f = bc
σv σv σv
Γ1 (1) (1)
(1)
,
− 1 −
√
3
,
√ 2 2
3 1
2 − 2
,
(1)
(1)
— 1
√
3
,
√2 2
3 1
— 2 − 2
(1) (1) (1)
(−1) (−1) (−1)
1 0
,
− 1 −
√
3
, ,
− 1
√
3
,
√2 2 √ 2 2
0 −1 − 3 1 3 1
2 2 2 2
Γ2 (1)
Γ3
1 0
n, θ (0, 0, 1), 0 (0, 0, 1), 2π/3 (0, 0, 1), 4π/3
− 1 −
√
3
!
− 1
√
3
!
√ 2 2 √2 2
3 1 3 1
2 − 2 − 2 − 2
1 1
− 1 −
√
3
!
− 1
√
3
!
√ 2 2 √2 2
3 1 3 1
2 − 2 − 2 − 2
1 1
√
3 1
√
3 1
(0, 1, 0), π (− 2 , −2 , 0), π (− 2 , 2 , 0), π
, 1
, − 1 −
√
3
!
− 1
√
3
!
−1 √2 2 √ 2 2
3 1 3 1
1
− 2 2 2 2
1 1
, −1
, 1
√
3
! 1 −
√
3
!
√2 2 2√ 2
1 3 − 1 − 3 − 1
−1 2 2 2 2
−1 −1
Γpol =
Γ1 + Γ3
, 1
1
1
,
Γax =
Γ2 + Γ3
, 1
1
1
,
Γeq =
Γ1 + Γ3
1
1
1
1
1
1
1
1
1
1
1 1
1
1
1
1 1 1

Transformation of QM States (cont’d)
) in general: representation {Ð(gi )} of states {|ν⟩} is reducible
) We have Ð′(gi ) = U−1 Ð(gi ) U
) ′
More explicitly: Ð (g )
i µ ν
Σ
µν
−1
µ' µ
' ' = U Ð(g )
` ˛¸ x
i µν νν
(µ| Pˆ(gi ) |ν⟩
U '
µν
Σ −1
µ' µ
ˆ
= ⟨µ|U P(g ) U
i νν ' |ν⟩
= ⟨µ′| P̂(gi ) |ν′⟩
′
Σ
with |ν ⟩ = Uνν' |ν⟩
ν
) Thus: block diagonalization
{Ð(gi )} → {Ð′(gi ) = U−1 Ð(gi ) U}
corresponds to change of basis
′ Σ
ν
{|ν⟩} → {|ν ⟩ = Uνν' |ν⟩}

Basis Functions for Irreducible
Representations
) matrices {ÐI (gi)} are fully characterized by basis functions
I
{ψ (r) : ν = 1, . . . n } transforming according to IR Γ
ν I I
P̂(gi
I −1
ν ν i
Σ
µ
) ψ (r) = ψ (g r) = Ð (g
I i µν
I
µ
) ψ (r)
) convenient if we need to spell out phase conventions for {ÐI (gi )} (→ Koster)
) identify IRs for (components of) polar and axial vectors
Example: Symmetry group C3v
z
y
x
identity
rotation
φ
=
120
◦
rotation
φ
=
240
◦
reflection
y
↔
−y
roto-
reflection
φ
=
120
◦
roto-
reflection
φ
=
240
◦
basis
functions
(1) (1) (1) (1) (1) (1)
x2
+ y2
;z;
L2
+ L2
Γ1
Γ2 (1) (1) (1) (−1) (−1) (−1)
x y
Lz
Γ3
1 0
0 1
— −
1 3
√ 2 2
3 − 1
2 2
− 1 3
√2 2
— −
3 1
2 2
, √ , , √ ,
1 0
0 −1
1 3
√2 2
3 − 1
2 2
, √ , ,
−
1 3
2√ 2
— −
3 1
2 2
√ ,
x, y
1 0
0 1
— −
1 3
√ 2 2
−
3 1
2 2
−
, √ , , √
1 3
√2 2
— −
3 1
2 2
,
−1 0
0 1
,
— −
1 3
−
√2 2
3 1
2 2
− 1 3
2 2
√
2
3 1
2
√ , , √ ,
Lx , Ly
r = (x, y, z) = polar vector, L = (Lx , Ly , Lz) = axial vector

) The block diagonal form of {Ð′
(gi )} implies that {|µ′
⟩ is reducible
Relevance of Irreducible Representations
Invariant
Subspaces
Definition:
) Let 5 = {gi } be a group of symmetry transformations.
Let H = {|µ⟩} be a Hilbert space with states |µ⟩.
A subspace S ⊂ H is called invariant subspace (with respect to 5) if
P̂(gi ) |µ⟩ ∈ S ∀gi ∈ 5, ∀|µ⟩ ∈ S
) If an invariant subspace can be decomposed into smaller invariant
subspaces, it is called reducible, otherwise it is called irreducible.
Theorem 10:
An invariant subspace S is irreducible if and only if the states in S
transform according to an irreducible representation.
Proof:
) Suppose {Ð(gi )} is reducible.
) ∃ unitary transformation U with {Ð′
(gi ) = U−1
Ð(gi ) U} block diagonal
) For {Ð′
(gi )} we have the basis {|µ′
⟩ =
Σ
µ Uµµ' |µ⟩}

Invariant Subspaces
(cont’d)
Corollary: Every Hilbert space H can be decomposed into irreducible
invariant subspaces SI transforming according to the IR ΓI
Remark: Given a Hilbert space H we can generally have multiple
(possibly orthogonal) irreducible invariant subspaces S α
I
α
I
S = |I να⟩ : ν = 1, . . . , nI
}
transforming according to the same IR ΓI
ˆ
Σ
µ
i I i µν
P(g ) |I να⟩ = Ð (g ) |I µα⟩
Theorem 11:
(1) States transforming according to different IRs are orthogonal
(2) For states |I µα⟩ and |I νβ⟩ transforming according to the same
IR ΓI we have
⟨I µα | I νβ⟩ = δµν ⟨I α||I β⟩
where the reduced matrix element ⟨I α||I β⟩ is independent of µ, ν.
Remark: This theorem lets us anticipate the Wigner-Eckart theorem

Invariant Subspaces
(cont’d)
Proof of Theorem 11
) Use unitarity of P̂(gj ):
) Then
†
ˆ ˆ
j j
1
h
Σ
i
ˆ † ˆ
i i
1 = P(g ) P(g ) = P(g ) P(g )
1
h
Σ
ˆ †
Σ
ˆ
i i
⟨I µα |Jνβ⟩ = ⟨I µα| P(g ) P(g ) |Jνβ⟩
i ` ˛¸ x ` ˛¸ x
Σ
µ ' (I µ' α|oI (gi )∗
µ'µ ν ' o J (gi )ν ' ν |J ν' β⟩
µ' ν'
′ ′ 1
Σ Σ
h i
∗
= ⟨I µ α |J ν β⟩ Ð (g ) Ð (g )
I i µ' µ J i ν'ν
` ˛¸ x
(1/nI ) δIJ δµν δµ ' ν '
= δ δ
IJ µν
Σ
µ'
′ ′
⟨I µ α |I µ β⟩
1
nI
` x
≡(I α
˛¸
|
|
Iβ⟩ qed

Discussion Theorem
11
) ΓJ × ΓI contains the identity representation Γ1 if and only if the IR ΓJ is the
complex conjugate of ΓI , i.e., Γ∗
J = ΓI ⇔ ÐJ (g)∗ = ÐI (g) ∀g.
) If the ket |Jµα⟩ transforms according to the IR ΓJ , the bra ⟨Jµα|
transforms according to the complex conjugate representation Γ∗
J.
) Thus: ⟨J µα|I νβ⟩ =
/ 0 equivalent to
• bra and ket transform according to complex conjugate representations
• ⟨Jµα|I νβ⟩ contains the identity representation
) Indeed, common theme of representation theory applied to physics:
Terms are only nonzero if they transform according to a representation
that contains the identity representation.
) Variant of Theorem 11 (Bir & Pikus):
If fI (x) transforms according to some IR ΓI , then
∫
fI (x) dx /= 0
only if ΓI is the identity representation.
) Applications
• Wigner-Eckart Theorem
• Nonzero elements of material tensors
• Our universe would be zero “by symmetry” if the apparently trivial identity
representation did not exist. Roland Winkler, NIU, Argonne, and NCTU 2011−2015

