group theory summary · group theory summary the universe is an enormous direct product of...
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Imperial College
Group Theory Summary
Lecture notes
January 14, 2009
A summary of Dr. A. Hanany’s lectures during the autumn term 2008
Matthieu Schaller
Group Theory Summary
The universe is an enormous direct product of representations of symmetry groups.Steven Weinberg
The picture on the title page is a 2-dimensionnal projection graph of E8, the largestcomplex exceptionnal Lie group. It has been discovered in 1889 by Wilhelm Killing. It hasdimension 248 and rank 8.
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Group Theory Summary
This document is not official
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CONTENTS Group Theory Summary
Contents
Foreword 5
1 Notation Conventions 6
2 Abstract Group Theory 72.1 Definiton . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.2 Basic properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.3 Finite and infinite groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.4 Multiplication Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.5 Subgroups and Cosets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.6 Classes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.7 Product Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.8 Factor Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
3 List of usual finite groups 13
4 Representation of groups 144.1 Homomrphisms and Isomorphims . . . . . . . . . . . . . . . . . . . . . . . . . 144.2 Representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144.3 Reductible and Irreducible Representations . . . . . . . . . . . . . . . . . . . 16
5 The Great Orthogonality Theorem 17
6 Characters and Character Tables 196.1 Orthogonality relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196.2 The decomposition Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . 206.3 The regular representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206.4 Character Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
7 Permutation Groups 227.1 Permutations and Cycles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227.2 Young Diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237.3 Young Tableaux and Sn irreps . . . . . . . . . . . . . . . . . . . . . . . . . . 24
8 Continous Groups and Lie Groups 268.1 Definitions and examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 268.2 Compact and Non-Compact Lie Groups . . . . . . . . . . . . . . . . . . . . . 278.3 Matrix Generators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 288.4 Exponential of infinitesimal Generators . . . . . . . . . . . . . . . . . . . . . 28
9 Introduction to Lie Algebras 309.1 Definition and examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 309.2 Generators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 309.3 Subalgebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 319.4 Structure constants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 329.5 Lie Algebras and Lie Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . 329.6 Lie Algebras of product Lie Groups . . . . . . . . . . . . . . . . . . . . . . . . 32
10 Orthogonal Groups 3410.1 O(n) groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3410.2 SO(n) groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3410.3 The SO(2) group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3510.4 The SO(3) group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
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CONTENTS Group Theory Summary
11 Unitary Groups 3711.1 U(n) groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3711.2 The U(1) group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3711.3 SU(n) groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3711.4 The SU(2) group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3811.5 The SU(3) group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3811.6 SU(n) groups and Young Tableaux . . . . . . . . . . . . . . . . . . . . . . . . 38
12 Other Lie Groups 3912.1 Sp(2n) groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3912.2 Exceptional groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
13 Group theory in Physics 40
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CONTENTS Group Theory Summary
Foreword
Important remarks
This document has not been checked by the lecturer and I cannot ensure that everything istrue.
For obvious reasons of ecology, please print this document only if you really need it.
About this document
This documents is a summary of Dr. Amihay Hanany’s lectures on Group Theory that Ifollowed during one term between October and December 2008. Some informations alsocome from the lecture notes form the same course at EPFL and from different sources inbooks or on the web.
Only a few explanations are given and the theorem’s proofs are left out. It contains alsosome elements of Lie Algebra and representation theory.
If you find a mistake, if you have some suggestions to improve this document or if youwant to correct my poor English, take contact with me. The last version of this documentcan always be found on my website1.
If this document has helped you or if you enjoyed (!) it, feel free to tell me.
Acknowledgements
Thanks to Fabien Margairaz for his help setting up the LATEX style of this document.
Matthieu [email protected]
1http://www.matthieu-schaller.no-ip.fr (Mathematics projects part)
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1 Notation Conventions Group Theory Summary
1 Notation Conventions
Sets
• N: Set of natural numbers. (0 ∈ N)
• Z: Set of integers.
• Z+ = N∗: Set of positive integers.
• R: Set of real numbers.
• C: Set of complex numbers
• Z: Either R or C.
• {A : B}: Set of all A such that the property B is true
• X ∩ Y = {z : z ∈ X and z ∈ Y }
• X × Y = {(x, y) : x ∈ X and y ∈ Y }
Symbols used
• ∀: for all.
• ∃: There exists and ∃! there exists a unique.
• ∈: belongs to.
• ⊂: is included in (is a subset of).
• ⇒: implies.
• ⇔: if and only if.
• ≡: is equivalent to.
Example 1.1. ∀n ∈ N,∃!n+ 1 ∈ N ⊂ R
Complex numbers
• z: Complex conjugate. a+ bi = a− bi
• |z|: Absolute value. |a+ bi| =√a2 + b2
Matrices
• Aij : A matrix.
• [A]ij : The element in row i and column j of the matrix A.
• Id: The identity matrix. [A]ij = δij
• AT : Transpose of a matrix. [AT ]ij = [A]ji
• A∗: The hermitian adjoint of a matrix. [A∗]ij = [AT ]ij = [A]ji
• Tr(A): Trace of a matrix. Tr(A) =∑i[A]ii
• det(A): Determinant of a matrix.
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2 Abstract Group Theory Group Theory Summary
2 Abstract Group Theory
2.1 Definiton
Definition 2.1 (Group). A group G is a set of elements {g1, g2, g3, ...} togetherwith a composition law which has the following four properties:
1. Closure. The composition of two elements in G, written gigj, is itself anelement of G. gigj ∈ G ∀gi, gj ∈ G
2. Associativity. The composition law is associative.gi(gjgk) = (gigj)gk ∀gi, gj , gk ∈ G
3. Identity. There exists an element, called identity and denoted as e, suchthat egi = gie = gi ∀gi ∈ G
4. Inverses. Every element gi in G has an inverse, denoted as g−1i , which is
also in G and such that gig−1i = g−1
i gi = e
The composition law is not always the usual multiplication but the notation is the same.By analogy, the notation g2
i means gigi. The associativity property implies that the paren-theses are superfluous.
Example 2.1. The set {0, 1, 2, 3} with the composition law being the addition modulo 4operation is a group, usually denoted as Z4.
Example 2.2. The set of all permutations of 3 elements is a group. Its composition lawcorresponds to perform successive permutations. It is usually denoted as S3. The 6 elementsof this group are:
e ≡(
1 2 31 2 3
), a ≡
(1 2 32 1 3
), b ≡
(1 2 31 3 2
)c ≡
(1 2 33 2 1
), d ≡
(1 2 33 1 2
), f ≡
(1 2 32 3 1
)The letters used here to describe these elements will be the same in the rest of this document.
