irreducible <=> G-module has no non-trivial submodules.
An irreducible representation (irrep) is a representation which can not be put into block form.
G abelian => irreps 1 dimensional.
Theorem: G finite with irreps Di, ni=dim Di order(G)=h =>
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.