Decomposition into Irreducible Invariant
Subspaces
) Goal: Decompose general state |ψ⟩ ∈ H into components from
irreducible invariant subspaces SI
) Generalized projection operator Π̂ I nI
µµ h
Σ
i
' := ÐI (gi )∗
µµ
ˆ i
' P(g )
) Theorem 12: (i)
(ii)
Π̂ I
µµ ' |Jνα⟩ = δIJ δµ'ν |Iµα⟩
Π̂ I J
µµ νν
' '
Π = δ δ
IJ µ'ν
ˆ Π̂
J
µν '
Σ
I µ
ˆI
µµ
(iii) Π = 1
Proof:
µµ
nI
h
Σ
i
∗
I i µµ
ˆ i
Σ
ν ' DJ (gi ) ν ' ν |Jν'α⟩
(i) Π̂ I
' |Jνα⟩ = Ð (g ) ' P(g ) |Jνα⟩ =
ν
nI
Σ Σ
h i
∗
Ð (g ) ' Ð (g ) '
I i µµ J i ν ν
˛¸ x ' ` ˛¸ x
`
δIJ δ µ ν ' δµ'ν
′
|Jν α⟩
ˆ
(ii) Π̂ I
' ΠJ
µµ νν ' =
=
i
n n
I J
h h
Σ Σ
j
∗
µµ
I i J j
∗
νν
ˆ ˆ
i j
Ð (g ) ' Ð (g ) ' P(g ) P(g ) subst. gigj = gk
n n
I J
h h
Σ Σ
i k
∗
µµ i
Ð (g ) ' Ð (g g
I i J k
−1 ∗ ˆ
νν k
) ' P(g )
=
kλ
n n
J I
h h
Σ Σ
i
∗
µµ
I i J i
Ð (g ) ' Ð (g )
−1 ∗
` ˛¸ x
DJ (gi )λ ν
` ˛¸ x
δIJ δ µ λ δ µ ' ν
∗ ˆ
ν λ J k λν k
Ð (g ) ' P(g )
[or use (i)]
Σ
Iµ
ˆ
(iii) Π
I
µµ = 1
h
Σ Σ Σ
µ I i
Ð (g )∗
µµ
χI
∗(gi )
i I
` ˛¸ x`˛¸x
χI (e)
ˆ
I i
Σ
i
' n P(g ) = δ i
g e i
ˆ ˆ
P(g ) = P(e) ≡ 1

Decomposition into Invariant Subspaces
(cont’d)
Discussio
n Σ
Jνα
) Let |ψ⟩ = cJνα |Jνα⟩ general state with coefficients cJµα (*)
µµ
) Diagonal operator Π̂ I projects |ψ⟩ on components |Iµα⟩:
µµ
ˆ ˆ
I 2 I Σ
α
µµ Iµα
• (Π ) |ψ⟩ = Π |ψ⟩ = c |I µα⟩
Σ
I µ
Î
µµ
• Π = 1
Î Σ
µ
ˆ
) Let Π ≡ ΠI
µµ = I
h
Σ
n ∗
i
I
ˆ
i i
χ (g ) P(g ) :
Î Σ
να
• Π |ψ⟩ = cIνα |I να⟩
• Π̂ I
projects |ψ⟩ on the invariant subspace SI (IR ΓI )
) For functions ψ(r) ≡ ⟨r|ψ⟩:
Π̂ I
µµ ' ψ(r) = nI
h
Σ
i
I i
Ð (g )∗ −1
µµ i
' ψ(g r)
we need not know
the expansion (*)

Irreducible Invariant Subspaces
(cont’d)
Example
:
) Group Ci = {e, i}
Ci e i
e e i
i i e
) character table
e = identity
i = inversion
Ci e i
Γ1
Γ2
1 1
1 −1
) P̂(e)ψ(x) = ψ(x), P̂(i) ψ(x) = ψ(−x)
) ˆI
Projection operator Π = I
h
Σ
n ∗
i
ˆ
I i i
χ (g ) P(g ) with nI = 1, h = 2
2
) Π̂1 = 1 [P̂(e) + P̂(i)] ˆ1 1
2
⇒ Π ψ(x) = [ψ(x) + ψ(−x)] even part
⇒
2 1
2
ˆ ˆ ˆ
Π = [P(e) −P(i)] ˆ2 1
2
Π ψ(x) = [ψ(x) −ψ(−x)] odd part

Product
Representations
) Let {|I µ⟩ : µ = 1, . . . nI } and {|Jν⟩ : ν = 1, . . . nJ} denote basis functions
for invariant subspaces SI and SJ (need not be irreducible)
Consider the product functions
{|I µ⟩ |Jν⟩ : µ = 1, . . . , nI ; ν = 1, . . . , nJ}.
How do these functions transform under 5?
) Definition: Let ÐI (g) and ÐJ (g) be representation matrices for g ∈ 5.
The direct product (Kronecker product) ÐI (g) ⊗ÐJ (g) denotes the matrix
[ÐI (g) ⊗ ÐJ (g)]µν,µ'ν' = ÐI (g)µµ' ÐJ (g)νν'
whose elements in row (µν) and column (µ′ν′) are given by
µ, µ′
= 1, . . . , nI
ν, ν′
= 1, . . . , nJ
x11 x12
x21 x22
,
J
) Example: Let ÐI (g) =
,
and Ð (g) =
y y
11 12
y21 y22
, ,
ÐI (g) ⊗ ÐJ (g) =
x11 ÐJ(g) 12 J
x Ð (g)
x21 ÐJ(g) x22 ÐJ(g)
, ,
=
x11y11 x11y12 x12y11 x12y12
x11y21 x11y22 x12y21 x12y22
x21y11 x21y12 x22y11 x22y12
x21y21 x21y22 x22y21 x22y22
) Details of the arrangement in the following not relevant

Product Representations (cont’d)
) Dimension of product matrix
dim[ÐI (g) ⊗ÐJ (g)] = dim ÐI (g) dim ÐJ (g)
) Let ΓI = {ÐI (gi )} and ΓJ = {ÐJ (gi )} be representations of 5. Then
ΓI × ΓJ ≡ {ÐI (g) ⊗ ÐJ (g)}
is a representation of 5 called product representation.
) ΓI × ΓJ is, indeed, a representation:
Let ÐI (gi ) ÐI (gj ) = ÐI (gk ) and ÐJ (gi ) ÐJ (gj ) = ÐJ (gk )
⇒ [ÐI (gi ) ⊗ ÐJ (gi )][ÐI (gj ) ⊗ ÐJ (gj )] µν,µ' ν'
Σ
κ λ
I i µ κ J i νλ I j κ µ J j λν
= Ð (g ) Ð (g ) Ð (g ) ' Ð (g ) '
→ ÐI (gk )µµ' → ÐJ (gk )νν'
= [ÐI (gk ) ⊗ ÐJ (gk)]µν,µ'ν'
) Let P̂(g) |Iµ⟩ =
Σ
µ ' ÐI (g)µ' µ |Iµ′⟩
P̂(g) |Jν⟩ =
Σ
ν ' ÐJ (g)ν'ν |Jν′⟩
ˆ
Then P(g) |Iµ⟩|
Σ
µ' ν'
I J
Jν⟩ = [Ð (g) ⊗Ð (g)]µ'ν',µν |I µ′⟩|Jν′⟩

Product Representations (cont’d)
) The characters of the product representation are
χI × J (gi ) = χI (gi ) χJ (gi )

Decomposing Product
Representations
K
I J
I J K
Γ × Γ = a ΓK
I J
K
where a =
) Let ΓI = {ÐI (gi)} and ΓJ = {ÐJ (gi )} be irreducible representations of 5
The product representation ΓI × ΓJ = {ÐI × J (gi)} is generally reducible
) According to Theorem 7, we have
N˜
Σ Σ h
k=1
k
h
∗
K
χ (C ) χ
k I × J k
(C )
` ˛¸ x
=χI (Ck ) χJ (Ck )
) The multiplication table for the
I
irreducible representations Γ of 5
lists
Σ
k K
aI J ΓK
ΓI × ΓJ Γ1 Γ2 . . .
Γ1
Γ2
.
) Example: Permutation group P3
χ(C) e a,b c, d, f
1
Γ1 1 1
Γ2 1 1 −1
Γ3 2 −1 0
ΓI × ΓJ Γ1 Γ2 Γ3
Γ1 Γ1 Γ2
Γ2 Γ1
Γ3
Γ3
Γ3
Γ1 + Γ2 + Γ3

(Anti-) Symmetrized Product Representations
Let {|σµ ⟩} and {|τν ⟩} be two sets of basis functions for the same n-dim.
representation Γ = {Ð(g )} with characters {χ(g )}. (again: need not be
irreducible)
(1) “Simple” Product:
) |ψµν ⟩ = |σµ⟩|τν ⟩,
(discussed previously)
µ = 1, . . . , n
ν = 1,. . . n
,
total: n2
) ˆ µν
P(g)|ψ ⟩ =
n
µ'=1 ν'=1
n
Σ Σ
Ð(g)µ'µÐ(g )ν'ν |σµ' ⟩|τν' ⟩
n n
Σ Σ
µ'=1 ν'=1
≡ [Ð(g) ⊗ Ð(g)]µ'ν',µν |ψµ' ν' ⟩
) Character tr[Ð(g) ⊗ Ð(g)] = χ2(g)