Definition 2.2 (Abelian Group). An Abelian group G is a group in which thecomposition law is commutative. gigj = gjgi ∀gi, gj ∈ G
Example 2.3. Z4 is an Abelian group and S3 is not. (It is the smallest non-Abelian group.)
2.2 Basic properties
Theorem 2.1 (Uniqueness of the identity). The identity element e in a group Gis unique.
Theorem 2.2 (Uniqueness of inverses). For each element gi in group G, there isa unique element b in G, such that gib = bgi = e
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2.3 Finite and infinite groups Group Theory Summary
These two theorems imply that one can talk form the inverse and the identity withoutambiguity.
Example 2.4. In Z4, the identity element is the element 0.
Theorem 2.3 (Cancellation). In a group G, the left and right cancellation lawshold.gigj = gigk ⇒ gj = gk ∀gi, gj , gk ∈ G andgjgi = gkgi ⇒ gj = gk ∀gi, gj , gk ∈ G
2.3 Finite and infinite groups
Definition 2.3 (Order of a group). The order of a group G, denoted as |G|, isthe number of elements contained in the group.
Example 2.5. The order of Z4 is 4, as this group contains 4 elements.Similarly, |S3| = 6
Definition 2.4 (Order of an element). The order of an element gi, denoted as|gi|, is the smallest positive integer such that g|gi|
i = e.
This implies that |e| = 1.
Definition 2.5 (Period of an element). The period of an element g is the collectionof elements {e, g, g2, g3, ...g|g|−1}.
The period of a group is generally different from the set of elements in the group.
Example 2.6. In Z4, |0| = 1, |1| = 4, |2| = 2, |3| = 4 and the different periods are:{0, 2}, {0, 1, 2, 3}, {0, 3, 2, 1}.
Definition 2.6 (Cyclic group). A cyclic group is a group based on the set ofn different objects {g, g2, g3, ..., gn}, where gn = e. The element g is called thegenerator of the group.
Cyclic groups of order n are denoted as Zn and all cyclic groups are Abelian.All groups whose order are prime numbers are cyclic.
Example 2.7. Z2 is the smallest non-trivial cyclic group.
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2.4 Multiplication Tables Group Theory Summary
2.4 Multiplication Tables
Definition 2.7 (Multiplication Table). The multiplication table of a group is asquared array whose rows and columns are labelled by the elements of the groupand the entries correspond to the products. The element gij is the product of theelement gi and gj.
The multiplication table fully defines a group.
Example 2.8. The multiplication table of Z4 is :
0 1 2 30 0 1 2 31 1 2 3 02 2 3 0 13 3 0 1 2
A group is Abelian if and only if its multiplication table is symmetric about the maindiagonal.
Theorem 2.4 (Rearrangement Theorem). If {e, g1, g2, ...g|G|−1} are the elementsof a group G and if gi is any element of the group, then the set of elements
Ggi = {egi, g1gi, g2gi, ..., g|G|−1gi}
contains each group element once and only once.
This theorem implies that each group element appears once and only once on each rowand column of the multiplication table and hence, gives an easy way to compute multipli-cation tables.
Example 2.9. The multiplication table of S3 is :
e a b c d fe e a b c d fa a e d f b cb b f e d c ac c d f e a bd d c a b f ef f b c a e d
2.5 Subgroups and Cosets
Definition 2.8 (Subgroup). A subset H of the elements of a group G is a sub-group of G if it is a group with the same composition law than G.
Example 2.10. The group Z4 has 3 subgroups:
{0}, {0, 2}, {0, 1, 2, 3}
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2.5 Subgroups and Cosets Group Theory Summary
The unit element {e} and G itself form subgroups of G. These two subgroups are calledimproper subgroups. The other one (if they exist) are called proper subgroups. Theidentification of proper subgroups is one of the importantest part of group theory.
Example 2.11. The group S3 has 4 proper subgroups:
{e, a}, {e, b}, {e, c}, {e, d, f}
Definition 2.9 (Coset). If H = {e, h1, h2, ..., h|H|−1} is a subgroup of a group Gand g is an element of G, then the set
Hg = {eg, h1g, h2g, ..., h|H|−1g}
is called a right coset of H. Similarly, the set
gH = {ge, gh1, gh2, ..., gh|H|−1}
is called a left coset of H.
A coset is a subgroup of G only if g ∈ H.Left and right cosets are in general diffrent.
Example 2.12. The subgroup H = {e, a} of S3 has 3 different right cosets:
{e, a}, {b, d}, {c, f}
Theorem 2.5. Two cosets of a subgroup either contain exactly the same elementsor have no common elements at all.
This theorem sets an upper limit on the number of different cosets of a given subgroup.
Definition 2.10 (Index). The number of distinct cosets of a subgroup H of a groupG is called the index of H and is denoted as |G : H|.
Example 2.13. The subgroup H = {e, a} of S3 has an index |S3 : H| = 3.
Theorem 2.6 (Lagrange’s theorem). The order of a subgroup H of a finite groupG is a divisor of |G|.
|G : H| = |G||H|
Example 2.14. The subgroup H = {e, a} of S3 is of order 2, which is a divisor of 6, the orderof S3.
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2.6 Classes Group Theory Summary
2.6 Classes
Definition 2.11 (Conjugation). Two elements gi and gj of a group G are conju-gate if there is a group element a, the conjugating element, such that gi = agja
−1
Definition 2.12 (Conjugacy classes). The set of all elements that are conjugatedto each other is called a conjugacy class.
Example 2.15. The group S3 has 3 conjugacy classes:
{e}, {a, b, c}, {d, f}
All elements in a conjugacy class have the same order. But the opposite is not true.In an Abelian group, all elements are in their own conjugacy class.
Example 2.16. As it is Abelian, the group Z4 has 4 conjugacy classes:
{0}, {1}, {2}, {3}
Definition 2.13 (Self-conjugate subgroups). A subgroup H of a group G is self-conjugate if the elements gHg−1 are identical with those of H for all elements gof G.
An alternative definition is to say that H is self-conjugate if gH = Hg ∀g ∈ G.A group with no proper self-conjugated subgroup is called simple.
Example 2.17. The subgroup {0, 2} of Z4 is self-conjugate.
All subgroups of an Abelian group are self-conjugate.
2.7 Product Groups
Definition 2.14 (Product group). A group G is a product group if there are twoproper subgroups Hi and Hj such that :
1. ∀g ∈ G, ∃ha ∈ Haand hb ∈ Hb such that g = hahb = hbha
2. Ha ∩Hb ≡ e
The product group is signified by G = Ha ×Hb.
All subgroups involved in a product group are self-conjugate. This implies that a simplegroup can not be expressed as a product of smaller groups.The order of a product group is given by: |Ha ×Hb| = |Ha||Hb|
Example 2.18. Z6∼= Z3×Z2 (The precise signification of ∼= will be given in the next section.)
