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.

source

jl@crush.caltech.edu index

orthogonal_group

dirac_spinor

reducible

character

Schur