Group Theory (PHYS 507) Solution Set #6
5/23/13
1. Georgi problem 6.B
We want to nd [E , E ] given [E , E ] = NE+ , We know
|[E , E ] = E |E = |E N ,
where in the last step we have used the facts that E raises the weight by
and that roots are unique. Thu
Group Theory (PHYS 507) Solution Set #7
5/30/13
1. This problem lls in some loose ends in the classications of compact, simple
Lie algebras:
(a) First some linear algebra. Explain why the determinant of the Cartan
matrix
2(i) (j )
Aij = (j ) (j )
vanish
Group Theory (PHYS 507) Solution Set #8
6/6/13
1. Georgi problem 9.C The 3 of SU(3) has highest weight with Dynkin indices
(1, 0), so the complete weights of the representation have Dynkin weights (1, 0),
(1, 1) and (0, 1):
10
1 1
0 1
Therefore the weight
Group Theory (PHYS 507) Solution Set #5
5/16/13
1. Georgi problem 2.A
Using A2 = diag(1, 0, 1) and A3 = A it is straightforward to sum the series to
nd
eiA = 1 + A2 (cos 1) + iA sin .
2. Georgi problem 2.B
One approach is to use Georgis eq. (2.44) in whic
Group Theory (PHYS 507) Solution Set #4
5/7/13
1. As discussed in class, the conductivity tensor of a crystal, ij , transforms in the
tensor product of the vector representation with itself. (The vector representation describes how vectors transform, v =
Group Theory (PHYS 507) Solution Set #3
4/25/13
1. Practice with Young tableaux. This topic will be discussed in lectures, but
a summary appears towards the end of the rst chapter of Georgi.
(a) Find all the conjugacy classes for the permutation group of
Group Theory (PHYS 507) Solution Set #2
4/18/13
1. Consider again the group C4v (or D4 ) of the transformations that leave a square
invariant. You worked on this in the rst HW. In the present HW you construct
the irreducible representations (irreps), usin
Group Theory (PHYS 507) Solution Set #1
4/11/13
1. Which of the following describe groups, and which do not? Of the ones which
do describe groups, which are Abelian? Justify your answers.
e
a
b
c
d
e
e
a
b
c
d
a
a
d
c
e
b
b
b
c
a
d
e
c
c
e
d
b
a
Table 1
d