(Anti-) Symmetrized Product Representations (cont’d)
(2) Symmetrized Product:
)
)
s 1
µν 2
|ψ ⟩ = (|σ ⟩|
µ ν ν µ
τ ⟩ + |σ ⟩|τ ⟩),
µ = 1, . . . , n
ν = 1,. . . µ
,
2
total: 1 n(n + 1)
ˆ s
µν
P(g)|ψ ⟩ = 1
2
n n
Σ Σ
µ'=1 ν'=1
Ðµ' µÐν' ν (|σµ⟩|τν ⟩ + |σν⟩|τµ⟩)
=
'
n µ −1
Σ Σ
µ'=1 ν'=1
(Ð Ð + Ð Ð
µ' µ ν'ν µ' ν ν' µ
s
µ' ν'
)|ψ ⟩ + Ðµ' µ µ'ν
Ð |ψ
s
µ' µ' ⟩
≡
n
µ'=1 ν'=1
µ'
Σ Σ
[Ð(g) ⊗ Ð(g)]
(s)
µ'ν',µν
s
|ψµ' ν' ⟩
) (s)
tr[Ð(g) ⊗ Ð(g)] =
n µ−1
Σ Σ
µ=1 ν=1
(ÐµµÐνν + Ðµν Ðνµ) + ÐµµÐµµ
1
= 2
n n
Σ Σ
µ=1 ν=1
[Ðµµ(g )Ðνν (g) + Ðµν (g)Ðνµ(g )]
1
n
Σ n
Σ
µ=1 ν=1
νν µµ
2
= 2 Ðµµ(g ) Ð (g) + Ð (g )
2
= 1 [χ(g)2 + χ(g2)]

(Anti-) Symmetrized Product Representations (cont’d)
(3) Antisymmetrized Product:
)
)
a 1
µν 2
|ψ ⟩ = (|σ ⟩|
µ ν ν µ
τ ⟩ −|σ ⟩|τ ⟩), µ = 1, . . . , n
ν = 1, . . . µ − 1
,
total: 1
2
n(n − 1)
ˆ a
µν
P(g)|ψ ⟩ = 1
2
n n
Σ Σ
µ'=1 ν'=1
Ðµ' µÐν' ν (|σµ⟩|τν ⟩ −|σν ⟩|τµ⟩)
=
n µ'−1
Σ Σ
µ'=1 ν'=1
µ' µ ν'ν µ' ν ν' µ
a
µ ν
' '
(Ð Ð −Ð Ð )|ψ ⟩
≡
n µ'−1
Σ Σ
µ'=1 ν'=1
[Ð(g) ⊗ Ð(g)]
(a) a
µ ν ,µν µ ν
' ' |ψ ' ' ⟩
) (a)
tr[Ð(g) ⊗ Ð(g)] =
n µ−1
Σ Σ
µ=1 ν=1
µµ νν µν νµ
(Ð Ð −Ð Ð )
1
= 2
n n
Σ Σ
µ=1 ν=1
[Ðµµ(g )Ðνν (g) − Ðµν (g)Ðνµ(g )]
1
n
Σ n
Σ
µ=1 ν=1
νν µµ
2
= 2 Ðµµ(g ) Ð (g) −Ð (g )
2
1 2 2
= [χ(g) −χ(g )]

Intermezzo: Material Tensors
to be added . . .

Discussio
n
) Representation – Vector Space
The matrices {Ð(gi )} of an n-dimensional (reducible or irreducible)
representation describe a linear mapping of a vector space V onto itself.
o(gi ) ′
1 n µ
Σ
u = (u , . . . , u ) ∈ V : u −→ u′ ∈ V with u = Ð(g ) u
i µν ν
ν
) Irreducible Representation (IR) – Invariant Subspace
The decomposition of a reducible representation into IRs ΓI corresponds to
a decomposition of the vector space V into invariant subspaces SI such that
o I (gi )
SI −
→ SI ∀gi ∈ 5 (i.e., no mixing)
This decomposition of V lets us break down a big physical problem
into smaller, more tractable problems
) Product Representation – Product Space
A product representation ΓI × ΓJ describes a linear mapping of the product
space SI × SJ onto itself
o I × J (gi )
SI × SJ −→ SI × SJ ∀gi ∈ 5
) I J
Σ
K K
The block diagonalization Γ × Γ = aIJ ΓK corresponds to a
decomposition of SI × SJ into invariant subspaces SK

Discussion
(cont’d)
Clebsch-Gordan Coefficients
(CGC)
) I J
Σ
K K
The block diagonalization Γ × Γ = aIJ ΓK corresponds
to a decomposition of SI × SJ into invariant subspaces SK
⇒ Change of Basis: unitary transformation
(
I J
K A=1
aIJ
K
Σ Σ
S
A
K
old basis { I J
µ ν
S × S −→
e e } −→ K A
n
new basis {e }
K A
n
Thus e =
Σ
µν
I J K l
µ ν n
e e
I J
µ ν
index l not needed
IJ
K
if often a ≤ 1
where
I J K l
µ ν n
= Clebsch-Gordan coefficients (CGC)
Clebsch-Gordan coefficients describe the unitary transformation for the
I J
decomposition of the product space S × S into invariant subspaces S
Æ
K

(cont’d)
Remarks
) CGC are independent of the group elements gi
) CGC are tabulated for all important groups (e.g., Koster, Edmonds)
) Note: Tabulated CGC refer to a particular definition (phase convention)
I
µ I i
for the basis vectors {e } and representation matrices {Ð (g )}
) Clebsch-Gordan coefficients C describe a unitary basis transformation
C†
C = C C†
= 1
) Thus Theorem 13: Orthogonality and completeness of CGC
Σ
µν
∗ ′ ′
I J K l I J K l
µ ν n µ ν n′ = δKK' δnn' δAA'
Σ
K An
I J K l
µ ν n
∗
I J K l
µ′ ν′ n
= δµµ' δνν'

(cont’d)
Clebsch-Gordan coefficients
block-diagonalize the representation
matrices (unitary transformation)
(1) I x J = C C†
(2) = C†
I x J C
More explicitly:
Theorem 14: Reduction of Product Representation ΓI × ΓJ
(1) ÐI (gi )µµ' ÐJ (gi )νν' =
Σ Σ
KA nn'
ÐK (gi )nn'
µ′ ν′
I J K l I J K l
µ ν n n′
∗
(2) Ð (g ) δ δ
K i nn' KK' AA'
=
Σ Σ
µµ' νν'
I J K l
µ ν n
∗
Ð (g )
I i µµ ' Ð (g ) '
I J
J i νν µ′ ν′
K′ l′
n′

Evaluating Clebsch-Gordan
Coefficients
) A group 5 is called simply reducible if its product representations
I J K
IJ
K
Γ × Γ contain the IRs Γ only with multiplicities a = 0 or 1.
) For simply reducible groups (⇒ no index l) according to Theorem 14 (1):
nK
h
Σ
∗
I i µµ J i νν K i n
˜
n
˜
Ð (g ) ' Ð (g ) ' Ð (g ) '
=
Σ
i Σ
K ' nn'
I J K ′
µ ν n
′ ∗
n
I J K K
µ′ ν′ n′
h
Σ
i
`
ÐK ' (gi )nn' ÐK
∗ (gi )n˜n˜'
˛¸ x
= δK' K δn˜n δn˜'n' (Theorem 4)
=
I J K I J K
µ ν n
˜ µ′ ν′ n
˜ ′
∗
) Choose triple µ = µ′ = µ0, ν = ν′ = ν0, and n
˜= ñ′ = n0 such that LHS /= 0
⇒
I J K
µ0 ν0 n0
r
=
nK
h
Σ
i
ÐI (gi )µ0µ0 ÐJ (gi )ν0ν0 ÐK
∗ (gi )n0n0 > 0
Given the representation matrices {ÐI (g)}, the CGCs are unique for each
triple I,J, K up to an overall phase that we choose such that I J K
µ0 ν0 n0
,
> 0
⇒
I J K
µ ν n
=
I J
µ0 ν0 κ0
1 nK
K h
Σ
i
Ð (g ) Ð (g ) Ð∗ (g )
I i µµ0 J i νν0 K i nn0
∀µ,ν, n
) IJ
K
If a > 1: CGCs not unique ⇒ trickier!

Example: CGC for group P3 ' C3v
IJ
K
This group is simply reducible, a ≤ 1, so we may drop the index l.
Here: For Γ3 use the representation matrices {Ð3(g)} corresponding to
the basis functions x, y.
1 1 1
1 1 1
=
1 2 2
1 1 1
=
2 2 1
1 1 1
= 1
1 3 3
1 µ ν
=
1 0
0 1 µν
2 3 3
1 µ ν
=
0 1
−1 0 µν
3 3 1
µ ν 1
=
1/ 2 0
√
0 1/ 2 µν
3 3 2
µ ν 1
=
√
0 1/ 2
−1/
√
2 0 µν
3 3 3
µ ν 1
√
√
=
1/ 2 0√
0 −1/ 2 µν
3 3 3
µ ν 2
=
0√ −1/ 2
−1/ 2 0
√
µν

Comparison: Rotation
Group
) Angular momentum j = 0, 1/2, 1, 3/2, . . . corresponds to the
irreducible representations of the rotation group
) For each j, these IRs are (2j + 1)-dimensional, i.e., the z component
of angular momentum labels the basis states for the IR Γj .
) Γj=0 is the identity representation of the rotation group
) The product representation Γj1 × Γj2 corresponds to the addition of
angular momenta j1 and j2;
Γj1 × Γj2 = Γ|j1−j2| + . . . + Γj1+j2
Here all multiplicities aj1j2
are one.
j3
) In our lecture, Clebsch-Gordan coefficients have the same meaning
as in the context of the rotation group:
They describe the unitary transformation from the reducible product
space to irreducible invariant subspaces.
This unitary transformation depends only on (the representation
matrices of) the IRs of the symmetry group of the problem so that
the CGC can be tabulated.

Symmetry of
Observables
) Consider Hermitian operator (observable) O
ˆ .
Let 5 = {gi } be a group of symmetry transformations
with {P̂(gi )} the group of unitary operators isomorphic to 5.
• For arbitrary |φ⟩ we have |ψ⟩ = Ô |φ⟩.
• Application of gi gives |ψ′
⟩ = P̂(gi )|ψ⟩ and |φ′
⟩ = P̂(gi)|φ⟩.
• Thus |ψ′
⟩ = Ô ′
|φ′
⟩ requires Ô ′
= P̂(gi ) O
ˆP̂(gi)−1
If Ô ′ = P̂(gi ) Ô P̂(gi )−1 = Ô ⇔ [P̂(gi), Ô] = 0 ∀gi ∈ 5
we call 5 the symmetry group of Ô which leaves Ô invariant.
Of course, we want the largest f possible.
) Lemma: If |n⟩ is an eigenstate of Ô, i.e., Ô |n⟩ = λn |n⟩,
and [P̂(gi), Ô] = 0, then P̂(gi ) |n⟩ is likewise an eigenstate of Ô
for the same eigenvalue λn .
As always P̂(gi) |n⟩ need not be orthogonal to |n⟩.
Proof: O
ˆ[P̂(gi ) |n⟩] = P̂(gi ) O
ˆ|n⟩ = λn [P̂(gi ) |n⟩]

Symmetry of Observables (cont’d)
) Theorem 15:
Let 5 = {P̂(gi )} be the symmetry group of the observable O
ˆ .
Then the eigenstates of a d-fold degenerate eigenvalue λn of Ô
form a d-dimensional invariant subspace Sn.
The proof follows immediately from the preceding lemma.
) Most often: Sn is irreducible
• central property of nature for applying group theory to physics problems
• unless noted otherwise, always assumed in the following
•Identify d-fold degeneracy of λn with d-dimensional IR of f.
) Under which cirumstances can Sn be reducible?
• f does not include all symmetries realized in the system, i.e., f ; f ′
(“hidden symmetry”). Then Sn is an irreducible invariant subspace of f′
.
Examples: hydrogen atom, m-dimensional harmonic oscillator (m > 1).
• A variant of the preceding case: The extra degeneracy is caused by the
antiunitary time reversal symmetry (more later).
• The degeneracy cannot be explained by symmetry: rare!
(Usually such “accidental degeneracies” correspond to singular points
in the parameter space of a system.) Roland Winkler, NIU, Argonne, and NCTU 2011−2015

Symmetry of Observables (cont’d)
Remarks:
) IRs of 5 give the degeneracies that may occur in the spectrum
of observable O
ˆ .
) Usually, all IRs of 5 are realized in the spectrum of observable Ô
(reasonable if eigenfunctions of O
ˆform complete set)

Application 1: Symmetry-Adapted
Basis
Let Ô = Ĥ = Hamiltonian
) Classify the eigenvalues and eigenstates of Ĥ according to the IRs ΓI
of the symmetry group 5 of Ĥ.
ˆ I α
α: distinguish different
Notation: H |Iµ, α⟩ = E |Iµ, α⟩ µ = 1, . . . , nI levels transforming
according to same ΓI
If ΓI is nI -dimensional, then eigenvalues EIα are nI -fold degenerate.
Note: In general, the “quantum number” I cannot be associated directly
with an observable.
) For given EIα, it suffices to calculate one eigenstate |Iµ0, α⟩. Then
{|I µ, α⟩ : µ = 1, . . . , nI } = {P̂(gi ) |Iµ0, α⟩ : gi ∈ 5}
(i.e., both sets span the same subspace of H)
) Expand eigenstates |Iµ, α⟩
in a symmetry-adapted basis
|Jν, β⟩ :
J = 1, . . . , N;
J
ν = 1, . . . , n ;
β = 1, 2, . . .
n ,
Jν,β ` ˛¸ x
Σ Σ
β
|Iµ, α⟩ = ⟨Jν, β|I µ, α⟩ |Jν, β⟩ = ⟨I α||I β⟩ |I µ, β⟩
=δIJ δµν (I α||I β⟩ see Theorem 11
⇒ partial diagonalization of Ĥ independent of specific details

Application 2: Effect of
Perturbations
0 1
) Let Ĥ = Ĥ + Ĥ , 0
Ĥ ˆ (0)
0 n
= unperturbed Hamiltonian: H |n⟩ = E |n⟩
Ĥ1 = perturbation
(0) ˆ
n n 1
) Perturbation expansion E = E + ⟨n|H |n⟩ +
ˆ1
Σ |⟨n|H |n ⟩|
′ 2
(0) (0)
n'=
/ n En −En'
+ . . .
ˆ ′ (0)
n
⇒ need matrix elements ⟨n|H|n ⟩ = E δ ˆ
nn' 1
′
+ ⟨n|H |n ⟩
) Let 50 = symmetry group of Ĥ 0
5 = symmetry group of Ĥ
,
usually 5 ; 50
)
)
)
0
I
The unperturbed eigenkets {|n⟩} transform according to IRs Γ of 5 0
0
I
{Γ } are also representations of 5, yet then reducible
0
I 0
Every IR Γ of 5 breaks down into (usually multiple) IRs { J
Γ } of 5
0
Σ
Γ = a Γ
I J J (see Theorem 7)
J
⇒ compatibility relations for irreducible representations
) Theorem 16: ⟨n = Jµα|Ĥ|n′ = J′µ′α′⟩ = δJJ' δµµ' ⟨Jα||Ĥ||J′α′⟩
j
ˆ ˆ ˆ ˆ j
† 1
Σ
h i
†
ˆ ˆ ˆ
i i
Proof: Similar to Theorem 11 with H = P(g ) H P(g ) = P(g ) H P(g )

Example: Compatibility Relations for C3v ' P3
) Character table C3v ' P3 C3v E 2C3 3σv
P3 e a, b c, d, f
Γ1
Γ2
Γ3
1 1 1
1 1 −1
2 −1 0
) 3v 3 3 3
2 −1
3 3 1
C ' P has two subgroups C = {E, C , C = C } ' G = { e, a,b}
Cs = {E, σv } ' G2 = {e, c} = {e, d} = {e, f }
) Both subgroups are Abelian, so they have only 1-dim. IRs
C3 E C3 C2
3
C2
E E C3
C3 C3 C2
C2 C2
3 3
C3 E C3 C2
3
3 Γ1 1 1 1
3 E Γ2 1 ω ω∗
E C3 Γ3 1 ω∗ ω
2πi /3
ω≡e
Cs E σi Cs E σi
E E σi Γ1 1 1
σi σi E Γ2 1 −1
) compatibility relations
C3v P3 Γ1 Γ2 Γ3
3
C G Γ Γ Γ + Γ
1 1 1 2 3
Cs G2 Γ1 Γ2 Γ1 + Γ2
2
 2−fold degen.
2
 nondegen.
2
 nondegen.
C 3v
2
1
3
C
2
2
C s

3
1
1
1

Discussion: Compatibility
Relations
Compatibility relations and Theorem 16 tell us how a degenerate level
0
I 0
transforming according to the IR Γ of 5 splits into multiple levels
transforming according to certain IRs {ΓJ } of 5 when the perturbation Ĥ 1
reduces the symmetry from 50 to 5 ; 50.
Thus qualitative statements:
) Which degenerate levels split because of Ĥ 1 ?
) Which degeneracies remain unaffected by Ĥ 1 ?
) These statements do not require any perturbation theory in the
conventional sense.
(For every pair 50 and 5, they can be tabulated once and forever!)
) These statements do not require some kind of “smallness” of Ĥ 1 .
) But no statement whether (or how much) a level will be raised
or lowered by Ĥ 1 .

according to Γ.
(Ir)Reducible Operators
) Up to now: symmetry group of operator Ô requires
P̂(gi ) Ô P̂(gi )−1 = Ô ∀gi ∈ 5
) More general: A set of operators {Q̂ ν : ν = 1, . . . , n} with
ˆ ˆ ˆ −1
P(g ) Q P(g ) =
n
Σ
µ=1
ˆ
Ð(g ) Q
i ν i i µν µ
∀ν = 1,. . . , n
∀gi ∈ f
is called reducible (irreducible), if Γ = {Ð(gi ) : gi ∈ 5}
is a reducible (irreducible) representation of 5.
Often a shorthand notation is used: gi Q̂ ν ≡ P̂(gi) Q̂ν P̂(gi)−1
) We say: The operators {Q̂ ν } transform according to Γ.
) Note: In general, the eigenstates of {Q̂ ν } will not transform

