##
A few more examples/exercises on representations of finite groups

1. Let G be a cyclic group of order N, with generator g.
Fix a primitive n-th root of unity w in **C**
(the standard choice is e^{2 Pi i/N}).
For each m in **Z**/N**Z**,
we then get a 1-dimensional (and thus necessarily irreducible)
representation of G by mapping g^{n} to w^{mn}.
These are all the irreducible representations of G (why?).
This gives the decomposition of the group algebra **C**[G]
as a direct sum of simple **C**-algebras,
which in this case are all isomorphic to **C**.
Interpret this, and the orthogonality theorems for characters
of finite groups, in terms of the ``discrete Fourier transform''
on complex-valued functions on G.
2. What are the irreducible representations of the same
(cyclic, order-N) group G over **Q**?
Over a finite field of characteristic not dividing N?

3. Fix a prime power q, and let and G
be the ax+b group over a finite field F of q elements.
Show that G has q-1 one-dimensional representations
over **C** (use the homomorphism b
from G to the multiplicative group of F).
Show also that G has q conjugacy classes.
Therefore G has just one more representation,
whose dimension must be q-1 by the sum-of-squares formula.
Find such a representation explicitly (Hint: start from
the permutation representation of G on **C**^{q}
coming from the action of G on F). Determine the character
of this extra representation, and verify that its inner product
with itself equals 1.

4. Determine the simple factors and irreducible representations of k[G]
where G is either of the two non-commutative groups of 8 elements
(viz., the dihedral and quaternion groups),
and k is either **C** or **R**.