Math 5114
Monday, February 4
Third Homework
Due 10:10 a.m., Monday February 11
1. Let : G GL(d , C) be a nite dimensional irreducible representation of the group
G, and let z Z(G) (the center of G). Prove that (z) = Id (where Id denotes the
d d identity m
Math 5114
Wednesday, March 20
Seventh Homework
Due 10:10 a.m., Wednesday March 27
1. Let G be a compact group, and let A = cfw_i j : G C denote all matrix functions
for all nite dimensional unitary representations : G GL(dim , C). Prove that A
is a subset
Math 5114
Wednesday, March 27
Eighth Homework
Due 10:10 a.m., Wednesday April 3
1. Let n be a nonnegative integer, let gl(n, K) denote the Lie algebra of n n matrices,
let sl(n, K) denote the subspace consisting of matrices with trace zero, and let s(n, K
Math 5114
Friday, March 8
Fifth Homework Solutions
2. Exercise 3.6 on page 45.
Let G be the group of complex matrices
ab
cd
such that |a| = 1, with the usual
topology.
(a) Is this group compact?
(b) Show that the fundamental representation of G on C2 is r
Math 5114
Wednesday, February 27
Fifth Homework
Due 10:10 a.m., Wednesday March 6
1. Exercise 1.5(a) on page 7.
cos sin
(where 0 < 2 ) and is isomorsin cos
10
phic U(1) = cfw_ei | 0 < 2 . Let =
. Then O(2) = SO(2) SO(2) .
0 1
(3 points)
SO(2) consists o
Math 5114
Wednesday, February 20
First Test Review
The test will cover chapter 2. Topics will include
1. Denition of representation, either as a homomorphism G GL(n, C) or as a homomorphism G GL(V ).
2. A representation : G GL(V ) is irreducible if and on
Math 5114
Friday, February 15
Sample First Test no. 2. Answer All Problems.
Please Give Explanations For Your Answers.
1. Let G be a nite group and let : G GL(n, C) be a representation of G with character . If (g) = 0 for all g G \ 1, prove that |G| divid
Math 5114
Wednesday, March 6
Sixth Homework
Due 10:10 a.m., Wednesday March 20
1. Exercise 3.3 on page 44.
(3 points)
2. Exercise 3.4 on page 44.
An operator P on a Hilbert space V is positive means (Pv | v) 0 for all v V . In
the case P = P , it has a un
Math 5114
Wednesday, January 23
Policy Sheet
Course
Prereq
Instructor
Book
Ofce
Telephone
E-mail
Room
Math 5114, Specialized Topics in Algebra CRN (index number) 14528
Math 4124 (Introduction to Abstract Algebra)
Peter A. Linnell
Groups and Symmetries by
Math 5114
Wednesday, February 13
Sample First Test no. 1. Answer All Problems.
Please Give Explanations For Your Answers.
1. Let : G GL(d , C) be a representation of the nite group G, and let denote the
character of . Suppose g G and (g) = d . Prove that
Math 5114
Wednesday, February 6
Second Homework Solutions
1. Let h G. Then (h)v = gG (h) (g)u = gG (hg) = gG (g) = v. It follows
that (h)Cv = Cv and hence Cv is an invariant subspace under .
If v = 0, then Cv is a nonzero invariant subspace of dimension 1
Math 5114
Wednesday, January 23
First Homework
Due 10:10 a.m., Monday January 28
1. Let cfw_e1 , e2 be the standard basis of C2 , and let : C2 C2 be a C-linear transformation satisfying (e1 + ie2 ) = 2e1 and (e1 ie2 ) = 2e2 . What is the matrix of
with
Math 5114
Monday, January 28
Second Homework
Due 10:10 a.m., Monday February 4
1. Let (E , ) be a representation of the nite group G (where E is a C-vector space, not
necessarily nite dimensional), let u E , and let v = gG (g)u. Prove that Cv is
an invari
Math 5114
Monday, February 11
Fourth Homework
Due 10:10 a.m., Wednesday February 20
1. Exercise 2.14 on page 30.
(5 points)
2. Compute the character table of the symmetric group S5 . Use this to compute the
character table of the alternating group A5 . Yo
Math 5114
Friday, March 29
Seventh Homework Solutions
1. First A is closed under multiplication. Let : G GL(d , C) and : G GL(e, C)
be unitary representations. Then the unitary representation : G GL(d + e, C)
has ( )(i, p),( j,q) = i j pq , which proves t