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2.8 Factor Groups Group Theory Summary
Definition 2.15 (Product Group). A product group Ha ×Hb of two groups Ha
and Hb is defined as the set of all distinct pairs (a, b) a ∈ Ha, b ∈ Hb with thecomposition law given by
(a1, b1)(a2, b2) ≡ (a1a2, b1b2)
These two definitions (2.14 & 2.15) are equivalent.
Example 2.19. Let Ha = {ea, a} and Hb = {eb, b} be two copies of Z2. We define
E ≡ (e1, eb), A ≡ (a, eb), B ≡ (ea, b), C ≡ (a, b)
The multiplication table of Z2 × Z2 is then
E A B CE E A B CA A E C BB B C E AC C B A E
2.8 Factor Groups
Definition 2.16 (Factor Group). The factor group of a self-conjugate subgroupH of a group G is the collection of cosets of H, with each coset being considered asa group element.The order of the factor group is equal to the index of the self-conjugate subgroup.The factor group is denoted as G/H.
Remark. Note that (G/H)×H is not always G.
Example 2.20. The subgroup H = {e, d, f} of S3 has two differents right-cosets:
{e, d, f}, {a, b, c}
and two differents left-cosets:{e, d, f}, {a, b, c}
Since these cosets are the same, H is self-conjugate. Multiplying the cosets together on anelement-by-element basis, one get:
{e, d, f}{e, d, f} = {e, d, f}, {e, d, f}{a, b, c} = {a, b, c}
{a, b, c}{e, d, f} = {a, b, c}, {a, b, c}{a, b, c} = {e, d, f}
These cosets are closed under multiplication rule. Defining E ≡ {e, d, f} and A ≡ {a, b, c},the multiplication table of this factor group is
E AE E AA A E
which is the multiplication of Z2. Hence, S3/H ∼= Z2
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3 List of usual finite groups Group Theory Summary
3 List of usual finite groups
All finite simple groups have been classified and it is possible to build any finite group byusing a product of the simple groups. All these groups are not interesting and the list showedhere only contains some of them.
Cyclic Groups A cyclic group is a group based on the set of n different objects{g, g2, g3, ..., gn}, where gn = e. The element g is called the generator of the group.These groups can be interpreted as the number 0, 1, ..., n− 1 with the addition modulo n ascomposition law.These groups are simple and their order is n.They are usually designated as Zn, Z/nZ or Cn.
Klein four-Group With only four elements, it is the smallest non-cyclic group. It is theproduct Z2 × Z2. It corresponds to the symmetry of a rectangle in 2D.It is usually designated as Z2 × Z2 or V .
Symmetric Groups They correspond to the sets of all bijection from X to X with thecomposition of function as composition law. If X is finite (X = {1, ..., n}), the group iscalled Sn and has order n!. These groups are not Abelian if n > 2.
Alterning Groups They correspond to the sets of all even-valued permutations of a finiteset X = {1, ..., n}. It has order n!/2. These groups are simple if n = 3 or if n ≥ 5.They are usually denoted as An.
Dihedral Groups They correspond to the sets of symmetries of a regular polygon, in-cluding reflexions and rotations. They are denoted as Dn where n is the number of sides ofthe polygon. The order of the group is 2n.
Quaternion Group It is the group made of the 8 elements Q = {1,−1, i,−i, j,−j, k,−k}with the composition law being:
• (−1)(−1) = 1
• (−1)a = a(−1) = −a ∀a ∈ Q
• i2 = j2 = k2 = ijk = −1
It is usually designated as Q8 or just Q.
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4 Representation of groups Group Theory Summary
4 Representation of groups
Group elements correspond to symmetry operations on spatial coordinates. They can berepresented as linear transformations with respect to a coordinate system. The resultingmatrices form a group which is equivalent to the group of symmetry operations. The matricesform a representation of the group and each matrix is an element of the group.
4.1 Homomrphisms and Isomorphims
Definition 4.1. Let G and G′ be two finite groups {e, g1, g2, ...} and {e′, g′1, g′2, ...}.Let ϕ be a mapping between the elements of G and G′ that preserves their compo-sition rules.
a′ = ϕ(a) and b′ = ϕ(b) ⇒ ϕ(ab) = ϕ(a)ϕ(b) = a′b′
If the two groups have the same order, then the mapping is called an isomorphismand the two groups are said to be isomorphic. This is denoted as G ∼= G′.If the orders are different, then the mapping is called an homomorphism and thetwo groups are said to be homomorphic.
Example 4.1. Consider the group G = −1, 1 with the usual multiplication as compositionlaw and the group Z2. Then the mapping
ϕ(1) = 0 ϕ(−1) = 1
establishes a one-to-one correspondance between G and Z2. Thus, these two groups areisomorphics.
An homomorphism loses some parts of the structure of the original group.
Example 4.2. Consider the mapping defined in the example 2.20 for the group S3:
E ≡ {e, d, f} A ≡ {a, b, c}
This is an homomorphism because three elements of S3 correspond to a single element ofZ2.
4.2 Representations
Definition 4.2 (Representation). A representation of dimension n of a groupG is a homomorphism or isomorphism ϕ : G→ GL(n,C). Where GL(n,C) is thegroup of non-singular (det 6= 0) n× n matrices with complex entries.
If the representation is an isomorphism, then it is said to be faithful, and if it is anhomomorphism, then it is called unfaithful.
Example 4.3. The representation of S3 given by:
{e, a, b, c, d, f} → 1
is called the identical representation. It is obviously unfaithful.
The identical representation exists for all groups.
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4.2 Representations Group Theory Summary
Example 4.4. The representation of S3 given by:
De =(
1 00 1
), Da =
12
(1 −
√3
−√
3 −1
), Db =
12
(1√
3√3 1
),
Dc =(−1 00 1
), Dd =
12
(−1 −
√3√
3 −1
), Df =
12
(−1
√3
−√
3 −1
)is a faithfull representation of the group of dimension 2.
Example 4.5. The representation of S3 given by:
{e, d, f} → 1 {a, b, c} → −1
is called the parity representation. It corresponds to the determinant of the matrices ofthe representation in the previous example (4.4). It is unfaithful.
Definition 4.3 (Similarity). Given a representation {De, Dg1 , Dg2 , ...} of a group{e, g1, g2, ...} and a non-singular matrix S of the same dimension than the repre-sentation the set
{SDeS−1, SDg1S
−1, SDg2S−1, ...}
is a similarity transformation of the representation.
The transformed matrices are also representations of the group G.
Definition 4.4 (Direct Sum). Let {Dg} and {D′g} be two representations of dimen-sions d and d′ of the same group. Then the direct sum of these two representationsis the set of block-diagonal matrices{(
De 00 D′e
),
(Dg1 0
0 D′g1
),
(Dg2 0
0 D′g2
), ...