(Ir)Reducible Operators (cont’d)
Examples:
) Γ1 = “identity representation”; Ð(gi ) = 1 ∀gi ∈ 5; nI = 1
⇒ P̂(gi ) Q̂ P̂(gi )−1 = Q̂ ∀gi ∈ 5
We say: Q̂ is a scalar operator or invariant.
) most important scalar operator: the Hamiltonian Ĥ
i.e., Ĥ always transforms according to Γ1
The symmetry group of Ĥ is the largest symmetry group
that leaves Ĥ invariant.
) position operator xˆν
momentum operator p̂ν = −ik ∂xν
ν = 1,2,3 (polar vectors)
⇒ {xˆν} and {p̂ν } transform according to 3-dim. representation Γpol
(possibly reducible!)
) composite operators (= tensor operators)
ˆ
ν
Σ
λ,µ
e.g., angular momentum l = ε x
ˆ p̂
λµν λ µ ν = 1,2,3 (axial vector)

Tensor Operators
ˆ I ˆI
µ I
) Let Q ≡ {Q : µ = 1, . . . , n } transform according to ΓI = {ÐI (gi )}
ˆ J ˆJ
ν J
Q ≡ {Q : ν = 1, . . . , n } transform according to ΓJ = {ÐJ (gi )}
) ˆ ˆ
Then Q Q :
I J µ = 1, . . . , nI
µ ν ν = 1, . . . , nJ
}
transforms according to the product
representation ΓI × ΓJ
) ΓI × ΓJ is, in general, reducible
⇒ The set of tensor operators Q
ˆ ˆJ
I
µ ν
}
Q is likewise reducible
) A unitary transformation brings ΓI × ΓJ = {ÐI (gi ) ⊗ÐJ (gi )}
into block-diagonal form
⇒ The same transformation decomposes Q
ˆ ˆ
I J
µ ν
}
Q into irreducible tensor
operators (use CGC)

Where Are We?
We have discussed
) the transformational properties of states
) the transformational properties of operators
Now:
) the transformational properties of matrix elements
⇒ Wigner-Eckart Theorem

Wigner-Eckart Theorem
Let {|I µ, α⟩ : µ = 1, . . . , nI } transform according to ΓI = {ÐI (gi)}
{|I ′µ′, α′⟩ : µ′ = 1, . . . , nI' } transform according to ΓI' = {ÐI ' (gi )}
ˆ ˆ
J J
ν J
Q = {Q : ν = 1, . . . , n } transform according to ΓJ = {ÐJ (gi)}
ˆ
′ ′ ′ J
ν
Then ⟨I µ , α | Q | Iµ, α⟩ =
Σ
A
′
J I I l
ν µ µ′
′ ′ ˆJ
⟨I α || Q || Iα⟩ A
A
where the reduced matrix element ⟨I′α′ || Q̂J || Iα⟩ is independent
of µ, µ′ and ν.
Proof:
) ˆJ
ν
Q |Iµ, α⟩ : µ = 1, . . . , nI
ν = 1, . . . , nJ
}
transforms according to Γ × Γ
I J
) ˆJ
ν
Thus CGC expansion Q |Iµ, α⟩ =
Σ
K l,Æ
J I K l
ν µ κ
, ,
|Kκ, l⟩
) ˆ
′ ′ ′ J
ν
⟨I µ , α | Q | Iµ, α⟩ =
Σ
K l,Æ
, ,
J I K l ′ ′ ′
ν µ κ
⟨I µ , α | Kκ, l⟩
` ˛¸ x
≡ δI'K δµ ' l ⟨I ′
α′
|| Q̂J
|| Iα⟩ Æ
Theorem 11

Discussion: Wigner-Eckart
Theorem
) Matrix elements factorize into two terms
• the reduced matrix element independent of µ, µ′
and ν
• CGC indexing the elements µ, µ′
and ν of ΓI , ΓI' and ΓJ .
(CGC are tabulated, independent of Q̂J
)
) Thus: reduced matrix element = “physics”
Clebsch-Gordan coefficients = “geometry”
) Matrix elements for different values of µ, µ′ and ν have a fixed ratio
independent of Q̂J
) If ΓI' is not contained in ΓI × ΓJ
Equivalent to: If ΓI
∗
' × ΓJ × ΓI does not
contain the identity representation
⇒
′
, ,
J I I l = 0 ′
ν µ µ′ ∀ν,µ, µ
ˆ
′ ′ ′ J
ν
⇒ ⟨I µ , α | Q | Iµ, α⟩ = 0 ∀ν, µ, µ′
Many important selection rules are some variation of this result.
) Theorems 11 and 16 are special cases of the WE theorem for
Q̂1 = 1 and Q̂1 = Ĥ (yet we proved the WE theorem via Theorem 11)

Discussion: Wigner-Eckart Theorem (cont’d)
Application: Perturbation theory
) Compatibility relations and Theorem 16 describe splitting of degenerate
levels using the symmetry group 5 of perturbed problem
) Alternative approach
Splitting of levels using the symmetry group 50 of unperturbed problem
(i.e., no need to know group f of perturbed problem)
• Let Q̂J
be tensor operator transforming according to IR ΓJ of f0
1
ˆ ˆ ˆ
• Often: perturbation H = F · Q = F Q
J J J J
ν ν i.e., Ĥ1 is proportional to
only νth component of tensor operator Q̂J
ˆ ˆν
Q projected on component Q via suitable orientation of field F
J J J
0 1 0
ˆ ˆ ˆ ˆJ
ν
• Symmetry group of H = H + H is subgroup f ⊂ f which leaves Q invariant.
• WE Theorem:
′ J
ˆ ˆ
1 ν ν
J ′ ′ ′ ′
⟨n|H |n ⟩ = F ⟨n=Iµα|Q |n =I µ α ⟩ =
Σ
,
Æ
′
J I I l
′
ν µ µ
,
Æ
⟨I α || Q̂J
|| I ′
α′
⟩ (∗)
J
• Changing the orientation of F changes only the CGCs in (∗)
ˆJ ′ ′
Æ
The reduced matrix elements ⟨Iα || Q || I α ⟩ are “universal”

Example: Optical Selection
Rules
Example: Optical transitions for a system with symmetry group C3v
(e.g., NH3 molecule)
) Optical matrix elements ⟨iI |e · r̂|fJ ⟩ (dipole approximation)
where |iI ⟩ = initial state (with IR ΓI ); |fJ ⟩ = final state (IR ΓJ )
e = (ex, ey, ez) = polarization vector
ˆ
r = (x̂, ŷ, ẑ) = dipole operator (≡ position operator)
) x̂,y
ˆ transform according to Γ3
z
ˆ transforms according to Γ1
) e.g., light xy polarized: ⟨i1|ex x
ˆ+ eyŷ |f3 ⟩
•transition allowed because Γ3 × Γ3 = Γ1 + Γ2 + Γ3
• in total 4 different matrix elements, but only one reduced matrix
element
) z polarized: ⟨i1|ez ẑ |f3 ⟩
•transition forbidden because Γ1 × Γ3 = Γ3
) These results are independent of any microscopic models for the
NH3 molecule!

Goal: Spin 1/2 Systems and Double
Groups Rotations and Euler Angles
) So far: transformation of functions and operators dependent on position
) Now: systems with spin degree of freedom
⇒ wave functions are two-component Pauli spinors
Ψ(r) = ψ↑(r) |↑⟩ + ψ↓(r) |↓⟩ ≡ ↑
ψ (r)
ψ↓(r)
) How do Pauli spinors transform under symmetry operations?
) Parameterize rotations via Euler angles α, β, γ
x
y

axis z, angle 
x’
x

z z z z

y’
y
x’’
x’
y’ z’ y’ z’
axis y’, angle 
x

y’
x’’
x’
z’ z

) Thus general rotation R(α, β, γ) = Rz' (γ) Ry' (β) Rz(α)
y’
y
x’’’

z
y’’
axis z’, angle 

Rotations and Euler Angles (cont’d)
) General rotation R(α, β, γ) = Rz' (γ) Ry' (β) Rz(α)
) Difficulty: axes y′ and z′ refer to rotated body axes (not fixed in space)
z
−1
y'
y' (β) R (γ) R (β)
) Use Rz' (γ) = R
Ry' (β) = Rz(α) Ry (β) Rz
−1(α)
preceding rotations
are temporarily undone
) Thus R(α, β, γ) = R
` ˛¸ x
−1
z y z
R (α) R (β) R (α)
rotations about
z axis commute
z
−1
y'
y' (β) R (γ) R (β) R
` ˛¸ x
=1
z
y' (β) R (α)
) Thus R(α, β, γ) = Rz(α) Ry (β) Rz(γ)
) R(2π, 0, 0) = R(0, 2π, 0) = R(0, 0, 2π) = 1 ≡ e
rotations about
space-fixed axes!
) More explicitly: rotations of vectors r = (x, y, z) ∈ R 3
z
cosα − sinα 0
R (α) = sin α cosα
0 0 1
0 , Ry(β) =
cosβ 0 − sinβ
0 1 0
sinβ 0 cosβ
etc.
) SO(3) = set of all rotation matrices R(α, β, γ)
= set of all orthogonal 3 × 3 matrices R with detR = +1.

