Appendix B

Exercises

B.1 Chapters 1 and

Problem 1:
Consider the groupS 3 of permutations of 3 objects. This group acts on the
set of 3 elements. Consider the representation (π,C^3 ) this gives on the vector
spaceC^3 of complex valued functions on the set of 3 elements (as defined in
section 1.3.2). Choose a basis of this set of functions, and find the matricesπ(g)
for each elementg∈S^3.
Is this representation irreducible? If not, can you give its decomposition
into irreducibles, and find a basis in which the representation matrices are block
diagonal?

Problem 2:
Use a similar argument to that of theorem 2.3 forG=U(1) to classify the
irreducible differentiable representations of the groupRunder the group law of
addition. Which of these are unitary?

Problem 3:
Consider the groupSO(2) of 2 by 2 real orthogonal matrices of determinant
one. What are the complex irreducible representations of this group? (A hint:
how areSO(2) andU(1) related?)
There is an obvious representation ofSO(2) onR^2 given by matrix multipli-
cation on real 2-vectors. If you replace the real 2-vectors by complex 2-vectors,
but use the same representation matrices, you get a 2-complex dimensional rep-
resentation (this is called “complexification”). How does this decompose as a
direct sum of irreducibles?

Problem 4: