“Enthusiasm is followed by disappointment and even depression, and then by renewed enthusiasm.”

― Murray Gell-Mann

About this course:

  • Lecturer: Prof. Dr. C. Hanhart, PD Dr. A. Wirzba
  • Year: 2017/2018
  • Difficulty:
  • Course page: ITKP
  • Tutor: M. Mikhasenko
  • Literature:

A good course on group theory. The concepts became pretty abstract and hard in the end. Unfortunately, it is not always easy to recognize the physical meaning behind all the math.

Exercise 1:

Keywords: Group Definition, Left identity Element, Left inverse Element, Multiplication Table, Finite Group, Cyclic Group, Cn, C2, C3, Zn, Z4, Dihedral Group, Dn, D2, D3, Rearrangement Theorem, Multiplication modulo ten, Vierergruppe, Klein Four-group, Z2 x Z2, Directed Polyhedra, Undirected Polyhedra

  1. Simple Facts about Groups
  2. Multiplication Tables
  3. The Rearrangement Theorem
  1. An Alternative for a Group with four Elements
  2. Vierergruppe (Klein Four-Group)
Presence 1
Homework 1

Exercise 2:

Keywords: Rotation around Axis, Polygon, Dihedral Group, Dn, D4, Mirror Reflections, Rotations, Symmetric Group, Sn, S3, S4, S5, Multiplication Table, Generators, (Cayley’s Theorem), Alternating Group, An, A4, Even Permutations, Odd Permutations, Cyclic Structure, Pertinent Disjoint Cycles, Partitions, Young Frames, Conjugacy Classes

  1. Dihedral Groups
  2. The Symmetric Group Sn
  3. (Cayley’s Theorem)
  1. The Alternating Group An
  2. The Symmetric Group Sn
PRESENCE 2
Homework 2

Exercise 3:

Keywords: Finite Group, Properties, Simple Groups, Abelian Groups, Normal Subgroups, Cosets, Invariant Subgroups, Direct Product, Quotient Groups, (Center of a Group), (Proper Center), (Pertinent Conjugating Element), (Cayley’s Theorem), (Klein Four-group), Alternating Group, An, Symmetric Group, Sn, S2, S3, S8, Dihedral Group, Dn, D4, Conjugacy Classes, Subgroups, Pauli Matrices, Multiplication Table, Order of a Group, Order of an Element

  1. Simple Facts about Groups and Subgroups
  2. Direct Product and Quotient Group
  3. (Center of a Group)
  4. (Cayley’s Theorem)
  5. Invariant Subgroup
  1. The Dihedral Group — Again
  2. Pauli Matrices
PRESENCE 3
Homework 3

Exercise 4:

Keywords: Poincaré Group, Lorentz Transformations, Normal Subgroup, Semisimple Group, Subgroups, Symmetric Group, Sn, S4, Conjugacy Classes, Order of Permutations, Regular Permutations, Invariant Subgroups, Factor Groups, Lagrange’s Theorem, Regular Subgroups, Dihedral Group, Dn, D4, Center of a Group, Inner Automorphisms, Partitions, Cyclic Structure, Pauli Matrices, Quaternion Group, Q4, Order of Elements, Multiplication Table, Proper Invariant Subgroups, Quotient Groups

  1. The Poincaré Group
  2. Subgroups of S4
  3. Center of a Group continued
  1. Partitions
  2. Pauli Matrices and the Quaternion Group Q4
PRESENCE 4
Homework 4

Exercise 5:

Keywords: Representations, Normal Proper Subgroup, Quotient Group, Faithful Representations, Inner Product, Inner Product Space, Unitary Representation, Two-dimensional Representation, Dihedral Group, Dn, D3, Characters, Conjugacy Class, Vector Space, Orthonormal Basis, Gram-Schmidt, Induction, Basis Functions, Homogeneous Functions, O(2) Transformations, Coordinate Space, Matrix Form

  1. Representations
  2. Inner Product Space
  3. Representations of D3
  1. Gram-Schmidt Revision
  2. Representation on Homogeneous Functions
PRESENCE 5
Homework 5

Exercise 6:

Keywords: Representations, Vector Space, Invariant Subspaces, Similarity Transformations, (Coordinate Transformations), (Wave Function), Unitary irreducible Representations, (Basis Functions), Dihedral Group, Dn, D3, (Three-dimensional Representation), Matrix Representation, (Characters), Reducible Representation, (Subspaces), (Two-dimensional Representation), One-dimensional Representation, (Transformation Matrix), Irreducible Representations, Abelian Group, Cyclic Group, Polynomials of Degree two, Six-dimensional Representation

  1. Similarity Transformations
  2. (Fundamental Orthogonality Theorem)
  3. (A reducible Representation of D3)
  1. Irreducible Representations of Abelian Groups
  2. A Representation of D3 on Polynomials
PRESENCE 6
Homework 6

Exercise 7:

Keywords: Representations, Rotation Group, Wigner d-matrix, Euler Angles, Wigner small-d-matrix, Irreducible Tensor Operator, Clebsch-Gordon Series, Irreducible Representation, Conjugacy Classes, Dirac Character, Characters, Orthogonality Theorem for Characters, Irreducible Spherical Tensor of Rank one, Reduced Matrix Element, Algorithm

