Summary of lecture 3
			Simple and semisimple rings
			---------------------------

Simple ring (definition): 
	A ring with exactly two two-sided ideals, namely, zero and itself.

	Example of a simple f.d. algebra: 
	End_k(V) for V a finite dimensional vector space 
	In other words,  the ring of n x n matrices with entries over any field k

Mentioned without proof:  For k an algebraically closed field,
	every simple finite dimensional algebra is of the form M_n(k) for some n.
	(Proof will be outlined in an item in the 4th set of exercises.)

Semisimple ring (definition):  A ring that is semisimple as a module over itself. 
	Comment:   We should be saying, more precisely, "left semisimple"
	because it is not clear that semisimplicity as left and right modules
	are the same.   (But they are in general!  In fact, this will be obvious
	in the case of finite dimensional algebras over an algebraically closed field
	by the structure theorem for them by Wedderburn.)

Examples of semisimple finite dimensional k-algebras:
	M_n(k) (or End_k(V) for V a k-vector space of fintie dimension n)
	finite products of such rings
	(As we will see, Wedderburn's structure theorem below says that, when
	k is algebraically closed, these are the only examples)

Maschke's theorem (below) assures us of the semisimplicity of the rings of interest
to us in this school, namely, the group rings of finite groups over the complex numbers.

Averaging process:   G finite,  char k does not divide |G|
	Then, on any kG-module V,  the map (\sum g)/|G| is a G-projection onto
	the submodule of G-fixed points.   (Such a projection is unique, after
	the fact to be proved by Maschke that V is semisimple:  it is the
	projection along the sum of the non-trivial isotypics.)   

New G-module structures from old:  V*, Hom(V,W), V direct sum W, V tensor W, etc.

Maschke's theorem:  kG for G finite and char k not dividing |G| is semisimple
	Proof: Let V be a kG-module and W a submodule.  We use the averaging
	process applied in the kG-module Hom(V,W).  Choose linear
	complement U (for W in V).   Let p be the projection
	of V to W along U. The average of p is a G-projection of V to W,
	the kernel of which is therefore a submodule and a complement to V.

Structure theorem of Wedderburn:   Let k be an algebraically closed field.
	Any semisimple finite dimensional k-algebra is
	a finite product of matrix rings M_n(k).

Uniqueness of decomposition:  the factors are in one-to-one correspondence
	with isomorphism classes of simple modules.   The size of each factor
	equals the dimension of the corresponding simple module.

The proof is based on the following elementary proposition:  for a ring R with 1,
	the endomorphism ring of module maps from R to itself as a left R-module 
	is "naturally" isomorphic to the opposite of R.

Proof of Wedderburn:   Combining above proposition with the structure of the	
	endomorphism of a semisimple module,   we deduce that the opposite of
	our semisimple algebra is a product of matrix algebras.   But then the
	opposite of a product P of matrix algebras is isomorphic to P
	(by transpose on each factor).

Corollaries: 
	number of isomorphism classes of simple modules =
		dimension of the centre of the algebra
	sum of the squares of the dimensions of its simple modules =
		dimension of the algebra 

Application to the ordinary representation theory of the finite groups:
	("ordinary" means that we are working over the complex numbers)
	number of irreps for any finite group =
		number of conjugacy classes in the group
	sum of the squares of the dimensions of all the irreps =
		order of the group

Special case of the symmetric group on n letters:
	number of irreps = number of partitions of n
	sum of the squares of f_\lambda (number of SYT of shape \lambda)
	as \lambda varies over all partitions of n equals n!

RSK (or, more precisely, just RS) gives a purely combinatorial proof of 
	this last representation theoretic equality 

		Other corollaries of Wedderburn
		-------------------------------
		(these will appear as items in exercise set 4)

Left semisimple implies right semisimple for a ring
	(at least for f.d. algerbas over algebraically closed fields)

Density: let R be a k-algebra,  k algebraically closed field
	(Note R is not necessarily assumed to be semisimple)
	given S_1, S_2, ..., S_k distinct isomorphism classes of 
	finite dimesnional simple modules,  and k-linear endomorphisms
	phi_1, phi_2, ..., phi_k respectively of S_1, S_2, ..., S_k
	there exists an element r in R such that r acts as phi_j on S_j
	(Proof of this will be outlined in an item in 4th set of exercises.)