Course 3413 Group Representations
Sample Paper III
Dr Timothy Murphy
2 hour paper
Attempt 3 questions. (If you attempt more, only the best 3 will
be counted.) All questions carry the same number of marks.
Unless otherwise stated, all groups are compact (o
Course 3413 Group Representations
Sample Paper II
Dr Timothy Murphy
December 2010
Attempt 6 questions. (If you attempt more, only the best 6 will
be counted.) All questions carry the same number of marks.
Unless otherwise stated, all groups are compact (o
Course 3413 Group Representations
Sample Paper I
Dr Timothy Murphy
December 2010
Attempt 6 questions. (If you attempt more, only the best 6 will
be counted.) All questions carry the same number of marks.
Unless otherwise stated, all groups are compact (or
Course 3413 Group Representations
Sample Paper III
Dr Timothy Murphy
2 hour paper
Attempt 3 questions. (If you attempt more, only the best 3 will
be counted.) All questions carry the same number of marks.
Unless otherwise stated, all groups are compact (o
Course 3413 Group Representations
Sample Paper II
Dr Timothy Murphy
December 2010
Attempt 6 questions. (If you attempt more, only the best 6 will
be counted.) All questions carry the same number of marks.
Unless otherwise stated, all groups are compact (o
Course 424 Group Representations
Sample Exam
Dr Timothy Murphy
April 2009
Attempt 7 questions. (If you attempt more, only the best 7 will
be counted.) All questions carry the same number of marks.
In this paper representation means nite-dimensional repres
Course 3413 Group Representations
Sample Paper I
Dr Timothy Murphy
December 2010
Attempt 6 questions. (If you attempt more, only the best 6 will
be counted.) All questions carry the same number of marks.
Unless otherwise stated, all groups are compact (or
Course 424
Group Representations
Dr Timothy Murphy
GMB
20 May 2009
09:0012:00
Attempt 7 questions. (If you attempt more, only the best 7 will
be counted.) All questions carry the same number of marks.
Unless otherwise stated, all groups are compact (or ni
Course 424 Group Representations
Sample Exam
Dr Timothy Murphy
April 2009
Attempt 7 questions. (If you attempt more, only the best 7 will
be counted.) All questions carry the same number of marks.
In this paper representation means nite-dimensional repres
Course 424
Group Representations
Dr Timothy Murphy
Sample Paper
Attempt 6 questions. (If you attempt more, only the best 6 will
be counted.) All questions carry the same number of marks.
Unless otherwise stated, all groups are compact (or nite), and
all r
Chapter 1
Group Representations
Denition 1.1 A representation of a group G in a vector space V over k is
dened by a homomorphism
: G GL(V ).
The degree of the representation is the dimension of the vector space:
deg = dimk V.
Remarks:
1. Recall that GL(V
17
Exercises 1
All representations are over C, unless the contrary is stated.
In Exercises 0111 determine all 1-dimensional representations of the given group.
1 C2
2 C3
3 Cn
6 Dn
7 Q8
8 A4
11 D = r, s : s2 = 1, rsr = s
4 D2
9 An
5 D3
10 Z
Suppose G is a
Chapter 1
Compact Groups
Most innite groups, in practice, come dressed in a natural topology, with respect to which the group operations are continuous. All the familiar groups
in particular, all matrix groupsare locally compact; and this marks the
natura
Course 3413
Exam Format
Timothy Murphy
19 April 2011
The format of the exam will be rather dierent from that of the two sample papers sample-2010-1/2, following a recommendation from the external
examiner that there should be a uniform format for 1-semest
Chapter 1
Compact Groups
Most innite groups, in practice, come dressed in a natural topology, with respect to which the group operations are continuous. All the familiar groups
in particular, all matrix groupsare locally compact; and this marks the
natura
Chapter 1
Linear groups
We begin, as we shall end, with the classical groupsthose familiar groups
of matrices encountered in every branch of mathematics. At the outset, they
serve as a library of linear groups, with which to illustrate our theory. Later
w
Chapter 1
Linear groups
We begin, as we shall end, with the classical groupsthose familiar groups
of matrices encountered in every branch of mathematics. At the outset, they
serve as a library of linear groups, with which to illustrate our theory. Later
w
Chapter 1
Group Representations
Denition 1.1 A representation of a group G in a vector space V over k is
dened by a homomorphism
: G GL(V ).
The degree of the representation is the dimension of the vector space:
deg = dimk V.
Remarks:
1. Recall that GL(V
Course 424
Group Representations III
Dr Timothy Murphy
EELT 3
Tuesday, 11 May 1999
14:0016:00
Answer as many questions as you can; all carry the same number
of marks.
In this exam, Lie algebra means Lie algebra over R, and representation means nite-dimens
Course 424
Group Representations IIc
Dr Timothy Murphy
Mathematics 1.8
Monday, 19 April 1999
16:0017:30
Answer as many questions as you can; all carry the same number
of marks.
All representations are nite-dimensional over C.
1. What is meant by a measure
Course 424
Group Representations IIb
Dr Timothy Murphy
Mathematics 1.8
Friday, 16 April 1999
16:0017:30
Answer as many questions as you can; all carry the same number
of marks.
All representations are nite-dimensional over C.
1. What is meant by a measure
Course 424
Group Representations IIa
Dr Timothy Murphy
Mathematics 1.8
Friday, 9 April 1999
16:0017:30
Answer as many questions as you can; all carry the same number
of marks.
All representations are nite-dimensional over C.
1. What is meant by a measure
Course 424
Group Representations Ia
Dr Timothy Murphy
Seminar Room
Thursday, 17 June 1999
16:3018:30
Attempt 6 questions. (If you attempt more, only the best 6 will
be counted.) All questions carry the same number of marks.
Unless otherwise stated, all gr
Course 424
Group Representations II
Dr Timothy Murphy
EELT3
Tuesday, 13 April 1999
16:0017:30
Answer as many questions as you can; all carry the same number
of marks.
All representations are nite-dimensional over C.
1. What is meant by a measure on a comp
Course 424
Group Representations I
Dr Timothy Murphy
Exam Hall
Monday, 13 January 1997
14:0016:00
Attempt 6 questions. (If you attempt more, only the best 6 will
be counted.) All questions carry the same number of marks.
Unless otherwise stated, all group
Course 424
Group Representations III
Dr Timothy Murphy
School of Mathematics Thursday, 8 May 1997 14:0016:00
Answer as many questions as you can; all carry the same number
of marks.
Unless otherwise stated, all representations are nite-dimensional
over C.
Course 424
Group Representations II
Dr Timothy Murphy
Exam Hall
Tuesday, 4 April 1997
15:1516:45
Answer as many questions as you can; all carry the same number
of marks.
All representations are nite-dimensional over C.
1. What is meant by an invariant mea
Course 424
Group Representations III
Dr Timothy Murphy
Arts Block 3051
Thursday, 17 June 1993
14:0016:00
Answer as many questions as you can1 ; all carry the same number
of marks.
Unless otherwise stated, all Lie algebras are over R, and all representatio
Course 424
Group Representations
Dr Timothy Murphy
G.M.B.
Friday, 22 May 1995
14:0016:00
Answer as many questions as you can; all carry the same number
of marks.
Unless otherwise stated, all representations are nite-dimensional
over C.
1. Dene a group rep