}The direct sum is denoted as ⊕. The representation can be rewritten as :{
De ⊕D′e, Dg1 ⊕D′g1 , Dg2 ⊕D′g2 , ...}
Example 4.6. The direct sum of the representations of S3 given in 4.3 and 4.4. D′i = Di ⊕ 1
D′e =
1 0 00 1 00 0 1
, D′a =12
1 −√
3 0−√
3 −1 00 0 2
, D′b =12
1√
3 0√3 1 0
0 0 2
,
D′c =
−1 0 00 1 00 0 1
, D′d =12
−1 −√
3 0√3 −1 0
0 0 2
, D′f =12
−1√
3 0−√
3 −1 00 0 2
It is also a faithful representation of S3.
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4.3 Reductible and Irreducible Representations Group Theory Summary
4.3 Reductible and Irreducible Representations
Definition 4.5. If a similarity S bring all matrices of a representation into thesame block diagonal form then the representation is said to be reducible. Other-wise, it is irreducible.
Irreducible representations are often called irreps. For a given group, there are generallymore than one irreducible representation.
Example 4.7. The representations in example 4.3, 4.4 and 4.5 are irreducible but the onefrom example 4.6 is redducible.
All one-dimensionnal representations are irreducible.
Theorem 4.1. Every representation of a finite group can be brought in to a unitaryform by a similarity transformation
This theorem implies that one can deal with unitary matrices instead of general matriceswithout loss of generality.
16
5 The Great Orthogonality Theorem Group Theory Summary
5 The Great Orthogonality Theorem
Lemma 5.1 (Schur’s first Lemma). A non-zero matrix that commutes with all ofthe matrices of an irrep is a constant multiple of the unit matrix.
Lemma 5.2 (Schur’s second Lemma). Given two irreps {D1, D2, ..., D|G|} and{D′1, D′2, ..., D′|G|} of a group G of dimensions d and d′. Then if there is a matrixM such that
MDi = D′iM ∀i
then:If d = d′, either M = 0 or the two representations differ by a similarity.If d 6= d′, M = 0.
Schuhr’s lemmas provide restrictions on the form of matrices which commute with all ofthe matrices of an irrep.
Theorem 5.1 (Great Orthogonality Theorem). Let {D1, D2, ..., D|G|} and{D′1, D′2, ..., D′|G|} be two different irreps of a group G that have dimensionalities dand d′. Then
|G|∑α=1
(Da)∗ij(D′a)i′j′ = 0 ∀i, j, i′, j′
For the element of a single unitary irrep :
|G|∑α=1
(Da)∗ij(Da)i′j′ =|G|dδi,i′δj,j′
This theorem can be summarized in the following way. Let D(k)α be the αth matrix of
the kth irrep of a group G, then
|G|∑α=1
(D(k)a )∗ij(D
(k′)a )i′j′ =
|G|dδi,i′δj,j′δk,k′
Example 5.1. Consider the three irreps of S3 given in the examples 4.3, 4.4 and 4.5. Let
D(k)ij =
{[D(k)
e ]ij , [D(k)a ]ij , [D
(k)b ]ij , [D(k)
c ]ij , [D(k)d ]ij , [D
(k)f ]ij
}be a vector of the differents elements of the irreps of S3. Then
D(k)ij ·D
(k′)i′j′ =
|G|dδi,i′δj,j′δk,k′
where d is dimension of the irrep.For example D(1) = {1, 1, 1, 1, 1, 1} and D(1) ·D(1) = 6 since d = 1.
A consequence of the Great Orthogonality Theorem is that∑k d
2k ≤ |G| where the
summation is over all irreps of a group G.
17
5 The Great Orthogonality Theorem Group Theory Summary
Example 5.2. The two one-dimensionnal irrep and the two-dimensionnal irrep of S3 given inexamples 4.3, 4.4 and 4.5 set in the previous equation give :∑
k
d2k = 12 + 12 + 22 = 6
which equals the order of the group. Hence, there are no other distinct irreducible represen-tations of this group.
18
6 Characters and Character Tables Group Theory Summary
6 Characters and Character Tables
Characters are a way to determine if a given representation is reducible or not and tosummarize all the property of a group in a table.
6.1 Orthogonality relations
Lemma 6.1. Two matrices in a given representation of group that corresponds totwo elements of the same conjugacy class have the same trace.
Example 6.1. For the representation of S3 given in example 4.4, the traces are given by:
χ1 = Tr(De) = 2, χ2 = Tr(Da) = Tr(Db) = Tr(Dc) = 0, χ3 = Tr(Dd) = Tr(Df ) = −1
Definition 6.1 (Character). The characters of a conjugacy class is the tracecorresponding to all the elements of this class in a given irreducible representation.
The character of the αth class of the kth irreducible representation of a group is denoted(in this document) as χkα.
Example 6.2. For the representation of S3 given in example 4.4, the characters are given by:
χ1 = 2, χ2 = 0, χ3 = −1
Theorem 6.1 (Orthogonality Theorem for Characters). The characters of the ir-reducible representations of a group obey the relation∑
α
nαχkαχ
k′∗α = |G|δk,k′
where nα is the number of elements in the αth class.
It is also possible to define a relation between the character of a given irrep:∑k
χkαχk∗β =
|G|nα
δα,β
Theorem 6.2. The number of irreducible representations of a group is equal to thenumber of conjugacy classes of the group.
19
6.2 The decomposition Theorem Group Theory Summary
6.2 The decomposition Theorem
Theorem 6.3 (Decomposition Theorem). The character χa for the αth class ofany irrep can be written uniquely in terms of the corresponding characters of theirreps of the group as
χα =∑k
αkχkα
whereαk =
1|G|
∑α
nαχk∗α χα
This theorem gives a criterion to determine whether a given representation is reducibleor not.
Theorem 6.4. A representation is irreducible if∑α
nα|χα|2 = |G|
and reducible if ∑α
nα|χα|2 > |G|
6.3 The regular representation
Definition 6.2. The regular representation of a group is a representation ob-tained directly from the multiplication table of the group. This is done in two steps:
1. Rearrange the table such that the unit element appears along the main diag-onal.
2. Regarde the table as a |G| × |G| matrix from which the matrix representationfor each group element is assembled by putting 1 where that element appearsand a zero elsewhere.
Example 6.3. The regular representation of Z4 is:
0 =
1 0 0 00 1 0 00 0 1 00 0 0 1
, 1 =
0 1 0 00 0 1 00 0 0 11 0 0 0
,
2 =
0 0 1 00 0 0 11 0 0 00 1 0 0
, 3 =
0 0 0 11 0 0 00 1 0 00 0 1 0
20
6.4 Character Tables Group Theory Summary
Theorem 6.5. The dimensionalities dk of the irreducible representations of a groupare related to the order of the group by∑
k
d2k = |G|
This gives a diophantine equation which for small groups has only a few solutions andthus implies that it is easy to determine the dimensions of the irreps.