Rotations: Spin 1/2
Systems
) Rotation matrices for spin-1/2 spinors (axis n)
n
i
2
R (φ) = exp − σ · nφ = 1 cos(φ/2) −in · σ sin(φ/2)
) R(α, β, γ) = Rz (α) Ry (β) Rz (γ)
⇒ SU(2) is called double group for
=
−i(α+γ)/2
e cos(β/2) −i(α−γ)/2
−e sin(β/2)
e−i(α−γ)/2 sin(β/2) ei(α+γ)/2 cos(β/2)
transformation matrix for spin 1/2 states
) SU(2) = set of all matrices R(α, β, γ)
= set of all unitary 2 × 2 matrices R with detR = +1.
rotation by 2π
) R(2π, 0, 0) = R(0, 2π, 0) = R(0, 0, 2π) = −1 ≡ ē
) R(4π, 0, 0) = R(0, 4π, 0) = R(0, 0, 4π) = 1 = e
is not identity
rotation by 4π
is identity
) Every SO(3) matrix R(α, β, γ) corresponds to two SU(2) matrices
R(α, β, γ) and R(α + 2π, β, γ) = R(α, β + 2π, γ) = R(α, β, γ + 2π)
= ēR(α, β, γ) = R(α, β, γ) ē

Double
Groups
) Definition: Double Group
Let the group of spatial symmetry transformations of a system be
5 = {gi = R(αi , βi , γi ) : i = 1, . . . , h} ⊂ SO(3)
Then the corresponding double group is
5d = {gi = R(αi , βi , γi ) : i = 1, . . . , h}
∪ {gi = R(αi + 2π, βi , γi ) : i = 1, . . . , h} ⊂ SU(2)
) Thus with every element gi ∈ 5 we associate two elements gi and
ḡi ≡ ēgi = gi ē ∈ 5d
) If the order of 5 is h, then the order of 5d is 2h.
) Note: 5 is not a subgroup of 5d because the elements of 5
are not a closed subset of 5d .
Example: Let g = rotation by π
2
• in f: g = e the same group element g is thus
interpreted differently in f and fd
2
• in fd : g = ē
) Yet: {e, ē} is invariant subgroup of 5d
and the factor group 5d /{e, ē} is isomorphic to 5.
⇒ The IRs of 5 are also IRs of 5d Roland Winkler, NIU, Argonne, and NCTU 2011−2015

Example: Double Group C3v
C3v E Ē 2C3 2C̄ 3 3σv 3σ¯v
Γ1 1 1 1 1 1 1
Γ2
Γ3
1
2
1
2
1
−1
1
−1
−1
0
−1
0
Γ4
Γ5
Γ6
1 −1 −1 1
2 −2 1 −1 0 0
1 −1 −1 1 i −i
−i i
) For Γ1, Γ2, and Γ3 the “barred” group elements have the same
characters as the “unbarred” elements.
Here the double group gives us the same IRs as the “single group”
) For other groups a class may contain both “barred” and “unbarred”
group elements.
⇒ the number of classes and IRs in the double group need not be
twice the number of classes and IRs of the “single group”

Time Reversal (Reversal of Motion)
) Time reversal operator θ
ˆ: t → −t
) Action of θˆ: θ̂r̂ θˆ−1 = ˆ
r
θ̂p̂ θˆ−1 =
−p̂ θ̂L̂ θˆ−1 =
−L̂ θ̂ Ŝ θˆ−1 =
−Ŝ
}
independent of t
linear in t
) Consider time evolution: Û(δt) = 1 −iĤ δt/k
⇒ Û(δt) θˆ|ψ⟩ = θ̂Û(−δt)|ψ⟩
⇔ −iĤ θˆ|ψ⟩ = θˆiĤ |ψ⟩ but need also [θˆ,Ĥ] = 0
⇒ Need θ
ˆ= UK
U = unitary operator
K = complex conjugation
Properties of θ
ˆ= UK:
) K c1|α⟩ + c2|β⟩ = c1
∗|α⟩ + c2
∗|β⟩ (antilinear)
) Let |α˜⟩ = θˆ|α⟩ and |β˜⟩ = θˆ|β⟩
⇒ ⟨β˜|α˜⟩ = ⟨β|α⟩∗
= ⟨α|β⟩
θ
ˆ = UK is
antiunitary
operator

Time Reversal (cont’d)
The explicit form of θ
ˆdepends on the representation
⇒ θˆψ(r) = ψ∗(r)
⇒ θˆψ(p) = ψ∗(−p)
) position representation: θ̂r̂ θˆ−1 = ˆ
r
) momentum representation: θ̂p̂θˆ−
1 = −p̂
) spin 1/2 systems:
• θ
ˆ= iσy K ⇒ θˆ2 = −1
• all eigenstates |n⟩ of Ĥ are at least two-fold degenerate
(Kramers degeneracy)

Time Reversal and Group Theory
) Consider a system with Hamiltonian Ĥ.
) Let 5 = {gi } be the symmetry group of spatial symmetries of Ĥ
[P̂(gi), Ĥ] = 0 ∀gi ∈ 5
) Let {|I ν⟩ : ν = 1, . . . , nI } be an nI -fold degenerate eigenspace of Ĥ
which transforms according to IR ΓI = {ÐI (gi )}
Ĥ |Iν⟩ = EI |Iν⟩ ∀ν
Σ
I i µν
P̂(gi ) |Iν⟩ = Ð (g ) |Iµ⟩
µ
) Let Ĥ be time-reversal invariant: [Ĥ, θ
ˆ
]= 0
⇒ θ
ˆis additional symmetry operator (beyond {P̂(gi )}) with
[θˆ,P̂(gi )] = 0
) ˆ ˆ ˆˆ ˆ
P(g ) θ |Iν⟩ = θ P(g )|Iν⟩ = θ Ð (g )
i i I i µν
Σ Σ ∗
I
|Iµ⟩ = Ð (g )
i µν
µ µ
) Thus: time-reversed states {θˆ|Iν⟩} transform according to
θ̂ |Iµ⟩
complex conjugate IR Γ∗
I = {ÐI
∗(gi )}

Time Reversal and Group Theory (cont’d)
) time-reversed states {θˆ|Iν⟩} transform according to
complex conjugate IR Γ∗
I = {ÐI
∗(gi )}
Three possiblities (known as “cases a, b, and c”)
(a) {|I ν⟩} and {θˆ|Iν⟩} are linear dependent
(b) {|I ν⟩} and {θˆ|Iν⟩} are linear independent
The IRs ΓI and Γ∗
I are distinct, i.e., χI (gi ) /= χ∗
I (gi)
(c) {|I ν⟩} and {θˆ|Iν⟩} are linear independent
ΓI = Γ∗
I , i.e., χI (gi ) = χ∗
I (gi ) ∀gi
Discussion
) Case (a): time reversal is additional constraint for {|I ν⟩}
e.g., nI = 1 ⇒ |ν⟩ reell
) Cases (b) and (c): time reversal results in additional degeneracies
) Our definition of cases (a)–(c) follows Bir & Pikus. Often (e.g., Koster)
a different classification is used which agrees with Bir & Pikus for spinless
systems. But cases (a) and (c) are reversed for spin-1/2 systems.

Time Reversal and Group Theory (cont’d)
) When do we have case (a), (b), or (c)?
Criterion by Frobenius &
Schur
1 Σ
h i
i
,
,
χI (g2) = 0
η case (a)
case (b)
−η case (c)
where η =
+1 systems with integer spin
−1 systems with half-integer spin
Proof: Tricky! (See, e.g., Bir & Pikus)

Example: Cyclic Group
C3
C3 e q q2
2
2 2
e e q q2
q q q e
q q e q
) C3 is Abelian group with 3 elements: C3 = {q, q2, q3 ≡ e}
) Multiplication table Character table
C3 e q q2 time
reversal
1
Γ 1 1 1
Γ2 1 ω ω∗
Γ3 1 ω∗ ω
a
b
b
2πi /3
ω≡e
) IR Γ1: no additional degeneracies because of time reversal
) IRs Γ2 and Γ3: these complex IRs need to be combined
⇒ two-fold degeneracy because of time reversal symmetry
(though here no spin!)