  1. Representations of the Rotation Operator
  2. Representations, Conjugacy Classes and Characters
  3. The Number of irreducible Representations
  1. The Wigner-Eckart Theorem
  2. Wigner d-matrix
PRESENCE 7
Homework 7

Exercise 8:

Keywords: Dihedral Group, Dn, D6, D6, Representations, Conjugacy Classes, Irreducible Representations, Character Table, Orthogonality Theorem for Characters, Benzene Molecule, Ring Molecule, Equilateral Hexagon, Molecular Symmetry Axis, Molecular Plane, Cyclic Permutation, Reflection, Unoriented Hexagonal equilateral Polygon, Symmetry Group, Six-dimensional Representation, Characters, Reducible Representations, Spectrum of Benzene, Degeneracies, (Symmetric Group), (Sn), (S4), (Alternating Group), (An), (A4), (Partitions), Two-dimensional Representations, Clebsch-Gordon Decomposition, Quaternion Group, Q4

  1. Benzene and its Symmetry Group D6
  2. (The Character Tables of S4 and A4)
  1. The Character Table of D4
  2. The Character Table of Q4
PRESENCE 8
Homework 8

Exercise 9:

Keywords: Lie Algebras, Basis, SU(2), (Euclidean Group), (E(2)), (Commutator Relation), Structure Constants, (Jacobi Identity), Adjoint Representation, (Killing Form), SL(2,C), SO(3), Euler Angles, Double Cover

  1. (The Lie Algebras of SU(2) and E(2))
  2. (Structure Constants)
  3. (Adjoint Representation)
  4. (Killing Form)
  1. The Lie Algebra of SL(2,C)
  2. The Homomorphism SU(2) → SO(3)
Not worked on
Homework 9

Exercise 10:

Keywords: (Highest Weight Construction), SU(2), (Raising Operator), (Lowering Operator), Commutators, (Eigenvalues), (Eigenstates), Spin Representations, Wigner-Eckart Theorem, (Spherically Symmetric Potential), (Adjoint Representation), (Lie Algebra Homomorphism), (Vector Spaces), (Lie Algebras), (Discrete Subgroup), (Kernel), Spin-3/2 Representation, Spin-1/2 Representation, Pauli Matrices, SO(3), Doule Cover

  1. (Spin s Representation of SU(2))
  2. (Wigner-Eckart Theorem)
  3. (Lie Algebra Homomorphism)
  1. More on SU(2) and Tensor Operators
  2. The Homomorphism SU(2) → SO(3) again
Not worked on
Homework 10

Exercise 11:

Keywords: Lie Algebra, Cartan Subalgebra, Cartan Generators, (Scalar Product), (Adjoint Representation), (Hermitian Cartan Elements), (Self-adjoint), (Cartan Algebra), Roots, (Complex conjugate Representation), Weights, (Real Representation), (Automorphism), (Pauli Matrices), (Four-dimensional Representation), Gell-Mann Matrices, Commutation Relations, Structure Constants, Root Chain, Weyl Reflection, Root Diagram, Weight Diagram

  1. (Roots and the Cartan Algebra)
  2. (Conjugate Representations)
  3. (Weights and Roots — A simple Example)
  1. The Gell-Mann Matrices
  2. Some Facts about Roots
Not worked on
Homework 11

Exercise 12:

Keywords: (Lie Algebras), (Complete Classification of all Lie Algebras), (Exceptional Algebras), (Dynkin Diagram), (Lines), (Roots), (G2), (Simple Roots), (Root Diagram), (Isospin Symmetry), (Nucleon-Nucleon Scattering), (Isosinglet), (Isodoublet), (Isotriplet), (Clebsch-Gordon Coefficients), (Cross-Section), (Wigner-Eckart Theorem)

  1. (Exceptional Algebra F4)
  2. (Dynkin Diagram for G2)
  3. (Isospin in Nucleon-Nucleon Scattering)
  1. No Homework
Not worked on
No Homework

Exercise 13:

Keywords: (Young Tableaux), (Young Tableau), (SU(3)), (Decomposition), (Dimensions of Representations)

  1. (Young Tableaux)
  1. No Homework
Not worked on
No Homework
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Vera

Hi Marvin
I study mathematics and I want to follow the course on Group theory next semester. Until now I have taken only an introductory course on groups (solvability, sylow theorems) and Theo1, Theo2. Is Theo3 really important for this course as a prerequisite and would it be hard for me to follow the course without it?
Thank you in advance

Last edited 2 years ago by Vera
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Marvin Zanke

Hey Vera, you will most likely not need the introductory course on groups (from your studies in mathematics?!) to be able to follow the group theory course that is part of a physics degree. Depending on the lecturer(s), the subject will be covered from scratch, i.e. you will potentially start with discrete groups. What I would definitely recommend, however, is Mathematics 1-3, which should — roughly — be equivalent to Analysis 1-2 and Linear Algebra 1-2 from the mathematics degree. Regarding the course on quantum mechanics: well, that (also) depends. I wouldn’t call it a prerequisite but it will certainly… Read more »

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