6.4 Character Tables
Character tables are a way to display informations on classes and entirely determine a group.It is a way to classify finite groups.
Definition 6.3. A character table is a table which colums are labelled with thedifferent conjugacy classes and the rows are labelled with the irreducible represen-tations (usually designated as Γi). The elements in position i, j is the character ofthe ith representation of the jth class.
To fill the different lines, the orthogonality theorem for characters is usefull. The “scalarproduct” of two rows of the table gives zero if the rows are different, and |G| if they are thesame. The same relations holds for columns.∑
α
nαχk∗α χ
k′
α = |G|δk,k′
Example 6.4. The character Table for the S3 group is
S3 {e} {a, b, c} {d, f}Γ1 1 1 1Γ2 1 −1 1Γ3 2 0 −1
where the first line is the trivial representation, the second line is the parity representationand the third line is the 2× 2 faithful matrix representation.
21
7 Permutation Groups Group Theory Summary
7 Permutation Groups
Permutation Groups play a central role in the theory of finite groups.
Theorem 7.1. All groups are a subgroup of a permutation group.
Example 7.1. Z3 is a subgroup of S3, which is a permutation group.
7.1 Permutations and Cycles
Definition 7.1. A cycle is a special type of permutation in which terms are justcylced, so the first object becomes the second, the nth becomes the n+ 1th and thelas object becomes the first. The notation used is
(12 . . . n)
to signify thatx1 → x2, x2 → x3, . . . , xn → x1
Definition 7.2. A k-cycle is a cycle with k elements in it.
Lemma 7.1. All permutations may be expressed as a product of cycles subsets,where each object appears once and only once.
Example 7.2. In S3 the diffrent permutations can be expressed in term of cycles as follow :
e ≡ (1)(2)(3)a ≡ (12)(3)b ≡ (1)(23)c ≡ (13)(2)d ≡ (231)f ≡ (321)
Each permutation can be expressed as k1 1-cycles, k2 2-cycles, ... , kn n-cycles where
n∑j=1
jkj = n
The set of {kj} is know as the cylce structure of a permutation.
Example 7.3. The cycle structure for the permutation a in S3 has the cycle-strucure, k1 =1, k2 = 1, k3 = 0. And
∑nj=1 jkj = 1 · 1 + 2 · 1 + 3 · 0 = 3.
22
7.2 Young Diagrams Group Theory Summary
Lemma 7.2. Permutations are in the same class of Sn if and only if they have thesame cycle structure.
Example 7.4. The cycle structures for the group S3 are:
{e} : k1 = 3, k2 = 0, k3 = 0{a, b, c} : k1 = 1, k2 = 1, k3 = 0{d, f} : k1 = 0, k2 = 0, k3 = 1
This gives a way to determine the number of elements in a class of Sn.
Lemma 7.3. The number of elements nα in the αth class of Sn is given by:
nα =n!
Πjkj !jkj
Example 7.5. In S3, the number of elements in each class is given by:
{e} : k1 = 3⇒ 3!3!13
= 1 element
{a, b, c} : k1 = 1, k2 = 1⇒ 3!(1!11)(1!21)
= 3 elements
{d, f} : k1 = 3⇒ 3!1!31
= 2 elements
Lemma 7.4. An n-cycle is equivalent to a product of (n− 1) 2-cycles.
This gives a natural division of permutations into two families, the odd-permutationsand even-permutations which are equivalent to an odd or even number of two-cycles.The even permutations form a group called the alterning group An. These two cosets alsoform a group, a representation of Z2. This implies Sn/An ∼= Z2.
7.2 Young Diagrams
Young diagrams are a way to represent easily Sn groups and to construct some of theirproperties.To construct the diagram for a given class of Sn, each class is represented by n boxes. Eachcycle is drawn as a column of boxes. Each column starts from the common horizontal lineand drops down. The largest cycles are put on the left.This gives a diagram which as one move from left to right is either constant in depth orshrinking. Each possible diagram is called a Young Tableau.
Example 7.6. These are the 5 valid Young Tableux for S4:
23
7.3 Young Tableaux and Sn irreps Group Theory Summary
These diagrams correspond to the classes (1234), (123)(4), (12)(34),(12)(3)(4) and (1)(2)(3)(4)respectively. As one cannot build other Young Tableaux with 4 boxes, these are the 5 classesof Sn.
From these diagrams, one can find the cycle structure. For each column, the depth ofthe colum corresponds to the length of the cycle. And the number of column of a givendepth ki corresponds to the number of ki-cycles in the group.
Example 7.7. The cycle structure from the second diagram in the previous exemple is k3 =1, k1 = 1.For the third diagram, k2 = 2 and for the fourth: k2 = 1, k1 = 2.
The number of elements in each class can then be found using the previously presentedformula or by considering the possible distinct cycles directly.
Example 7.8. From the diagrams in the previous example, one can find:
n1 =4!
1!41= 6, n2 =
4!(1!31)(1!11)
= 8, n3 =4!
(2!22)= 3, n4 =
4!(1!21)(2!12)
= 6, n5 =4!
4!14= 1
The sum of the ni is 24 as expected (|Sn| = n!).
7.3 Young Tableaux and Sn irreps
The regular representation plays an important role in determining the dimensions of theirreps of a permutation group. The regulare representation is the |G| × |G| matrix given by:
[Areg(g)]ij = 1⇔ gi(gj)−1 = g, 0 elsewhere
One can then define an orhtonormal basis ~u(g) given by [~u(g)]i = 1⇔ gi = g.Each Young Diagram corresponds to one Sn irrep whose basis may be constructed fromits tableaux. Conversly, each irrep µ in constructed just once. This implies that one cancompue the dimension of each irrep.
Definition 7.3 (Hook factor of a box). The Hook factor of a box hi is thenumber of boxes that a hook crosses in a given box i of a Young Tableau. The hookis made of two lines: one dropping from the box and one other going to the right,both starting from the box.
Example 7.9. The hook from the top-left box of the third Young Tableau of S3 is:
As the hook crosses 3 boxes, the hook factor of this box is 3.
Definition 7.4 (Hook factor of a Young Tableau). The Hook factor of a YoungTableau HY D is the product of all box’s hook factor of this Young Tableau.
HY D =n∏i
hi
24
7.3 Young Tableaux and Sn irreps Group Theory Summary
Hook factors play an important role in the following lemma:
Lemma 7.5 (Hook-length formula). The dimension dµ of an irreducible µ is:
dµ =n!
HY D
where HY D is the Hook factor of the Young diagram associated with the irrep µ.
