REPRESENTATION THEORY – EXERCISES 2
No credits will be awarded for doing the exercises, but I encourage you to do them.
1. Characters
(1) Prove that if χ is an irreducible character and θ an irreducible 1-dimensional character,
then χθ(g) = χ(g)θ(g) is an irreducible character. (Hint: use that roots of unity x
satisfy xx = 1.)
(2) A certain group of order 12 has six conjugacy classes with representatives g1, . . . , g6
where g1 = e. It has two irreducible characters


g g1 g2 g3 g4 g5 g6
χ1 1 −i i 1 −1 −1
χ2 2 0 0 −1 −1 2


Calculate its full character table using results from the course.
(3) Consider the standard Sn-module C[1, . . . , n]. Find a formula for the value of its
associated character χ at a permutation π ∈ Sn.
(4) In this question we will calculate the character table of S4.
• Describe the five conjugacy classes of S4.
• Describe two distinct 1-dimensional characters.
• By Maschke’s theorem there exists a 3-dimensional G-module U for which
C[1, . . . , 4] = C[1 + . . . + 4] ⊕ U.
Using the previous question, or otherwise, calculate its character and prove that
it is irreducible.
• Using results from the course, conclude that there exist two other irreducible
characters, of dimensions 3 and 2 respectively.
• Find the other 3-dimensional irreducible character.
• Finally, complete the character table.
2. Maschke’s theorem via the projection formula
(1) In Maschke’s theorem, one starts with a G-module V , and linear map p : V → V .
The key step is to show that the averaged linear function
q(v) =
1
|G|
g∈G
g−1
p(gv)
is a G-module homomorphism.
On the other hand, the projection formula used in the course says that given a
G-module V the map
k(v) =
1
|G|
g∈G
g.v
is a projection with image V G
= {v ∈ V : gv = v}.
By considering the internal hom G-module [V, V ] = V ect(V, V ) with its conjugation
action described in the course, show how the key step in Maschke’s theorem, namely
that q is a G-module map, follows from the projection formula.
Date: April 23, 2019.
1
2 REPRESENTATION THEORY – EXERCISES 2
3. Tensor products, homs and the trace
(1) In the course we proved that for finite G-modules V and W there exists an isomorphism
of G-modules θV,W : [V, W] ∼= V ∗
⊗W indirectly. Describe such an isomorphism
directly.
(2) Check that the composite
[V, V ]
θV,V
// V ∗
⊗ V
ev // k
gives a (matrix free) interpretation of the trace homomorphism. (If it doesn’t find an
isomorphism so that it does!)