PROBLEM SET #4
TOM GARRITY, PCMI SUMMER SCHOOL 2015
1) a) Find all the conjugacy classes of D8 (the group of symmetries of the square).
b) Write out the character table for D8 (use the 5 representations found yesterday).
2) a) Find all the conjugacy classes of S4 .
b) Write down the Young diagrams corresponding to n = 4.
c) Write out the character table for S4 .
x
∈ R2 . Show that the multipliy
cation (a + ib)(x + iy) can be captured by the matrix operation
r cos θ −r sin θ
x
,
r sin θ r cos θ
y
3) Let x + iy ∈ C, and consider it as a vector
for
r=
p
b
a2 + b2 and tan θ = .
a
4) Prove that (a + ib)n = 1 implies (a, b) lies on the unit circle.
1
= a − bi. Observe how this is used to
5) Suppose a2 + b2 = 1. Show that a+ib
immediately find the inverse of a + ib on the unit circle.
6) Master the proof of Schur’s Lemma: if (ρ1 , V ) and (ρ2 , W ) are irreducible
representations of a group G, and if φ : V → W is a linear map such that
ρ2 (g) · (φ(v)) = φ(ρ1 (g) · v)
for all g ∈ G, v ∈ V , then φ is either the zero map or an isomorphism. (We call
such a map an intertwining operator).
1
2
TOM GARRITY, PCMI SUMMER SCHOOL 2015
7) What can you conclude from the proof of Schur’s Lemma if W is not required
to be irreducible?
8) Show that, in the case of Schur’s Lemma where V = W (and V is irreducible),
that φ must act as a scalar. That is, φ(v) = λv for any v ∈ V and a fixed scalar λ.
9) Prove this corollary to Schur’s Lemma: if (ρ1 , V ) and (ρ2 , W ) are two irreducible
representations of G, and if φ : V → W is a nonzero intertwining operator of ρ1 , ρ2 ,
then the two representations must be equivalent.
10) Suppose you are given an orthonormal basis {v1 , v2 , ..., vn } of a vector space V .
Then, we can write
n
X
αi vi .
v=
i=1
Show that, for each i, αi = hv, vi i. You may want to start by looking at the
standard orthonormal basis (1, 0, 0, 0, ...), (0, 1, 0, 0, ...), etc.