25
8 Continous Groups and Lie Groups Group Theory Summary
8 Continous Groups and Lie Groups
Most of the theorems presented so far are only valid for finite groups because the invovle asum over all elements of a group. Some notions need to be redefined and some theorems tobe changed to accord with infinite and continuous groups. The first part of the work is tocreate a mix between between group theory and analysis, because analysis is good to dealwith continuous functions.
8.1 Definitions and examples
Definition 8.1 (Continuous Group). A continuous group (or topological group)is a group in which, each element R depends on a set of parameters R(~a) ≡(a1, a2, a3, ..., ad) and in which the composition law
R(~a)R(~b) = R(~c)
is continuous. The ~a are called coordinates and if there are d parameters ~a =(a1, a2, a3, ..., ad), the group is said to be a d−parameter continous group or tohave a dimension d.
The continuity of the composition law implies that R(~a) and R(~a+ δ~a) are both in thegroup for an arbitrarly small value of δ.Each group element can be represented as a point by associating it with its ~a d-dimensionnalvector. The space in which the elements lie is called the group manifold.
Example 8.1. Rn with the usual vector-addition law is an (Abelian) n-parameter continuousgroup.
Example 8.2. C∗ (complex numbers without 0) with the usual multiplication forms an(Abelian) continuous group.
It is easier to deal with continuity in the context of calculus.
Definition 8.2 (Group structure). R(~a)R(~b) = R(~c) is true if and only if
~c = ~f(~a,~b) or equivalently if ci = fi(~a,~b)
The set of fi is called the structure of the group and determine the whole groupin the same way as the multiplication table for discrete groups.
The properties of the composition law give some restrictons on the fi. The associativityrequires that
~f(~a, ~f(~b,~c)) = ~f(~f(~a,~b),~c)
The existence of an identity element ~a requires that
f( ~a0,~a) = f(~a, ~a0) = ~a
And finally the existence of an inverse ~a′ for each ~a implies that
f(~a′,~a) = f(~a, ~a′) = ~a0
Definition 8.3 (Lie Group). A continuous group in which all functions of thestructure are analytical is called a Lie Group.
26
8.2 Compact and Non-Compact Lie Groups Group Theory Summary
Example 8.3. Rn with the usual vector-addition law is an (Abelian) n-parameter Lie group.
Example 8.4. C∗ (complex numbers without 0) with the usual multiplication forms an(Abelian) Lie group.
Example 8.5. The set of invertible matrices GL(n,R) of size n × n with the usual matrixmultiplication is a Lie group called the general linear group. It has dimension n2.
Example 8.6. The set of invertible matrices SL(n,R) of size n× n and determinant 1, withthe usual matrix multiplication is a Lie group called the special linear group. It has alsodimension n2.
In the two last examples, the matrices are mapping n−dimensionnal vectors into othern−dimensionnal vectors. The space of vectors being mapped is called the representationspace and the dimension of this space is called the dimension of the representation.This can take many different values for the same group. It is, in general, different from thedimension of the group. The relationship between these two dimensions is different for eachkind of group.
8.2 Compact and Non-Compact Lie Groups
For each Lie Groups there are different possible representation. This document will only dealwith the matrix representation.
Definition 8.4 (Fundamental Representation). The smallest (in term of dimen-sion) faithful irreducible representation of a Lie Group is called the fundamentalrepresentation.
Example 8.7. For GL(n,R), the fundamental representation is the one given previously: Theset of matrices n× n.
Definition 8.5 (Compact Lie Group). A Lie Group is compact if there exists afaithful representation in which all the elements are alwas finite.
|Dij | <∞
The complete definition is more complex but this one is sufficient for this course.
Example 8.8. The set of phases {exp(iθ)| − π < θ ≤ π} forms a compact group called U(1).
The importance of compact Lie Groups arises from the following thorem.
Theorem 8.1. (i) All compact Lie Groups have a finite dimensional unitary rep-resentation.
∃{D(g)} such that D∗(g)D(g) = Id
(ii) Finite dimensional representations of compact Lie groups are reducible, theycan be made block-diagonal.(iii) Any irreducible representation of a compact Lie group is finite dimensional.
27
8.3 Matrix Generators Group Theory Summary
It is easy to classify all possible compact Lie Groups. And there is only one Abeliancompact Lie Group (U(1)). The one that are not product groups are called simple groups.Products of simple groups with other non-Abelian simple groups are called semi-simplegroups.There are three main families (SO(n), SU(n), Sp(2n)) of compact simple Lie Groups and5 exceptionnal cases (E6,E7,E8,F4 and G2).
8.3 Matrix Generators
Definition 8.6 (Differential of a Matrix). The infinisetimal change in a matrixnear the origin ∂D
∂ai(~0) is defined as
∂D∂ai
(~0) = limai→0
D(0, 0, ..., ai, ..., 0)−D(~0)ai
For an infinitesimal ai (ai � 1) the following holds:
Dai≡ D(0, 0, ..., ai, ..., 0) = D(0, 0, ..., 0) +
∂D∂ai
(~0)ai +O(a2i )
This can be summarized as:∂D(~a)∂ai
= DaiD(~a)
By using the same process, one can define higher order derivatives.
∂nD(~a)∂ani
= Dai
∂n−1D(~a)∂an−1
i
This implies that one can define the Taylor expansion of D(~a) around ~0.
Lemma 8.1 (Taylor Expansion). Any element in the Lie Group (which represen-tation has a dimension n) can be expressed as a Taylor Serie:
D(~a) = Id +n∑i=0
aiDai +12
n∑i=0
a2iD
2ai
+ · · ·
=n∑i=0
∞∑j=0
1j!
(ai)j
= exp (a1Da1 + a1Da1 + · · ·+ anDan)
to be checkedUnfortunately, the last equality cannot trivially be broken in the product of n exponentials.
8.4 Exponential of infinitesimal Generators
Definition 8.7 (Commutator). The commutator of two matrices is defined as:
[A,B] = AB −BA
where the product and substraction are the usual one for matrices.
28
8.4 Exponential of infinitesimal Generators Group Theory Summary
Lemma 8.2 (The Cambell-Baker-Hausdorff formula).
exp(A) exp(B) = exp(A+B +
12
[A,B] +112
[A, [A,B]] +112
[[A,B], B] + · · ·)
This reduces to the usual case exp(A) exp(B) = exp(A+B) when [A,B] = 0.As one want exp (a1Da1 + a1Da1 + · · ·+ anDan) = expϕ1Da1 expϕ2Da2 · · · expϕnDan ,
one apply the previous formula and get:
(a1Da1 + a1Da1 + · · ·+ anDan) = ϕ1Da1 + ϕ2Da2 + · · ·ϕnDan
+
+12
n∑i=0
n∑j=0
ϕiϕj [Dai ,Daj ]
+ · · ·
The only important parameters are the different commutators, which play an importantrole in Lie Algebras. The commutators entirely define the Lie Group because they entirelydefine the infinitesimal generators, which define all the elements by exponentiation.
