Introduction to Representation Theory of Finite Groups
MATH 267

Fall 2013
Representation Theory: Homework 4 (due Mon.,
10/28/13)
1. Read FultonHarris, Section 3.1.
2. FultonHarris, Exercises 3.1, 3.2, 3.3.
3. (a) Prove that Jordans theorem is equivalent to the following: Let H be
a proper subgroup of the nite group G. Then th
Introduction to Representation Theory of Finite Groups
MATH 267

Fall 2014
Representation Theory of Finite Groups, Fall 2014: Material you should know
before the rst day of class
Professor: Benson Farb
I thought I would let people know what I expect them to know before starting class. I
will assume that everyone has a solid know
Introduction to Representation Theory of Finite Groups
MATH 267

Fall 2014
Representation Theory of Finite Groups, Fall 2014: Syllabus
Professor: Benson Farb
Prerequisites
Please see the handout for course prerequisites.
The Chalk Website
If you take this course you will need to have access to the Chalk website. If you register
Introduction to Representation Theory of Finite Groups
MATH 267

Fall 2014
Representation Theory: Homework 6, (due Mon.,
11/24/14)
1. Let : R C be a continuous map satisfying for all s, t R:
(a) (s + t) = (s)(t).
(b) (t) = 1 for all t = 2n, n Z.
Prove that there exists c C and C so that (t) = cet for all t.
2. Let Vm,n denote th
Introduction to Representation Theory of Finite Groups
MATH 267

Fall 2014
Representation Theory: Homework 8, (due Monday
12/1/14)
1. Prove that the space Vn of homogeneous degree n polynomials in C[x, y]
is isomorphic as an SU(2)representation to Symn C2 .
2. As usual, let Vn be the space of homogeneous degree n polynomials in
Introduction to Representation Theory of Finite Groups
MATH 267

Fall 2014
Representation Theory: Homework 6 (due 11/17/14)
1. The support supp(P ) of a probability distribution P : G [0, 1] is dened
as
supp(P ) := cfw_g G : P (g) = 0
Let P and Q be probability distributions on G.
(a) Prove that the convolution P Q is a probabil
Introduction to Representation Theory of Finite Groups
MATH 267

Fall 2014
Representation Theory: Homework 5 (due Mon.,
11/10/14)
1. Read FultonHarris, 3.3, 3.4.
2. FultonHarris, Exercises 3.16, 3.23, 3.26, 3.30.
3. Let H be a subgroup of G, and let C[G/H] be the standard permutation
representation on cosets.
(a) Prove that C[
Introduction to Representation Theory of Finite Groups
MATH 267

Fall 2014
Representation Theory: Homework 4 (due Mon.,
10/27/14)
1. In class we dened the kernel of a character ker(V ) := cfw_g G : V (g) =
dim V . In class we showed that the the intersection V ker(V ) over all
irreducible Grepresentations V is trivial.
(a) Prov
Introduction to Representation Theory of Finite Groups
MATH 267

Fall 2014
Representation Theory: Homework 2 (due Mon.,
10/14/13)
Reading. Read FultonHarris, Lecture 2.
Problems.
1. FultonHarris Exercise 2.4 (see the hint that I posted on the Chalk website), 2.5, 2.21, 2.25, 2.37 [Note: Hints/answers to starred problems are
gi
Introduction to Representation Theory of Finite Groups
MATH 267

Fall 2014
Representation Theory: Homework 3 (due Monday,
10/20/14)
Reading. FultonHarris, Section 3.1.
Problems.
1. FultonHarris, Exercises 3.1, 3.2, 3.3.
2. (a) Prove that Jordans theorem is equivalent to the following: Let H be
a proper subgroup of the nite gro
Introduction to Representation Theory of Finite Groups
MATH 267

Fall 2013
Representation Theory: Homework 5 (due Mon.,
11/11/13)
1. Read FultonHarris, Lecture 3.
2. FultonHarris, Exercises 3.23, 3.24, 3.26, 3.31, 3.32
3. The purpose of this exercise is to show that induction of characters is
really just a special case of indu
Introduction to Representation Theory of Finite Groups
MATH 267

Fall 2013
Representation Theory: Homework 7 (due Mon.,
11/25/13)
1. Read FultonHarris Section 5.2.
2. FultonHarris, Exercises 5.7, 5.8, 5.9, 5.11.
3. Recall that the projective line over Fq , denoted P1 (Fq ), is dened to be
the set of lines in Fq Fq . This is th
Introduction to Representation Theory of Finite Groups
MATH 267

Fall 2013
Representation Theory: Homework 3 (due Mon.,
10/21/13)
1. Read for the second time FultonHarris, Lecture 2.
2. FultonHarris, Exercises 2.2, 2.3, 2.26.
3. For fun (dont hand in  answers are at the back of FultonHarris): 2.37,
2.38, 2.39.
Character poly
Introduction to Representation Theory of Finite Groups
MATH 267

Fall 2013
Representation Theory: Homework 6 (due Mon.,
11/18/13)
1. Read FultonHarris, Lecture 4.
2. FultonHarris, Exercises 4.6, 4.14.
3. Fix n 2. Any element Sn has a cycle type, which is the collection of
cycle lengths = (1 , . . . , d ) of , where the cycle l
Introduction to Representation Theory of Finite Groups
MATH 267

Fall 2013
Representation Theory: Homework 2 (due Mon.,
10/14/13)
1. Read FultonHarris, Lecture 2.
2. FultonHarris, Exercises 2.5, 2.21, 2.23, 2.25, 2.27[Note: Hints/answers to
starred problems are given in the back of the book.]
3. How much does the trace of a ma
Introduction to Representation Theory of Finite Groups
MATH 267

Fall 2013
Representation Theory: Homework 1
(due Monday 10/07/13)
Note: All representations will be assumed over C unless otherwise specied.
1. Read FultonHarris, Chapter 1 and Appendix B (at the back of the book).
2. Let G be a nite group and let H be a proper, n
Introduction to Representation Theory of Finite Groups
MATH 267

Fall 2013
Representation Theory: Homework 8 (ocially due Wed.,
12/04/13, but you can hand it in to Sean Howe by 12:30
on Friday 12/06/13)
1. Fix a group G. For each irreducible representation : G GLn (C) we
get a function ij : G C given by taking i j(g) to be the i
Introduction to Representation Theory of Finite Groups
MATH 267

Fall 2014
BULLETIN (New Series) OF THE
AMERICAN MATHEMATICAL SOCIETY
Volume 40, Number 4, Pages 429440
S 02730979(03)009923
Article electronically published on July 17, 2003
ON A THEOREM OF JORDAN
JEANPIERRE SERRE
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Introduction to Representation Theory of Finite Groups
MATH 267

Fall 2014
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Introduction to Representation Theory of Finite Groups
MATH 267

Fall 2014
Representation Theory: Homework 1
(due Monday 10/06/14)
Reading. FultonHarris, Lecture 1 and Appendix B.1 and B.2 (at the back
of the book).
Problems. Note: All representations will be assumed over C unless otherwise
specied. Similarly, all groups G are