MT4 2016 Exam Final
MT4 2016 Exam Final
MT4 2016 Exam Final
YOU ARE NOT PERMITTED TO READ THE CONTENTS OF THIS QUESTION PAPER UNTIL
INSTRUCTED TO DO SO BY AN INVIGILATOR.
Instructions:
Answer ALL questions from Section A. Answer ONLY TWO questions from Section B.
Section A carries 50 marks, each question in section B carries 25 marks.
If you answer more questions than specified, only the first answers (up to the specified
number) will be marked. Cross out any answers that you do not wish to be marked.
Only non-programmable calculators are permitted in this examination. Please state on your
answer book the name and type of machine used.
Complete all rough workings in the answer book and cross through any work that is not to be
assessed.
Important note: The academic regulations state that possession of unauthorised material at
any time when a student is under examination conditions is an assessment offence and can lead
to expulsion from QMUL.
Please check now to ensure you do not have any notes, mobile phones or unauthorised elec-
tronic devices on your person. If you have any, raise your hand and give them to an invigilator
immediately. It is also an offence to have any writing of any kind on your person, including on
your body. If you are found to have hidden unauthorised material elsewhere, including toilets and
cloakrooms it will be treated as being found in your possession. Unauthorised material found on
your mobile phone or other electronic device will be considered the same as being in possession
of paper notes. A mobile phone that causes a disruption is also an assessment offence.
Examiners:
Dr S Ramgoolam and Dr D Young
c Queen Mary University of London, 2016
Page 2 SPA6324 (2016)
Question A1
Given a finite group G and a subgroup H, define the space of cosets G/H.
[5 marks]
Question A2
Two permutations in S4 , the symmetric group of permutations of {1, 2, 3, 4}, are given, in cycle
notation, as g1 = (1, 2, 3, 4) and g2 = (1, 3, 4, 2). Show that they are in the same conjugacy class
by finding a g ∈ S4 such that g1 = gg2 g −1 . For your chosen g, compute gg2 as well as gg2 g −1 .
[6 marks]
Question A3
Show that the set of N × N anti-hermitian matrices form a Lie algebra. Your answer should
include a formula for the Lie bracket of two anti-hermitian matrices A, B, along with a proof that
the bracket satisfies the required properties.
[6 marks]
Question A4
Question A5
Show that any complex differentiable function obeys the Cauchy-Riemann equations.
[6 marks]
Question A6
For a circular contour of unit radius enclosing the origin traversed counterclockwise, calculate
the integral
I
1
dz
(z − 1/4)(z − 3)
[6 marks]
SPA6324 (2016) Page 3
Question A7
sin z
Find the Laurent expansion of the function 3 around the point z = 0. You may assume the
z
Taylor expansion of ez .
[6 marks]
Question A8
[5 marks]
Question A9
at z = 1 and z = −2.
[5 marks]
Turn over
Page 4 SPA6324 (2016)
Question B1
[5 marks]
(ii) The natural representation of the symmetric group Sn has a basis {e1 , e2 , · · · , en } and is
given by
[6 marks]
(iii) Show that the character of a permutation σ in the natural representation is given by the
number of cycles of length 1 in the cycle decomposition of the permutation, which we will denote
as F1 (σ).
[4 marks]
(iv) The natural representation decomposes into two irreducible representations, the trivial one-
dimensional representation and another of dimension n − 1. Use this fact along with character
orthogonality
X
χR (σ)χS (σ) = n! δ RS
σ∈Sn
to show that
1 X
(a) F1 (σ) = 1
n! σ
1 X
(b) (F1 (σ))2 = 2
n! σ
[5+5 marks]
SPA6324 (2016) Page 5
Question B2
The tensor product of two representations of spin j1 and j2 , Vj1 ⊗ Vj2 , can be decomposed into a
direct sum
[6 marks]
(ii) Calculate, in the general case of Vj1 ⊗ Vj2 , the sum of dimensions of the representations
on the right hand side of the above equation and show that it agrees with the dimension of the
tensor product.
[8 marks]
(iii) Show that the symmetric subspace of the tensor product V1/2 ⊗ V1/2 , invariant under permu-
tations of the two factors in the tensor product, has dimension 3.
[3 marks]
(iv) Calculate, as a function of j and n, the dimension of subspace of the n-fold tensor product
Vj⊗n , which is completely symmetric under all permutations of the n factors.
[8 marks]
Turn over
Page 6 SPA6324 (2016)
Question B3
(i) State the Cauchy integral formula giving the value of an analytic function in the interior of a
region enclosed by a contour, in terms of an integral over the contour.
[4 marks]
(ii) Use the Cauchy integral formula to show that if the complex derivative f (z) exists at a point
inside a region of analyticity, then all the higher order complex derivatives also exist.
[5 marks]
derive a formula for the coefficients an in the expansion as an integral around a circle surrounding
the point z0 .
[7 marks]
(iv) For the functions below, about the specified points, give the Laurent expansion where it ex-
ists, explaining why it fails to exist if that is the case
zez
(a) about z = 1
(z − 1)
z −2 ez
(b) about z = 0
sin(1/z)
z −2 sinh(1/z)
(c) about z = 0
e1/z
[9 marks]
SPA6324 (2016) Page 7
Question B4
[5 marks]
(ii) For f (x) analytic on the real line, define the principal value integral
Z ∞
f (x)
P dx
−∞ x
[4 marks]
[6 marks]
[6 marks]
[4 marks]
End of Paper
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