29
9 Introduction to Lie Algebras Group Theory Summary
9 Introduction to Lie Algebras
Lie Algebras are linked with Lie groups, hence this short introduction.
9.1 Definition and examples
Definition 9.1 (Lie Algebra). A Lie Algebra A ia a vector space over a fieldF together with a binary operation [·, ·] (called the Lie bracket) which obeys thefollowing rules:
1. Closure [x, y] ∈ A ∀x, y ∈ A
2. Bilinearity
[ax+ by, z] = a[x, z] + b[y, z][z, ax+ by] = a[z, x] + b[z, y]
∀x, y, z ∈ A and ∀a, b ∈ F.
3. Alterning [x, y] = −[y, x] ∀x, y ∈ A
4. Jacobi Identity [x, [y, z]] + [y, [z, x]] + [z, [x, y]] = 0 ∀x, y, z ∈ A
The alterning rule implies that [x, x] = 0 ∀x ∈ A.
Example 9.1. Any vector space V with the Lie bracket identical to zero ([x, y] = 0 ∀x, y ∈V) is an (Abelian) Lie Algebra.
Example 9.2. The Euclidian space R3 together with the usual cross product as Lie bracketis a Lie Algebra.
Example 9.3. The vector space of unitary matrices n × n together with the Lie bracketdefined as
[x, y] = xy − yx
(where the multiplication and the substraction are the usual ones) is a Lie Algebra. The Liebracket defined in this example is called the commutator.
As this document only deal with the matrix representation of Lie Groups, the focus willbe set on parts of Lie Algebra that are relevent for these kinds of vector spaces.
9.2 Generators
Definition 9.2 (Generators). The generators of a Lie Algebra A are the set ofmatrices {Ta} (basis vectors) of the vector space included in A.
This implies that A =∑a caTa ∀A ∈ A, ca ∈ R and that T ∗a = Ta.
Example 9.4. The generators of R3 are (1, 0, 0), (0, 1, 0), (0, 0, 1).
The choice of generators is not unique.
30
9.3 Subalgebras Group Theory Summary
Definition 9.3 (Dimension). The dimension of a Lie Algebra A is the numberof generators of A.
Example 9.5. dim(R3) = 3
Lemma 9.1 (Orthonormal generators). If the vector space included in the LieAlgebra A has a scalar product 〈·, ·〉, then the Lie Algebra has an orthonormal setof generators.
〈Ta, Tb〉 = δa,b ∀a, b, ..., n = dim(A)
The orthonormal generators can be obtained from any generators by the Grahm-Schmidtprocedure.
Example 9.6. The generators of R3 given in 9.4 are orthonormals.
For matrices, the usual scalar product is
〈Ta, Tb〉 =12
Tr(TaTb)
where the factor 12 is just here for normalization.
9.3 Subalgebras
Definition 9.4 (Cartan Subalgebra). The Cartan subalgebra C of an algebra Ais the largest set of elements which commute with themselves.
C = {A1, A2 : [A1, A2] = 0, A1, A2 ∈ A}
(It is the biggest subalgebra of A.)
For matrices, the Cartan Subalgebra is the largest number of elements of A which canbe diagonalised simultaneously. The standard choice for generators of A is to have thelargest number of diagonalised elements. Thus these generators are the basis of the CartanSubalgebra.
Example 9.7. For A = R3 the Cartan Subalgebra, is C = {(1, 0, 0)}.
Definition 9.5 (Rank of an Algebra). The rank of a Lie Algebra A is the dimen-sion of the Cartan subalgebra of A.
Example 9.8. The rank of A = R3 is 1.
31
9.4 Structure constants Group Theory Summary
9.4 Structure constants
Definition 9.6 (Structure constants). The structure constants fabc ∈ F aredefined as
[Ta, Tb] =dim(A)∑c=1
fabcTc, a, b = 1, 2, ..., dim(A)
where Ta are the generators of the Algebra.
The structur constants are totally antisymmetric
fabc = −f bac = −facb
and they totally define the Lie Algebra. The choice of structure constants for a given LieAlgebra is not unique.
The Jacobi identity can be rewritten in term of the structure constants as:
dim(A)∑e=1,g=1
fabefecg + f bcefeag + f caefebg = 0
Definition 9.7 (Adjoint Representation). The adjoint representation of a givenLie Algebra A is the Algebra generated by the sets of generators:
{(Xa)ij} = −faij , a, i, j = 1, 2, ..., dim(A)
where the faij are the structure constants of A.
Remark. The importance of this representation will be obvious later in the discussion of LieGroups.
9.5 Lie Algebras and Lie Groups
The lemma 8.1 gives a link between Lie Algebras and Lie Groups. All element U of a LieGroup G can be written as an exponential of anti-Hermitian matrices A.
U = exp(A), A∗ = −A
The generators {Ta} are the basis vectors of the Lie Algebra A =∑dim(A)i=a caTa, and the
derivatives with respect to the parameters of the Lie Groups ca are given by:
Da =dUdca
This imply that the knowledge of the structure constnats is enough to caracterise a LieGroup. They play the same role than multiplication table for finite groups.
9.6 Lie Algebras of product Lie Groups
If G = H×K is a product Lie Group, then the elements of G can be split into two commutingparts
U = PQ = QP ∈ G, P ∈ H,Q ∈ K
This implies that the Lie Algebra AG is the sum of the subalgebras AH and AK .
32
9.6 Lie Algebras of product Lie Groups Group Theory Summary
Using the formula 8.2, whith P = exp(B) and Q = exp(C), one can find that
exp(B) exp(C) = exp(C) exp(B) = exp(B + C)⇔ [B,C] = 0
This implies that the generators of the Lie Algebra AG obey the following rules:
[Ya, Yb] = fabcH Yc ∈ AH ⊂ AG[Za, Zb] = fabcK Yc ∈ AK ⊂ AG[Ya, Zb] = 0
where {Ya} are the generators of AH and {Za} the generators of AK .
The rest of this document will present the different simple Lie Groups.
33
10 Orthogonal Groups Group Theory Summary
10 Orthogonal Groups
Orthogonal groups play a central role in geometry.
10.1 O(n) groups
Definition 10.1 (O(n)). The group O(n) is the compact Lie group of all realorthogonal matrices n × n with the composition law being the usual matrix multi-plication.
O(n) = {D ∈ GL(n,R) : DTD = DDT = Id}
This group is called the Orthogonal Group in n dimensions.
This definition implies that all matrices in O(n) have a determinant equal to ±1.
Example 10.1. The group O(1) = {−1; 1} is isomorphic to Z2 and not a Lie Group.
The O(n) groups correspond to the set of all linear transformations which preserve thelength of the vectors in Rn. This implies that the group is Abelian.The dimension of the Lie Group O(n) is 1
2n(n − 1) because the matrices need to be sym-metric (modulo a change of sign).
