If you are a student in this class you can access your grades (provided you remember your NetId and password) via this link. You should be using Netscape or Microsoft Explorer of sufficiently recent vintage.
If you have questions or want to talk to me send me email at bergv@uiuc.edu. (In particular calling my office and leaving a message is not recommended if you want a reaction any time soon.) My official office hours are announced here.
Homework assignements and other information:
lecture #
Date
Topics covered
Homework assigned
1
01/14
Introduction: Lie Groups, Lie Algebras and representations.
.
2
01/16
Irreducuble modules, completely reducible (=semi-simple)
modules, Fourier theory as the representation theory of the
circle group, group algebra.
.
3
01/18
Invariant Hermitian Forms and semi-simplicity;
Finite Groups reps are semi simple; Schur's Lemma and
applications. The regular representation .
Homework: First assignement here.
4
01/23
Finite Abelian groups: all irreducibles are
one dimensional. Example of cyclic group of order 4. General
finite group: Every irreducible occurs as a summand of the
regular reps, so there are a finite \# of irreps. Decomposition
of $R$ in isotypical components, subrings of $R$. Characters.
.
5
01/25
Characters are class functions. Characters and
sum product dual of reps. Eigenvalues of finite group actions
are roots of unity. Projection on the trivial component on a
reps. Dimension of the trivial component and characters.
.
6
01/28
$\Hom$ and interwiners ($=G$-module
morphisms). Action of $G$ on $\Hom$. Orthogonality and
independence of simple characters, inner product of class
functions. Iso class of reps determined by
character. Permutation represenations. Fixed point
lemma. Character of the regular representation. Calculus of
simple characters.
.
7
01/30
The simple characters form a basis for the
class functions .
Homework: Second assignement here.
8
02/1
Number of irreducibles is number of conjugacy
classes. Order of the group is sum of squares of dimensions of
irreps. Projections on isotypical components. The dual of an
irreducible is irreducible. Matrix coefficients.
.
9
02/4
Generalities on duality. Duality between group
algebra and Functions, convolution.
.
10
02/6
Review of classical Fourier Transform for
S^1. Fourier coefficients,
Fourier expansion/inversion, Parseval.
.
11
02/8
Linear Groups. .
.
12
02/11
.
Homework: Third assignement here.
13
02/13
.
.
14
02/15
.
.
15
02/18
.
.
16
02/20
.
.
02/21
First Hour Exam, 243 Altgeld, Finite Groups and Characters.
5:00-6:30PM.
17
02/22
No class today.
As compensation for the evening exam.
Maarten Bergvelt