Group Theory in Solid State
Physics
First: Some terminology
) Lattice: periodic array of atoms (or groups of atoms)
) Bravais lattice:
R = n a + n a + n a
n x x y y z z
3
n = (n1, n2, n3) ∈ Z
ai linearly independent
Every lattice site Rn is occupied with one atom
Example: 2D honeycomb lattice is not a Bravais lattice
) Lattice with basis:
•Every lattice site Rn is occupied with z atoms
• Position of atoms relative to Rn: τ i , i = 1, . . . , q
• These q atoms with relative positions τ i form a basis.
• Example: two neighboring atoms in 2D honeycomb lattice

Symmetry Operations of Lattice
) Translation t (not necessarily by lattice vectors Rn)
) Rotation, inversion → 3 × 3 matrices α
) Combinations of translation, rotation, and inversion
⇒ general transformation for position vector r ∈ R3:
r′ = αr + t ≡ {α|t} r
) Notation {α|t} includes also
• Mirror reflection = rotation by π about axis perpendicular
to mirror plane followed by inversion
• Glide reflection = translation followed by reflection
• Screw axis = translation followed by rotation
Symmetry operations {α|t} form a group
` ˛¸ x
r'=αr+t
) Multiplication {α′|t′} {α|t} r = α′r′ + t′ = α′αr + α′t + t′
= {α′α|α′t + t′} r
) Inverse Element {α|t}−1 = {−α−1| −α−1t}
because {α|t}−1
{α|t} = {α−1
α|α−1
t − α−1
t} = {1|0}

Classification
Symmetry Groups of
Crystals
to be added . . .

Symmetry Groups of Crystals
Translation Group
Translation group = set of operations {1|Rn}
) {1|Rn' } {1|Rn} = {1|Rn' + 1 Rn} = {1|Rn'+n}
⇒ Abelian group
) associativity (trivial)
) identity element {1|0} = {1|R0}
) inverse element {1|Rn}−1 = {1| −Rn}
Translation group Abelian ⇒ only one-dimensional IRs

Irreducible Representations of Translation
Group
(for clarity in one spatial dimension)
) Consider translations {1|a}
) Translation operator T̂a = T̂ {1|a} is unitary operator
⇒ eigenvalues have modulus 1
) eigenvalue equation T̂a |φ⟩ = e−iφ |φ⟩ −π < φ ≤ π
more generally T̂na |φ⟩ = e−inφ |φ⟩ n ∈ Z
) ⇒ representations Ð({1|Rna}) = e−inφ −π < φ ≤ π
Physical Interpretation of φ
(r −a|
` ˛¸ x
e−i φ |φ⟩
Thus: Bloch Theorem
Consider ⟨
¸
r
x
|
`
T̂
˛
a | φ⟩ ⇒ ⟨r −a|φ⟩ = e−iφ ⟨r|φ⟩
(for φ = ka)
) The wave vector k (or φ = ka) labels the IRs of the translation group
) The wave functions transforming according to the IR Γk are
ikr
Bloch functions ⟨r|φ⟩ = e uk(r) with
eik(r−a)uk (r −a) = eikr uk(r)e−ika
or uk(r −a) = uk(r)

Irreducible
Representations of
Space Groups
to be added . . .

Theory of
Invariants
Luttinger (1956)
Bir & Pikus
Idea:
) Hamiltonian must be invariant under all symmetry transformations
of the system
) Example: free particle
Ĥ = Ekin =
p̂ 2
2m
`
sc
˛
a
¸
la
x
r
z z z
`˛¸x ` ˛¸ x
symmetric reversal
+ c1z
p̂ + c2 ˆ
r·zp̂ + c4 p̂4
not inversion breaks time
`
sc
˛
a
¸
la
x
r
+ . . .
) Crystalline solids:
Ekin = E(k) = kinetic energy of Bloch electron
with crystal momentum p = kk
⇒ dispersion E(k) must reflect crystal symmetry
E(k) = a0 + a1 k +a2 k2 + a3 k3 + . . .
only in crysta
`
l
˛
s
¸
w
x
ithout inversion
`
s
˛
y
¸
m
x
metry

Theory of Invariants (cont’d)
More generally:
) Bands E(k) at expansion point k0 n-fold degenerate
) Bands split for k /= k0
) Example: GaAs (k0 = 0)
) Band structure E(k) for small k via diagonalization
of n × n matrix Hamiltonian H(k).
E
k
conduction
band
valence band
HH
LH
SO
) Goal: Set up matrix Hamiltonian H(k) by exploiting the symmetry
at expansion point k0
) Incorporate also perturbations such as
• spin-orbit coupling (spin S)
• electric field E, magnetic field B
• strain ε
• etc.

Invariance
Condition
) Consider n × n matrix Hamiltonian H(K)
) K = K (k, S, E, B, ε, . . .) = general tensor operator
where k = wave vector E = electric field ε = strain field
S = spin B = magnetic field etc.
) Basis functions {ψν (r) : ν = 1, . . . , n} transform according to
representation Γψ = {Ðψ (gi )} of group 5. (Γψ does not have to be IR)
) Symmetry transformation gi ∈ 5 applied to tensor K
K → gi K ≡ P̂(gi ) K P̂(gi )−1
⇒ H(K) → H(gi K)
i
) Equivalent to inverse transformation g−1
applied to ψν(r):
Σ
µ
−1
i
ψν(r) → ψ (g r) = Ð (g )
ν i ψ µν µ
ψ (r)
ψ i ψ
−1
i
→ Ð (g ) H(K) Ð (g )
⇒ H(K)
) Thus i
−
1
ψ i ψ i
Ð (g ) H(g K) Ð (g ) = H(K) ∀gi ∈ 5
really n2
equations!

Invariance Condition (cont’d)
Remarks
) If −k0 is in the same star as the expansion point k0,
additional constraints arise from time reversal symmetry.
in particular: −k0 = k0 = 0
) If Γψ is reducible, the invariance condition can be applied to each
“irreducible block” of H(K) (see below).

ν
Invariant
Expansion
Expand H(K) in terms of irreducible tensor operators and basis matrices
) Decompose tensors K into irreducible tensors KJ
transforming
according to IR ΓJ of 5
J J Σ
K → g K ≡ Ð (g
ν i ν J i µν
) K J
µ
µ
) n2 linearly independent basis matrices {Xq : q = 1, . . . , n2}
transforming as
q i q ψ
−1
i
Σ
p
X → g X ≡ Ð (g ) X Ð (g ) = Ð (g ) X
q ψ i X i pq p
with “expansion coefficents” ÐX (gi)pq
) Representation ΓX ' Γ∗
ψ × Γψ is usually reducible.
We have IR Γψ for the ket basis functions of H and
IR Γ∗
ψ (i.e., the complex conjugate IR) for the bras
2
⇒ from {X : q = 1, . . . , n } form linear combinations X I ∗
q ν
transforming according to the IRs Γ∗
I occuring in Γ∗
ψ × Γψ
∗
µ
I I
i µ
∗ Σ
I
X → g X = Ð (g )
i µν X
∗ I ∗
µ

Invariant Expansion
(cont’d)
) Consider most general expansion
Σ Σ ∗
H(K) = b X K
IJ I J
µν µ ν
IJ µν
) Transformations gi ∈ 5:
bIJ
µν = expansion coefficients
∗ Σ
I ∗
µ I
X → Ð (g )
∗
I J Σ
µ' ν'
Xµ' , K → Ð (g
i µ' µ ν J i )ν'ν
J
ν
K '
) Use invariance condition (must hold ∀gi ∈ 5)
∗
IJ I J
µν µ ν
µν µν
b X K = bµν
µ' ν'
h
Σ Σ Σ Σ
IJ 1 ∗
i
I i µ' µ J i ν'ν
` ˛¸ x
δ δµ ν δ
IJ ' ' µ ν
∗
Ð (g ) Ð (g ) X K
I J
ν
µ' '
IJ
µ
= δ bII
µµ
`
≡
˛¸
aI
x
Σ Σ
µ'
∗
X K
I I
µ
µ' '
) Then
I
I
ν
Σ Σ ∗
H(K) = a X K
I I
ν ν

Irreducible Tensor Operators
Construction of irreducible tensor operators K = K(k, S, E, B, ε)
) i i i i ij I
Components k , S , E , B , ε transform according to some IRs Γ of 5.
⇒ “elementary” irreducible tensor operators K I
) Construct higher-order irreducible tensor operators with CGC:
K
n
K =
Σ
µν
I J K l
µ ν n
K K
I J
µ ν
If we have multiplicities aIJ
c 1
K
we get different tensor operators
for each value l
) Irreducible tensor operators KI
are “universally valid” for any matrix
Hamiltonian transforming according to 5
) Yet: if for a particular matrix Hamiltonian H(K) with basis functions
{ψν } transforming according to Γψ an IR ΓI does not appear in
ψ ψ
∗ I
Γ × Γ , then the tensor operators K may not appear in H(K).

