Summary of lecture 4
			     Character theory
			---------------------------
Throughout G denotes a finite group and V a complex linear rep of G.
	
Existence of a G-invariant hermitian inner product:
	Given any hermitian inner product on V,  we can average it over G
	to obtain a G-invariant hermitian inner product on V.

Corollary:  Any complex linear representation of G is semisimple.
	Proof:  Given a G-submodule W,   its orthogonal with respect to a
	G-invariant inner product on V is a G-complement.
	
From now on we assume that V is a finite dimensional complex linear rep of G.

Consider the set of all complex number valued functions on G.
This forms a complex vector space by pointwise addition and scalar multiplication.
We turn it into an (Hermitian) inner product space by defining an inner product as
follows:
< f , h > = average over G of the the product of the conjugate of f(g) and  h(g)

Now consider the subspace CF of all (complex valued) class functions on G. 
(A class function is a function that is constant on every conjugacy class.)
The space CF with the restricted inner product becomes a finite dimensional
complex Hilbert space.  Its dimension equals the number of conjugacy classes in G.

The main theorem of this section (see below) gives an orthonormal basis for CF.

Definition of characters:   Let V be a represenation of G and r: G --> GL(V)
the group homomorphism defining the representation.    Define the character
of V to be complex valued function on G given by g |--> Trace (r(g)).
Note that this is a class function.

Properties of characters:  character of a direct sum = sum of the characters
character of a tensor product = product of characters
character of the dual (or contragredient) = conjugate of the character
Hom(V,W) = V* tensor W
character of Hom(V,W) = conjugate char(V) . char(W)

Theorem:  The characters of the irreducible representations of G form an
orthonormal basis for the Hilbert space of class functions on G.

Before we prove the theorem,  let us note some of its consequences.

Corollary: The multiplicity of an irrep S in a representation V is given by
<char(S), char(V)>.
Proof:  V being completely reducible,  we have V = S^n direct sum ...
(where S denotes a simple module and n its multiplicity in V)
It follows that char(V) = n char(S) + ....  The other factors being orthogonal
to char(S),   the number n is determined as <char(S), char(V)>.   QED

Corollary:  Representations are determined by the their characters.
Proof:  Let n be the multiplicity of an irrep S in a representation V.
Since V is completely reducible,  these multiplicities, as S varies
over all irreps, determine V.  But n is determined as <char(S),char(V)>,
by the previous corollary.  QED

Corollary:  Every irrep occurs in the (left) regular representation
as many times as its dimension.
Proof:  The character of the regular representation takes value |G| on the
identity element of the group and is otherwise zero.    Thus
<char(S),char(reg)> = dim(S), where char(reg) denotes the character of
the regular representation.   Now apply the first corollary.  QED


Proof of the theorem: 
Denote by char(V) the character of the representation V.   We observe
that <char(V), char(W)> is the average of the character of Hom(V,W).  
Since trace is a linear function,  it follows that <char(V), char(W)> is the
trace of the average over g in G of r(g) where r: G --> GL(Hom(V,W)) is the
group homomorphism defining the representation of G on Hom(V,W).
But this average is just the G-projection to the trivial-isotpyic component of
Hom(V,W).  

When V and W are irreducible,  this isotypic component is either 1-dimensional
(in case V and W are isomorphic) or 0-dimensional (in case V is not isomorphic
to W).   Since the trace of a projection equals the dimension of the image of
the projection,  the orthonormality of char(V), as V runs over irreps, follows.

To show that the char(V), as V runs over irreps, span the space of class
functions,   we let c be a class function that is orthogonal to char(V) for
all irreps V.   We will show that c is identically 0.

Let C denote the element of the group ring that is the sum over g in G
of cbar(g)g where cbar(g) denotes the conjugate of c(g).  Since c is a class
function, so is cbar,  and it follows that C belongs to the centre of the
group ring.    Thus, by Schur, it acts like a scalar on any irrep of G.  Let V 
be an irrep.   We can determine the scalar by which C acts on V by taking the
trace of C on V and dividing by dim(V).   But this trace is precisely
<c,char(V)>,  which we have assumed to be zero.    Thus C kills every irrep.
By the complete reducibility of representations,  it follows that C kills every
representation.  In particular it kills the regular representation.  But C
acting on 1 of the group ring gives back C.   It follows that C=0 and that 
c is identically zero.