REPRESENTATION THEORY – EXERCISES 1
No credits will be awarded for doing the exercises, but I encourage you to do them.
(1) (a) Consider S3 and its permutation representation C[1, 2, 3]. In the lectures we used
Maschke’s theorem to show that C[1, 2, 3] = C[1 + 2 + 3] ⊕ C[2 − 1, 3 − 2]. Prove
that C[2 − 1, 3 − 2] is an irreducible S3-module.
(b) What are the other irreducible S3-modules over C?
(c) Show that, for n ≥ 2, the symmetric group Sn always has at least two nonisomorphic
1-dimensional representations.
(2) (a) Consider the cyclic group Cp = a : ap
= 1 for p a prime, and the matrix
representation Cp → Gl(2, Zp) sending aj
to the matrix
1 j
0 1
Show that the corresponding Cp module (Zp)2
has a 1-dimensional Cp-submodule,
but cannot be written as a direct sum of 1-dimensional submodules. This shows
that the conditions in Maschke’s theorem are necessary.
(b) Using a similar argument, show that the countably infinite cyclic group has a
2-dimensional module over C which is not completely reducible.
(3) Prove that any finite simple group has an irreducible faithful representation over C.
Is the converse true?
(4) Consider the quaternion group Q8, which can be presented as a, b : a4
= 1, a2
=
b2
, b−1
ab = a−1
. This has a 2-dimensional matrix representation given by
a →
i 0
0 −i
b →
0 −1
1 0
Show that the corresponding Q8-module structure on C2
is irreducible, and calculate
all of the irreducible representations of Q8.
Date: March 13, 2019.
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