Young tableaux, groups and representations
Update: Due to the Corona/COVID-19 crisis this course will take place as an online seminar - details in ILIAS.
We learn about Young tableaux based on the first few chapters of Fulton's book. Then we study Young operators and their applications in representation theory (for symmetric groups and for special unitary groups) using birdtrack diagrams.
Topics will be given to small groups of two to four students who will work on them together and prepare two to four seminar sessions.
Main reference
William Fulton
Young Tableaux
Cambridge University Press, 1997
available online (from inside the university network or via VPN) at
http://search.ebscohost.com/login.aspx?direct=true&scope=site&db=nlebk&db=nlabk&AN=570403
We read mainly Part I (Chapters 1-5) and Appendix A.
Schedule
| # |
Date | Topic | Chaps./Secs. in Fulton | Other literature | Speaker(s) |
| 1 | 15.04.20 | Bumping and sliding | 1 | all | |
| 2 | 22.04.20 | Birdtracks and permutations | doi:10.21468/SciPostPhysLectNotes.3 | SK | |
| 3 | 29.04.20 | Tableaux multiplication | 2.1 & 3 | SM,GG | |
| 4 | 06.05.20 | ||||
| 5 | 13.05.20 | Robinson-Schensted-Knuth correspondence & Schur polynomials |
2.2 & 4 | HJ,PK,JM | |
| 6 | 20.05.20 | ||||
| 7 | 27.05.20 | ||||
| 8 | 10.06.20 | Littlewood-Richardson rule | 5 | JE,KKW | |
| 9 | 17.06.20 | ||||
| 10 | 24.06.20 | Birdtracks for Young operators | doi:10.21468/SciPostPhysLectNotes.3 | SK | |
| 11 | 01.07.20 | Hermitian Young operators | doi:10.1063/1.4865177 | arXiv:1307.6147 doi:10.1063/1.4983478 | arXiv:1610.10088 |
BT,CR,GR | |
| 12 | 08.07.20 | ||||
| 13 | 15.07.20 |
How to prepare your sessions?
(1) Learning the topics of your sessions.
Study the chapters assigned to you:
- Initially, read the text once. Study the examples.
- Reread extracting all definitions, lemmas, propositions etc.
- Make sure you understand all statements.
- Study the proofs.
- Solve the exercises.
Discuss all statements, proofs, examples and exercises with the other members of your team.
If there are questions that you want to discuss with me, make an appointment via email, clearly stating the problem you want to discuss - the latter is an important part of your learning experience and must not be skipped.
You should have completed step (1) two weeks before your sessions at the latest.
(2) Arranging your sessions.
- What are the main questions / topics / theorems you will cover?
- What are good examples illustrating your main points?
- Decide which parts you want to present in which way (blackboard presentation, slides, developing s.th. together with the audience, building on the handout, using a worksheet, etc. - everything is allowed).
- Structure the material. What should go in which session? Who will speak at what time?
- Prepare a handout for your fellow students. This can be just a short summary with central results or take-home messages. Or it can be longer if you want to actively work with it in the sessions.
- Let your fellow students know how you expect them to prepare before your sessions and, in particular, between the two or three sessions taught by your team (central questions to think about, little exercises, puzzles, short text to read, a video to watch, etc.).
You should have completed step (2) about one week before your sessions.
What else?
Contribute to the other sessions. Ask questions!
More detailed syllabus
15.04.20 Bumping and sliding
Everybody studies Chapter 1 before the session, including the exercises. We explain the contents of Chapter 1 to each other, play with the bumping and sliding algorithms and discuss the solutions to the exercises.
22.04.20 Birdtracks and permutations
I give an introduction to the birdtrack notation for permutations and a brief summary of the representation theory of the symmetric group.
This is for the moment independent from the previous session and from the next few sessions.
Literature: SK, Birdtracks for SU(N), SciPost Phys. Lect. Notes 3 (2018), mainly Sec. 3, doi:10.21468/SciPostPhysLectNotes.3 | arXiv:1707.07280
28.04.20 & 05.05.20 Tableaux multiplication (Sec. 2.1, Chap. 3)
We proof the three claims from Chapter 1. To this end we have to learn about (row) words and related material.
13.05.20, 20.05.20 & 27.05.20 Robinson-Schensted-Knuth correspondence & Schur polynomials (Chap. 4 & Sec. 2.2)
We learn about the Robinson-Schensted-Knuth correspondence (RSK) and how to prove it. Some applications of RSK require the notion of Schur polynomials which will also be introduced.
10.06.20 & 17.06.20 Littlewood-Richardson rule (Chap. 5)
We state and prove the Littlewood-Richardson rule (LR). We study various guises of LR.
24.06.20 Birdtracks for Young operators
Some more birdtracks and the connection between Session 2 and the other sessions.
01.07.20, 08.07.20 & 15.07.20 Hermitian Young operators
We learn about different variants of Young operators, construct Hermitian Young operators according to two different algorithms, and practice calculations in birdtrack notation.
Main references:
- S. Keppeler and M. Sjödahl
Hermitian Young operators
J. Math. Phys. 55 (2014) 021702
doi:10.1063/1.4865177 | arXiv:1307.6147 - J. Alcock-Zeilinger and H. Weigert
Compact Hermitian Young projection operators
J. Math. Phys. 58 (2017) 051702
doi:10.1063/1.4983478 | arXiv:1610.10088
Related material:
- J. Alcock-Zeilinger and H. Weigert
Simplification rules for birdtrack operators
J. Math. Phys. 58, 051701 (2017)
doi:10.1063/1.4983477 | arXiv:1610.08801 - H. Elvang, P. Cvitanović and A. D. Kennedy
Diagrammatic Young projection operators for U(n)
J. Math. Phys. 46, 043501 (2005)
doi:10.1063/1.1832753 | arXiv:hep-th/0307186 - P. Cvitanović
Group Theory: Birdtracks, Lie’s, and Exceptional Groups
Princeton University Press (2008)
www.birdtracks.eu