Physics 129b, Winter 2016
Problem set 1 Solution
Thom Bohdanowicz
January 23, 2016
Problem 1. Kernel is a Normal Subgroup
Consider a general homomorphism of groups : G H. First we will prove that ker() = cfw_g
G|(g) = eH (where eH denotes the identity e
Physics 129b
5
Lecture 6
Caltech, 01/21/16
Irreducible Representations
Nick name: irreps.
5.1
Examples
(1) C2 = cfw_e, c: the cyclic group of order two has two irreducible representations.
D(e) = 1, D(c) = 1 and D(e) = 1, D(c) = 1
(1)
Comments:
a. The rst
Physics 129b
Homework 3
Due 01/28/16 by 4pm
1. Given a normal subgroup N of a group G, a representation DG/N of the quotient group G/N
can be lifted to give a representation DG of the full group G by the following denition:
DG (g) := DG/N (gN )
(1)
That i
Physics 129b
Homework 1
Due 01/14/16 by 4pm
1. Which of the following dene groups? For those that do not explain the reasons
The set of all complex n n matrices when the group operation is matrix multiplication.
The set of all complex n n matrices, with
Physics 129b
6
6.1
Lecture 9
Caltech, 02/02/16
Finite group in classical and quantum few body systems
Vibration of a triangular molecule
In the last lecture we reviewed how to reduce the problem of finding the vibrational dynamics of a
molecule to the pro
Physics 129b
2
Lecture 2
Caltech, 01/07/16
More examples with details
2.1
The cyclic group Cn
2.2
The Dihedral Group Dn
The symmetry group of rotations of a regular polygon with n undirected sides.
C
C
D
b3
b2
c
A
b1
B A
B
Figure 1: Undirected n-gon
Dn co
Physics 129b
Lecture 1
Caltech, 01/05/16
Introduction
What is a group? From Wikipedia: A group is an algebraic structure consisting of a set of elements
together with an operation that combines any two elements to form a third element.
Example: Set of ele
Physics 129b
Final Exam
Due 03/17/16 by 4pm
This is a take home exam and you are under the honor system!
Answer all four questions as much as you can. The points for each question is given in the
[ ].
Four hours should be sufficient, but you may take u
Frank Porter
Ph 129b
January 3, 2009
Chapter 1
The Basics
Def: A group is a pair (G, ), where G is a set, and is a binary operation
(multiplication) defined on G such that:
1. G is closed under :
a b G a, b G.
2. is associative:
(a b) c = a (b c) a, b, c
Physics 129b
3
3.1
Lecture 3
Caltech, 01/12/16
Basic concepts in group theory
Conjugacy class
Conjugacy, denition: two elements a and b of a group G are conjugate if there exists an element
g G such that a = gbg 1 . The element g is called the conjugating
Physics 129b
5
5.1
Lecture 8
Caltech, 01/28/16
Irreducible representations
Direct product of representations
Starting from two representations of dimensions m and n, we can form a bigger representation
of dimension m + n by taking a direct sum of them. We
Physics 129b, Winter 2016
Problem set 3 Solution
Thom Bohdanowicz & Hsiao-Yi Chen
January 24, 2016
Problem 1. Reps DG from lifted DG/N
From the property of normal group: the left coset gN is equal to the right coset N g and the product
rule of representat
Physics 129b
Homework 2
Due 01/21/16 by 4pm
1. Suppose that f is a group homomorphism from group A to group B. Show that the kernel
of f is a normal subgroup of A. (First show that it is a subgroup then show that it is a normal
subgroup.)
2. Consider the
Physics 129b
Homework 4
Due 02/04/16 by 4pm
1. Show that the character of the direct product representation equals the product of the characters
of the component representations. That is, if
D() = D() D()
(1)
() (g) = () (g) () (g)
(2)
then
2. Recall all
Physics 129b
Homework 5
Due 02/11/16 by 4pm
1. The point group C3v is, among other things, the symmetry group of the ammonia molecule N H3 ,
which forms a right pyramid on an equilateral triangle base, as shown below. AB = BC = CA. O
is the center of the
Physics 129b, Winter 2016
Problem set 4 Solution
Thom Bohdanowicz & Hsiao-Yi Chen
February 5, 2016
Problem 1. Character of the direct product representation
First, it would be more clear when we rewrite the direct product as:
()
()
()
Dab,cd = Da,c
Db,d
Physics 129b, Winter 2016
Problem set 1 Solution
January 16, 2016
Problem 1. dening a group
The denition of a group G is that if a G, b G and c G then :
(closure) ab G
(identity) e G such that ea = ae = a
(inverse) a1 G such that a1 a = e
(associative
Physics 129b
5
Lecture 7
Caltech, 01/26/16
Irreducible Representations
Nick name: irreps.
5.6
Regular representation and its decomposition into irreps
To see that the inequality
d2 |G|
(1)
is saturated, we need to consider the so-called regular representa
Physics 129b
6
Lecture 10
Caltech, 02/04/16
Finite group in classical and quantum many body systems
A many-body system refers to systems with a large number of particles. You may ask, how
large is large ? Well, in a macroscopic material, we know that the
Physics 129b
Homework 6
Due 02/18/16 by 4pm
1. The group O(2) can be represented as a linear transformation on a two dimensional real vector
space. In addition to rotation operations R() in the x y plane, it also contains the reflection
operation S which