Sheet 1 - Hu

HU Berlin
Summer semester 2014
Exercises for the course
Representation Theory
Sheet 1
1. Let D2n = {α, β : αn = β 2 = 1, βαβ = α−1 } be the Dihedral group with 2n elements.
Consider the map σ : D2n −→ GL2 (C) given by
σαr (xe1 + ye2 ) = ζ r xe1 + ζ −r ye2 ,
σαr β (xe1 + ye2 ) = ζ r ye1 + ζ −r xe2
for 1 ≤ r ≤ n, where σg = σ(g) and ζ = e2π/n . Show that σ defines a representation of
D2n of degree 2 and that this representation is irreducible. Determine Ker σ.
2. Consider the C- vector space
V {(x1 , x2 , x3 ) ∈ C3 : x1 + x2 + x3 = 0}.
Show that the symmetric group S3 admits a representation ρ : S3 −→ GL(V ) defined
by
ρσ (x1 , x2 , x3 ) = {xσ−1 (1) , xσ−1 (2) , xσ−1 (3) }
for σ ∈ S3 . Show that this representation is irreducible.
3. If p is a prime number and G is a nontrivial finite p-group (that is, G is of order pn ),
show that there is a non-trivial 1-dimensional representation of G.
4. Let S3 act on the finite set X = {1, 2, 3}. Consider the complex permutation representation of S3 associated to this action with underlying vector space V = C[X] (of
dimension 3). Show that the invariant subspace
V S3 = {v ∈ V : σ · v = v, ∀σ ∈ S3 }
is 1-dimensional. Find a 2-dimensional S3 -stable vector space W ⊂ V such that V =
V S3 ⊕ W . Show that W is irreducible.
5. Let G be a finite group acting on a finite set X and let C[X] be the associated complex
permutation representation.
(i) If G is transitive (i.e. there is only one orbit), show that there is a 1-dimensional
G-stable subspace V and a G-stable complement W , such that C[X] = V ⊕ W .
(ii) Show that for every G-orbit Y in X there is a 1-dimensional G-stable subspace VY
and a WY of dimension |Y | − 1 such that
C[X] = VY1 ⊕ WY1 ⊕ · · · ⊕ VYr ⊕ WYr
where Y1 , . . . , Yr are distinct G-orbits of X.
6. If ρ : G → GL(V ) is an irreducible representation, prove that the dual representation
ρ∗ : G → GL(V ∗ ) is also irreducible.

Download Report