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".
Some interesting links
- 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
have any "official" office hours, so please just e-mail me if you'd like
- Homework 1 , due Tuesday January 17.
- An optional problem set
review of advanced linear algebra.
- 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
(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
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.
Take-home Final Essay
(due Tuesday April 26).
- 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
- 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.
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 ,
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
(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
- 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
- March 10: Sections 7.1, 7.2 -- Induction as base change grom R[H] to
(including a brief review of the notion of adjoint functors). Frobenius
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.
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
"Automorphic Forms and Representations" by D. Bump
(see Theorem 4.1.1). For SL(2), see
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.