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.