irreducible <=> G-module has no non-trivial submodules.
An irreducible representation (irrep) is a representation which can not be put into block form.
Lemma:
G abelian => irreps 1 dimensional.
Theorem: G finite with irreps Di, ni=dim Di order(G)=h =>
* Sum(i=1,h,Dj(gi)(Dk(gi))*)=h/nj * delta(j,k)
(actually, the theorem applies element wise so if lm,np are the entries of Dj,Dk respectively => add on delta(l,n)delta(m,p)
* Sum(i,ni*ni)=h
* Sum(i=1,h,Xj(gi)(Xk(gi))*)=h* delta(j,k) (characters of irreps are orthogonal)
This can be extend to compact continuous groups.
Example: S3 has 3 inequivalent irreps dim=1,1,2
D1 D2 D3
e 1 1 2x2 corresponding to the movement of points.
c 1 1
cc 1 1
b 1 -1
bc 1 -1
bcc 1 -1
D1 equivalent to identity
D2 shows reflections
D3 shows rotations
Note: every row is column is orthogonal
Any function f(g) on G can be decomposed in terms of irreps:
f(g)=Sum(i,a,b,(AD(g))iab) where Aiab=ni/h * Sum(k=1,h,(Diab(gk)*)f(gk))
If you use the character instead, Xi(g) is a complete set of functions on the
conjugacy classes.