Basis
Matrices
) In general, the basis functions {ψν (r) : ν = 1, . . . , n} include
several irreducible representations ΓJ
⇒ Decompose H(K) into nJ × nJ' blocks HJJ' (K), such that
• rows transform according to IR Γ∗
J (dimension nJ)
• columns transform according to IR ΓJ' (dimension nJ' )
H(K) = H†
cv H
Hvv vv'
Hc
†
v' Hv
†
v' Hv' v'
,
Hcc Hcv Hcv'
c
v
v’
E
k
) I∗
ν
∗
I
∗
J
Choose basis matrices X transforming according to the IRs Γ in Γ × ΓJ'
XI∗
ν J i
−1 I ∗
ν i
Σ
µ
∗
I
→ Ð (g ) X ÐJ' (g ) = Ð (g )
i µν XI∗
µ
) More explicitly: (XI∗
)λµ =
ν
′
I J J l
ν µ λ
∗
ν
I ′
which reflects the transformation rules for matrix elements ⟨Jλ|K |J µ⟩
I
JJ' I
ν
Σ Σ ∗
⇒ For each block we get H (K) = a X K
I I
ν ν

Time Reversal
) time reversal θ
ˆconnects expansion point k0 and —
k0
) often: θˆψk0λ(r) and ψ−k0λ(r) linearly dependent
θ̂ ψ 0
k λ
Σ
λ'
(r) = f λλ' −k 0
ψ λ ' (r)
) thus additional condition
f −1 H(ζK) f = H∗(K) = Ht (K)
K even under θ
ζ = +1 :
ζ = −1 : K even under θ
applicable in particular for k0 = −k0 = 0
) k0 /= —
k0: often k0 and —
k0 also connected by spatial symmetries R
−k k
0 0
−1 −1
H (K) = Ð(R) H (R K) Ð (R).
−1
⇒ f Hk0
(R K)f
−1 ∗
k0
t
k0
= H (ζK) = H (ζK) kx
ky
K
K’ 
graphene: k0 = K

Example:
Graphene
) Electron states at K point: point group C3v
strictly speaking
D3h = C3v + inversion
) “Dirac cone”: IR Γ3 of C3v
It must be Γ3 because
this is the only 2D IR of C3v
(with Γ∗
3 = Γ3)
) We have Γ∗
3 × Γ3 = Γ1 + Γ2 + Γ3
1 2
⇒ basis matrices X1 = 1; X1 = σy ; 3 3
1 z 2
X = σ , X = —
σx
) Irreducible tensor operators K up to second order in k:
2 2
Γ1 : k2 + k2 Γ3 : kx, ky ; k —k , 2k k
x y y x x y Γ2: [kx, ky ] ∝ Bz
) Hamiltonian here: basis functions |x⟩ and |y⟩
2 2 2
31 x z y x 11 x y 32 y
2
x z x y x
H(K) = a (k σ —k σ ) + a (k + k )1 + a [(k —k )σ —2k k σ ]
) More common: basis functions |x —iy⟩ and |x + iy⟩
1 2 3 3
⇒ basis matrices X1 = 1; X1 = σz ; X1 = σx , X2 = σy
2 2 2 2
31 x x y y 11 x y 32 y x x x y y
H(K) = a (k σ + k σ ) + a (k + k )1 + a [(k —k )σ + 2k k σ ]
) additional constraints for H(K) from time reversal symmetry

Graphene: Basis Matrices (D3h)
Symmetrized matrices for the invariant expansion of the blocks Hαβ
for the point group D3h.
Block Representations Symmetrized matrices
H55 Γ∗
5 × Γ5
= Γ1 + Γ2 + Γ6
Γ1 : 1
Γ2 : σz
Γ6 : σx , σy
no spin
H77 Γ∗
7 × Γ7
= Γ1 + Γ2 + Γ5
H99 Γ∗
9 × Γ9
= Γ1 + Γ2 + Γ3 + Γ4 with spin
H79 Γ∗
7 × Γ9 1, —
iσz
Γ1 : 1
Γ2 : σz
Γ5 : σx , —
σy
Γ1 : 1
Γ2 : σz
Γ3 : σx
Γ4 : σy
Γ5 :
Γ6 :
= Γ5 + Γ6 σx , σy

Graphene: Irreducible Tensor Operators (D3h)
K
55
Terms printed in bold give rise to invariants in H (K) allowed by
1
2
time-reversal invariance. Notation: {A, B} ≡ (AB + BA).
2 2
Γ1 1; k + k ; { 2 2
x y x y x x x y y xx yy
k , 3k − k }; k E + k E ; s + s ;
(syy − sxx)kx + 2sxyky; (ϵyy —ϵxx )Ex + 2ϵxy Ey ;
sxBx + syBy; szBz; (sxky − sykx)Ez; sz(kxEy − ky
Ex); 2 2
y x y
Γ2 {k , 3k —k } z x y y x
; B ; k E − k E ;
(ϵxx —ϵyy )ky + 2ϵxy ky ; (sxx + syy)Bz; (sxx − syy)Ey + 2sxyEx;
sz; sxBy —syBx ; (sxkx + syky )Ez ; sz(sxx + syy);
Γ3 Bx kx + By ky ; Ex Bx + Ey By ; Ez Bz; (ϵyy —ϵxx )Bx + 2ϵxy By ;
sxkx + syky ; sxEx + syEy ; szEz ; sx(ϵyy —ϵxx ) + 2syϵxy
Γ4 Bx ky —By kx; Ez ; Ex By —Ey Bx ; (ϵxx —ϵyy )By + 2ϵxy Bx ;
(ϵxx + ϵyy )Ez ; sxky —sykx; sxEy —syEx ; sy(ϵxx —ϵyy ) + 2sxϵxy
Γ5 Bx , By ; By ky —Bx kx, Bx ky + By kx; ky Ez , —
kx Ez ;
Ey By —Ex Bx , Ey Bx + Ex By ; (ϵxx + ϵyy )(Bx , By );
(ϵxx —ϵyy )Bx + 2ϵxy By , (ϵyy —ϵxx )By + 2ϵxy Bx ; 2ϵxy Ez , (ϵxx —ϵyy )Ez ;
sx, sy; syky —sxkx, sxky + sykx; syBz, —
sxBz; szBy , —
szBx ;
syEy —sxEx , sxEy + syEx ; (sx, sy)(ϵxx + ϵyy );
sx(ϵxx —ϵyy ) —2syϵxy , sy(ϵyy —ϵxx ) —2sxϵxy

Graphene: Irreducible Tensor Operators (cont’d)
x y y x y x
Γ6 k , k ; {k + k , k − k } x y
, 2{k , k };
2 2
{k , k + k }, { 2 2
x x y y x y z y z x
k , k + k }; B k , —
B k ;
Ex , Ey ; ky Ey —kxEx , kxEy + ky Ex ;
EyBz, −ExBz; EzBy, −EzBx;
syy − sxx, 2sxy; (sxx + syy)(kx, ky);
(sxx − syy)kx + 2sxyky , (syy − sxx)ky + 2sxykx;
2ϵxy Bz, (ϵxx —ϵyy )Bz ;
(ϵxx —ϵyy )Ex + ϵxy Ey , (ϵyy —ϵxx )Ey + ϵxy Ex ;
(ϵxx + ϵyy )(Ex , Ey ); szky , —
szkx;
syBy − sxBx, sxBy + syBx; szEy, −szEx;
syEz, −sxEz; sz(kxEy + kyEx), sz(kxEx − ky
Ey);
(sxky + sykx)Ez, (sxkx −
syky)Ez; 2szϵxy , sz(ϵxx —ϵyy );

Graphene: Full Hamiltonian Winkler and Zülicke,
PRB 82, 245313
H(K) = a61(kx σx + ky σy ) “Dirac term”
nonlinear + anisotropic
corrections
orbital Rashba term
orbital Zeeman term
strain-induced terms
2 2 2
12 x y 62 y
2
x x x y y
+ a (k + k )1 + a [(k —k )σ + 2k k σ ]
+ a22(kx Ey —ky Ex )σz
+a21Bz σz
+ a14(ϵxx + ϵyy )1 + a66[(ϵyy —ϵxx )σx + 2ϵxy σy ]
+a15[(ϵyy —ϵxx )kx + 2ϵxy ky ]1
+a67(ϵxx + ϵyy )(kx σx + ky σy ) isotropic velocity
renormalization
anisotropic velocity
renormalization
strain - orbital Zeeman
strain - orbital Rashba
intrinsic SO coupling
Rashba SO coupling
a68{[(ϵxx —ϵyy )kx + 2ϵxy ky ]σx
+[(ϵyy —ϵxx )ky + 2ϵxy kx]σy }
+a23(ϵxx + ϵyy )Bz σz
+a24[(ϵxx —ϵyy )Ey + 2ϵxy Ex ]σz
+ a21 szσz
+ a61 sz(Ey σx —Ex σy ) + a62 Ez (syσx —sxσy )
+a63 sz[(kxEy + ky Ex )σx + (kx Ex —ky Ey )σy ]
+a64 Ez [(sxky + sykx )σx + (sxkx —syky )σy ]
+a11 (sxky —sykx)Ez + a12 sz(kx Ey —ky Ex )
+ a26(ϵxx + ϵyy )szσz strain-mediated SO coupling

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g-lecture.pptx