Example 10.2. The matrix
D(ϕ) =(
cosϕ − sinϕsinϕ ± cosϕ
), 0 ≤ ϕ < 2π
are two matrices in O(2).
All O(n) groups are related together by the relation
O(n− 1) ⊂ O(n)
The group O(n) can be split in two sets according to the determinant of the matrices.The subset with determinant −1 is not a group, because it doesn’t contain the identity.
10.2 SO(n) groups
Definition 10.2 (SO(n)). The SO(n) group is the subgroup of O(n) in which thematrices have a determinant +1.
SO(n) = {D ∈ GL(n,R) : detD = 1} (10.1)
It is called the special orthogonal group in n dimensions.
Example 10.3. The matrix
D(ϕ) =(
cosϕ − sinϕsinϕ cosϕ
), 0 ≤ ϕ < 2π
is in SO(2).
The SO(n) groups correspond to the set of all proper rotations in Rn.
34
10.3 The SO(2) group Group Theory Summary
Definition 10.3. Let U ∈ O(n) and R ∈ SO(n), then U = Z±R, where
Z+ = Id
Z− ={
−Id if n odddiag(−1,+1,+1, ...,+1) if n even
where diag indicates the entries on the diagonal of the matrix.
Example 10.4. {Z+,Z−} forms a group isomorphic to Z2.
The matrix −Id commutes with everything this implies that
O(2m+ 1) ∼= Z2 × SO(2m+ 1) m ∈ N
If n is even , this is not true and the only relation that exists is for n = 2
O(2)/SO(2) ∼= Z2
10.3 The SO(2) group
The dimension of the SO(2) group is 1. And the fundamental representation is given by:
D(ϕ) =(
cosϕ − sinϕsinϕ cosϕ
), 0 ≤ ϕ < 2π
The character of this represenation is
χ(ϕ) ≡ Tr{D(ϕ)}= 2 cosϕ= eiϕ + e−iϕ
= χ(1)(ϕ) + χ(−1)(ϕ)
This implies that the representation is a direct sum of two irreps. This can be extended tothe following irreps:
χ(m)(ϕ) = eimϕ, m ∈ ZIf we add the identity representation (which always exists), the character table of SO(2)
is:SO(2) E D(ϕ)
Γm 1 eimϕ
The generator of the corresponding Lie Algebra (called D1) is given by
dDdϕ
=(
0 −11 0
)= X
10.4 The SO(3) group
The dimension of the SO(3) group is 3 and the generators of the fundamental representationare:
Dx =
1 0 00 cosϕ − sinϕ0 sinϕ cosϕ
, Dy =
cosϕ 0 sinϕ0 1 0
− sinϕ 0 cosϕ
, Dz =
cosϕ − sinϕ 0sinϕ cosϕ 0
0 0 1
With the same process than SO(2), one get the character table:
SO(3) E D(ϕ)
Γl 1sin[(l+ 1
2 )ϕ]sin( 1
2ϕ
35
10.4 The SO(3) group Group Theory Summary
The generators of the Lie Algebra (called B1) are:
X1 =
0 0 00 0 −10 1 0
, X2 =
0 0 10 0 0−1 0 0
, X3 =
0 −1 01 0 00 0 0
The computation of the structure constants leads to fabc = εabc, where εabc is the Levi-
Civita tensor.
36
11 Unitary Groups Group Theory Summary
11 Unitary Groups
11.1 U(n) groups
Definition 11.1 (U(n)). The group U(n) is the compact Lie group of all complexunitary matrices n× n with the composition law being the usual matrix multiplica-tion.
U(n) = {D ∈ GL(n,C) : DTD = DDT = Id}
This group is called the Unitary Group in n dimensions.
This definition implies the matrices in U(n) obey the equation |detD| = 1.The dimension of this group in n2.
11.2 The U(1) group
It is the only Abelian simple Lie Group. Hence, all Abelian Lie Groups are made of productsof U(1).
Definition 11.2. The U(1) group (or circle group) is the multiplicative group ofall complex numbers with absolute value 1.
U(1) = {z ∈ C : |z| = 1}
U(1) can be represented as a circle of radius 1 around the origin in the complex plane.This circle is the group manifold. This is the same manifold than SO(2), hence:
U(1) ∼= SO(2)
11.3 SU(n) groups
Definition 11.3 (SO(n)). The SU(n) group is the subgroup of U(n) in which thematrices have a determinant +1.
SU(n) = {D ∈ GL(n,C) : detD = 1} (11.1)
It is called the special unitary group in n dimensions.
The dimension of the group is n2 − 1.These groups are important because they are simple groups and because U(n) groups
can be considered as the product group:
U(n) = SU(n)× U(1)
Definition 11.4 (Adjoint Representation). Let χ = {H} be the set of all n × nhermitian matrices. The application ϕS defined as:
ϕs : χ→ χ : H → H ′ = ϕSH = SHS∗ (11.2)
where S ∈ SU(n), is called the adjoint representation of SU(n).
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11.4 The SU(2) group Group Theory Summary
Every hermitian matrix S can be split into two parts.
S′ = tId + J, t =1n
Tr{S}, Tr{J} = 0 (11.3)
This implies, that all irreducible SU(n) representations are traceless.
11.4 The SU(2) group
The SU(2) group plays an important role in Quantum Mechanics.A matrix in SU(2) has the form(
a b
−b a
), |a|2 + |b|2 = 1
An obvious set of generators for this group is then the set of Pauli matrices.(0 11 0
), σ2 =
(0 −ii 0
), σ3 =
(1 00 −1
)and the structure constants are fabc = εabc, the Levi-Civita tensor. These are the same
structure constants than SO(3) and hence, their Lie Algebras are identical. One can showthat
SU(2)/Z2∼= SO(3)
11.5 The SU(3) groupto be com-pleted11.6 SU(n) groups and Young Tableaux
Each irrep of SU(n) corresponds to a distinct Young diagram whit no more than n boxesin any column.
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12 Other Lie Groups Group Theory Summary
12 Other Lie Groups
12.1 Sp(2n) groups
Definition 12.1 (Group Sp(2n)). The group Sp(2n) is the set of real matrices Dthat satisfy MJMT = J , where J is the 2n × 2n matrix given in terms of n × nblocks:
J =(
0 Id−Id 0
)It is called the symplectic group of n-dimension.
This group is involved in classical Hamiltonian mechanics and its dimension is n(2n+ 1) to be com-pleted
12.2 Exceptional groups
These 5 groups are called exceptionnal because they don’t belong to an infinite serie (SO(n),SU(n) or SP (2n)). to be com-
pleted
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13 Group theory in Physics Group Theory Summary
13 Group theory in Physicsto be com-pleted
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