Representation theory of finite and compact groups, and applications
MATH 843

Fall 2011
Math 843 Notes, 9/6 and 9/8
October 5, 2011
Motivation  Why study RT?
Let V be a vector space over C, and let A : V V be an operator on V . Many problems
can be interpreted as the study of an operator like this, so wed like good techniques for
studying A
Representation theory of finite and compact groups, and applications
MATH 843

Fall 2011
Math 843
Representation Theory, Fall 11 Problems Set 4
Character of permutation representation, Irr(S4 )
October 10, 2011
1. Character of permutation representations.
Denition. Let ( ; G; V ) be a nitedimensional representation a nite group G: We dene th
Representation theory of finite and compact groups, and applications
MATH 843

Fall 2011
HOMEWORK3
DONGNING WANG, FAN ZHENG
1. P ROBLEM 2:I NTERTWINING N UMBER
BETWEEN PERMUTATION PREPRESENTATIONS
1.1. (a). G acts on Hom(C(X ), C(Y ) by g G, T Hom(C(X ), C(Y )
(g T )(a) := g (T (g 1a)
a C (X ).
i.e.
g T := Y (g ) T X (g 1).
G acts on C(X Y )
Representation theory of finite and compact groups, and applications
MATH 843

Fall 2011
Math 843
Representation Theory, Fall 11 Problems Set 3
Weil representation, intertwining numbers, Irr(S3 ); Irr(A4 )
October 8, 2011
1. Construction of the Weil representation of SL2 (k ), k = Fp ; p 6= 2:
(a) Let (V; ! ) be the twodimensional symplectic
Representation theory of finite and compact groups, and applications
MATH 843

Fall 2011
Math 843
Representation Theory, Fall 11 Problems set 2
Schur lemma, complete reducibility
s
October 1, 2011
1. Schur lemma.
s
(a) Recall the denition of irreducible representation.
(b) State Schur lemma about the intertwining space Hom( ; ) for two irredu
Representation theory of finite and compact groups, and applications
MATH 843

Fall 2011
Symmetry Groups of the Platonic Solids
Silas Johnson
September 26th, 2011
In this seminar, we will determine the symmetry groups of the Platonic
solids. Note that we need only consider the tetrahedron, cube, and dodecahedron, since the octahedron and icos
Representation theory of finite and compact groups, and applications
MATH 843

Fall 2011
Math 843
Representation Theory, Fall 11 Problems set 1
Symmetries of the platonic solids
September 18, 2011
1. Action of a group on a set.
(a) Dene the notion of a group G acting on a set X:
(b) Denote the action of a group G on a set X by : Show that we
Representation theory of finite and compact groups, and applications
MATH 843

Fall 2011
Regular representation, group algebra, character theory, and
induced representation
Math 843 Notes from October 2011
In the following we shall consider only nite groups and nitedimensional representations
over the complex eld C. The algebras always conta
Representation theory of finite and compact groups, and applications
MATH 843

Fall 2011
Notes from September 29, 2011
1
Projects
By now, everyone should be thinking about their end of the semester projects. The week
of November 1st you should be ready with a report on your progress.
2
Motivation
Suppose T : V V is a transformation that we wa
Representation theory of finite and compact groups, and applications
MATH 843

Fall 2011
Representation Theory Notes
September 27, 2011
1
Note from Last Time
Last time, we wrote
C(Fp Fp \ 0) =
H
where : Fp C , (ab) = (a)(b). The dimension of C(Fp Fp \ 0) is p2 1. Each
H has dimension p + 1 and there are p 1 of them, so our dimensions match.
2
Representation theory of finite and compact groups, and applications
MATH 843

Fall 2011
Math 843
Representation Theory of Finite & Compact Groups, and Applications
Fall 2011
Instructor: Shamgar Gurevich, 317 VV.
Time and Location: TueThu 9:3010:45, Room VVB333.
Tuesday 11 & 1
12
2pm
O ce Hours:
Texts: The course notes. In addition the foll