Groups and Representations

Due to the COVID19 pandemic this course will be taught online.

You will get instructions (text & video) for the preparation of the synchronous sessions (lectures).
In the synchronous sessions (lectures) we will discuss questions, work with quizzes or study examples.
You can ask questions at any time in ILIAS and of course during the synchronous sessions (lectures).
After studying the instructions you will be ready to work on the weekly homework problems (exercises).
You can submit your solutions to the homework problems and they will be graded.
You will discuss solutions to the homework problems in weekly propblem classes (exercises).
You can ask questions about the homework problems at any time in ILIAS and of course during the synchronous sessions (lectures and/or problem classes).

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Objectives

To know basic concepts of group and representation theory.
To be able to explain proofs learned in the lectures and use methods and results in order to solve exercises. To be able to apply representation theory also in novel contexts.
To know examples for the application of group representations in physics.

Contents

groups: subgroups, homomorphisms, isomorphisms, group actions, orbits, stabilisers, equivalence classes, normal subgroups, cosets, factor groups
representations (reps): faithful reps, unitary reps, irreducible reps (irreps), reducibility, characters, Schur's lemma(s), orthogonality of irreps
applications: symmetries and degeneracies in quantum mechanics, selection rules
representations of finite groups: group algebra, regular representation, ideals, idempotents
symmetric groups: Young tableaux, Young operators, dimensions and characters
application: identical particles in quantum theories
Lie groups: Haar measure, representations, Lie algebras
tensor reps of classical groups: symmetry classes, Young tableaux
applications: SU(2) and SU(3) in particle physics (spin, isospin, flavour)
maybe: irreps of the Lorentz and Poincaré groups
application: notion of particles in quantum theories
maybe: roots and weights; Killing-Cartan classification of semi-simple Lie algebras

Registration

Please join the course on Ilias and register for the exercises on URM.

Internet forum

You find a discussion forum for this course within Ilias (access with zdv login). Feel free to discuss course related questions or the homework problems here with your fellow students. Your course instructors will also give advive or answer questions. If we happen to effectively solve a homework problem through discussion in the internet forum that's fine.

Literature

Note: You do not need to buy a specific book for this course. You should work with your notes. Check out the books for additional information or alternative presentations. Do not buy any book unless you already know that you like it and that you can work with it.

Irene Verona Schensted, A course on the Application of Group Theory to Quantum Mechanics, NEO Press, 1976.
Barry Simon, Representations of Finite and Compact Groups, American Mathematical Society, 1996.
Wu-Ki Tung, Group Theory and Physics, World Scientific, 1985.
William Fulton and Joe Harris, Representation Theory - A First Course, Springer-Verlag, 1991.
Predrag Cvitanović, Group Theory - Birdtracks, Lie's and Exceptional Groups, Princeton University Press, 2008, http://birdtracks.eu/.