§ Burnside lemma by representation theory.
Recall that burnside asks us to show that given a group
acting on a set , we have that the average
of the local fixed points is
equal to the number of orbits (global fixed points) of , .
Let us write elements of as acting on the vector space , which is
a complex vector space spanned by basis vector . Let
this representation of be called .
Now see that the right hand side is equal to
Where we have:
- is the charcter of the trivial representation
- The inner product is the -average inner product over -functions :
So, we need to show that the number of orbits is equal to the
multiplicity of the trivial representation in the current representation
, given by the inner product of their characters .
let whose orbit we wish to inspect. Build
the subspace spanned by the vector .
This is invariant under and is 1-dimensional. Hence, it corresponds
to a 1D subrepresentation for all the elements in the orbit of .
(TODO: why is it the trivial representation?)