MATH 423/502: Linear Representations of Finite Groups.

Instructor: Julia Gordon.
Where and when : Tuesday 2-3:30pm, in Math Annex 1102 ; Thursday 8-9:30am, in Math building, Room 225
My office: Math 217.

e-mail: gor at math dot ubc dot ca


The textbook: Jean-Pierre Serre, "Linear representations of finite groups".

• Course description (including marking policy, prerequisites, etc.)

Some interesting links

• "The origin or representation theory" by Keith Conrad
• Keith Conrad's page contains many notes on relevant topics, see especially the notes on characters of abelian groups.
• Introduction to representation theory by P. Etingof et al. (See Chapters 3 and 4, and Chapter 1 for some background material).

Announcements:

• Monday April 25: The change to office hours time on April 26: I'll be in the office 12:30-3pm, not 11am - 1pm as announced earlier.
• Final exam or essay is due April 26. I will be at the office on the 26th from 12:30 till 3pm. You are welcome to come in person to hand it in, or to e-mail it, drop it off in my mailbox, etc. Please make sure that I get the exam and all outstanding work that you might have had an extension for by April 26. This is a firm deadline, since the course marks are due shortly afterwards, and as you know, I am not a fast marker :-).
• I am available and happy to make appointments, answer questions, etc. any time between Tuesday April 19 and April 26. I will not have any "official" office hours, so please just e-mail me if you'd like to meet.

Homework assignments

• Homework 1 , due Tuesday January 17.
• An optional problem set covering some review of advanced linear algebra.
  Helpful references:
  • The best reference for all the topics we need: new Notes by W. Casselman (up to Section 7).
  • A rather standard reference for tensor and exterior products. (This is part of Chapter 4 of the textbook "Linear Algebra and Geometry" by A.I. Kostrikin and Yu.I. Manin). The most relevant sections here are: 1.1 -- 1.6; 5.1 -- 5.9; 6.1 -- 6.6. If you'd like a paper copy, please ask.
  • "Abstract Algebra" by D. Dummit and R. Foote, 3d edition (but not the earlier editions), has a very nice short section about dual vector spaces, and a short section on tensor products of vector spaces. However, the tensor products section relies, to some extent, on the preceding chapter about modules.
    
• Homework 2 , due Tuesday February 1.
• Homework 3 , due Tuesday February 22. Correction: in Problem 6, the representation should be assumed irreducible.
• Midterm This is the midterm; due Thursday March 3. The relevant sections from Dummit and Foote, in case you need to consult this reference.
• Two optional (slightly harder) problems .
• Homework 4 ; due Tuesday March 15.
• Homework 5 (due Thursday April 7): from Serre: Exercises 7.4, 8.6, 8.11, 9.1, 9.8. Optional: 9.3.

Take-home Final Essay

(due Tuesday April 26). Possible topics:

• Representation theory of the symmetric group S_n.
• Peter-Weyl theorem.
• Heisenberg group and Stone-von Neumann theorem.
• Representations over the field of real numbers (see Serre, Chapter 13).
• Suggestions are welcome, but please discuss them with me.
• The "default option" -- this is a continuation of what we did in class for SL(2). You might find the character tables useful. (This is a copy of a page of "Representations of Finite Groups of Lie type" by Digne and Michel; we used a completely different method, and therefore different notation, to construct the representations; however, the representations should be recognizable by their dimensions).
• Weil representation for SL(2). The easiest reference for it is Bump . More specifically, see exercises 4.1.13 -- 4.1.16, and my e-mail from April 13.

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Approximate syllabus (reliable only for the dates in the past)

• January 4-6: Sections 1.1 -- 1.6, 2.1. Definition of representations; irreducibility; Maschke's Theorem; symmetric square; definition and basic properties of characters.
• January 11-13: Sections 2.2 - 2.4. Orthogonality relations for matrix coefficients and characters; decomposition of the right regular representation.
• January 18 and 25: Sections 2.5 -- 2.7, and Sections 3.3, 5.7, 5.8. More precisely, the content of these two lectures is covered in the Notes by W. Casselman , Sections 8-14. The topics covered: characters form an orthonormal basis of the space of class functions; canonical decompostion of a representation (Section 2.6); the notion of Hom_G(V, W) -- the vector space of homomorphisms between representations, and how it relates to the decompositions of V and W into a direct sum of irreducible representations; induced representation (see the notes for the definition); Frobenius reciprocity (see the notes); Mackey criterion of irreducibility of an induced representation (the case of a normal subgroup -- see Proposition 20 in the notes). Examples: symmetric and alternating groups in 4 variables; the group of upper-triangular 2x2 matrices with entries in Z/pZ, and a "1" on the diagonal (see the notes).
• Thursdays January 20, 27: NO CLASS. Make-up classes will be scheduled later.
• February 1: review of all the topics listed for January 18-25; computation of the character of an induced representation (please see Section 3.3, and Casselman's notes , pp.19-20).
• February 3: Sections 3.1, 3.2 of Serre: products of groups. Representations of abelian groups. Also, definition and discussion of semidirect product of groups (if you missed the class and don't know what it is, please see any text on abstract algebra, e.g. Dummit and Foote).
• February 8: Abelian groups: Pontryagin duality and Fourier transform. Fourier inversion formula; Plancherel formula.
• February 10: Chapter 4: compact groups, and connections with Fourier analysis. The notion of a dual representation. Statement of Peter-Weyl theorem.
• February 15-17: READING WEEK
• February 22: Trace formula for compact groups. Plancherel formula for compact groups. The reference for this lecture is Section 1 of the article "Harmonic analysis on reductive p-adic groups and Lie algebras" by R.E. Kottwitz, in Clay Mathematics Proceedings (see pp. 399-403).
• February 24: The group algebra: Sections 6.1, 6.2. If you missed the class, please make sure that you are familiar with the notions of "simple algebra", "semisimple algebra", and the statement of Wedderburn's Theorem. A good source for this is, for example, Section 1 in Chapter IV of James Milne's notes on Class Field Theory (though there are many other references covering this).
• March 1: Discussion of the problem set on linear algebra.
• March 3: Integrality properties of characters (Sections 6.2, 6.3, 6.4). Take-home midterm due March 3.
• March 8: (by Guillermo Mantilla-Soler) Degrees of irreducible representations divide the index of the centre (an application of integral properties of characters) -- Section 6.5; optional related reading: a proof of quadratic reciprocity via character theory (please see this article ).
• March 10: Sections 7.1, 7.2 -- Induction as base change grom R[H] to R[G]. (including a brief review of the notion of adjoint functors). Frobenius reciprocity revisited. Note: we almost skipped Section 7.2 since the character of induced representation was computed earlier.
• March 15: Restriction of induced representations to arbitrary subgroups; Mackey's irreducibility criterion (Sections 7.3 7.4).
• March 17: Degrees of irreducible representations divide the index of any normal abelian subgroup. The main ingredient of the proof is Proposition 24 in 8.1, which is also used in many other results in the sequel. Detailed discussion of Proposition 24. Section 8.2 skipped; Discussion of the definitions of solvable, supersolvable, and nilpotent groups (section 8.3); Section 8.4 skipped (assumed to be well-known to everyone in the class).
• March 22: Section 8.5 -- representations of supersolvable groups. Example: Heisenberg group (and a very special case of Stone - von Neumann Theorem). The Grothendieck ring R(G) (Section 9.1).
• March 24: Artin's Theorem. Chapter 9.
• March 29 and 31: Brauer's Theorem and some applications. Chapter 10. Also, sketch of sections 11.3, 11.2, and 12.3.
• April 5: Structure of the groups SL(2, F_p) and GL(2, F_p). Bruhat decomposition. Conjugacy classes. A nice discussion of conjugacy classes appears in Paul Garret's notes. (Note, however, that we did very little compared to what is done in these notes).
• April 7: Principal series representations. These are discussed in many sources; for example, for GL(2), see the book "Automorphic Forms and Representations" by D. Bump (see Theorem 4.1.1). For SL(2), see Garrett's notes , Sections 2,5,7.
• Make-up lecture: Tuesday April 12, 10-11:20am. Finishing classifications of the representations of SL(2, F_p). Weil representation. We followed Bump .
• Take-home final due Tuesday April 26. Please see possible topics above, or discuss suggestions with me.