Presentation topics, Representation Theory, Math 9140

All students are expected to attend all presentations and to arrive on time.

Possible Topics: These are suggestions, but you can also propose other topics. Topics need to be discussed with me and approved. You should choose a topic that is not something you already know about. I can give more information about the topics and can suggest further references. You should also do some research about the topics.

Some of these may be covered in class, as the final choice of course material is still being worked out.

Sternberg's book is available here.

• The representation ring of a group, which organizes information about direct sum and tensor product of representations. The wikipedia page gives some references, and sample computations. Working out some of these computations could be part of the presentation. See also Broecker-tom Dieck, Section II.7, which defines R(G) using class functions.
• The Burnside ring of a finite group, which encodes information about the actions of the group on finite sets. Chapter 5 of Tom Dieck's online Representation theory notes covers this material.
• Representations of semidirect products of groups. 3.8 in Sternberg is one possible source, but I believe there are slicker ways to do it. See Section 8.2 of Serre's book on representation theory and/or Section 3.7 of Andy Baker's notes for brief treatments.
• The representations of the Poincare group, using the previous item. This gives Wigner's classification of particles by mass and spin. Sternberg does this in 3.9, but some parts aren't clear. Other references are given in this mathoverflow question, e.g. Varadarajan's book (Thm 9.4, 2nd ed.). This article by Straumann might be helpful. There's also a long, inconclusive discussion at the n-category cafe.
• Representations of the symmetric group S_n (Steinberg, Chapter 10). This is too much for one presentation, but it could be split into two, with one focusing on the combinatorics of Young diagrams/tableaux/tabloids and the other on the representation theory. Even then, there would only be time for highlights. Chapter 7 of the book Diaconis, 1988 is probably an even better source than Steinberg, as it is more efficient, covering the main result in four pages. Many other sources cover this material as well.
• Representations and characters of GL₂(F_q) and SL₂(F_q), e.g. Fulton and Harris, Section 5.2.
• Schur functors and their characters, e.g. Fulton and Harris, Section 6.1.
• An application of Fourier analysis to graph theory, as in Section 5.4 of Steinberg. Remark 5.4.13 contains pointers to further work, which might be good to incorporate.
• Voting and Fourier analysis on the symmetric group S_n, e.g. via the references mentioned in Example 5.5.8 of Steinberg (e.g. Diaconis, 1989).
• Probability and random walks on groups (Steinberg, Chapter 11, especially 11.4).
• The fast Fourier transform; most treatments don't have much representation theory, so this is a bit tangential. (One exception is a survey paper by D. Maslen and D. Rockmore available here, but it's fairly intricate.)
• Representations of SU(2) (e.g. Sternberg 4.3).
• Representations of compact topological groups or Lie groups, possibly covering Haar measure and the Peter-Weyl theorem (e.g. Sternberg 4.1, 4.2 and Appendix E; or J.F. Adams, Lectures on Lie Groups; or Bump, Lie Groups; or Bröcker and tom Dieck, Representations of Compact Lie Groups; etc.)
• Representations of Lie algebras and su(2) / sl(n,C). Possible sources are Sternberg 4.10 and 4.11, Humphreys, Introduction to Lie algebras, Fulton and Harris, Section 15, or others.
• Brauer's theorem (every character of a finite group is an integer linear combination of monomial characters) and its applications (Serre, Chapters 10 and 11).
• To what extent does the representation theory of a finite group G determine the group? The answer depends on what you remember about the representations. Tannaka duality provides one answer, and there are other interesting things to discuss. Ask me for references.
• Group cohomology (e.g. Carlson's book, Weibel's book on homological algebra, this expository article by Isaksen, etc.)
• Modular representation theory, i.e. representations over finite fields; maybe also the stable module category of a group (e.g. Carlson's book).
• Infinite representation type (tame or wild). E.g. Peter Webb's book A Course in Finite Group Representation theory, Chapter 11.
• Applications to particle physics (e.g. Sternberg 3.10, 3.12, 5.9 to 5.12, ...).
• Applications to crystalography (e.g. Sternberg 1.5, 1.8, 1.9, 1.10).
• Other applications (e.g Sternberg 4.5 to 4.9).
• Representations of quantum groups/Hopf algebras, or just SL_q(2). Connections between representation theory of quantum groups and knot theory.
• The Hurwitz Theorem about sums of squares, proved using representation theory. This relates to the existence of division algebras over the reals. See these notes by Keith Conrad.
• Connections between representation theory and complexity theory. For example, see Section 10.3 of Invitation to Nonlinear Algebra, Michalek and Sturmfels.

Duration: 45-50 minutes. The presentations are not long, so you will need to carefully select the appropriate amount of material to present. You should focus on the key ideas, with illustrative examples, motivation, necessary background, and history (e.g. attributions and years). You aren't expected to prove everything, but should give one or two short proofs. It should be regarded more like a seminar talk than a course lecture.

Grading: The presentations will be worth 1/3 of the overall mark in the course. They will be graded on:

• Outline and organization: Well-organized; appropriate choice of topics and amount of material; good outline, handed in on time.
• Knowledge of material. Be prepared to answer questions.
• Clarity and style of presentation: speaking clearly, looking at audience, giving clear explanations, etc.
• Blackboard use: use boards in order, don't erase what you've just written, don't stand in front of what you've written, legible, enough words written, use coloured chalk when appropriate, use the side board for things you want to leave up, etc.
• Duration: End within the correct duration and go at an appropriate pace. You might want to build some flexibility into the end of your presentation so you can adjust on the fly. And take into account that there may be questions during your talk.

Timeline:

• May 21-May 28: look over topics and read about a couple of them.
• In class on May 29 we'll decide on topics and select dates. Bring two possible choices of topic to class.
• Give me a brief outline (1 to 2 pages, point form) ≥ 2 weeks ahead of your date. Here is a sample outline. It could also be grouped into sections and could contain time estimates for each section.
• Talks will take place during the last three weeks of classes, after the break.