Definition of a tensor:
To construct a tensor, start with irreps of GL(m,C). construct the kronecker product of n irreps. This is an n-rank tensor defined as T(12...n).
Let P be a projection operator, D(g) a representation of a group element. PD(g)T=D(g)PT
Operation of P on Tijk
A general n rank tensor of GL(m,C) has n boxes and m rows.
A general n-rank tensor of SU(m) has n boxes and m-1 rows.
A general n-rank tensor of SO(m)
m odd => n boxes in (m-1)/2 rows with traces removed. m even =>
<m/2 rows => describe irreps with traces removed. m/2 rows => gives reps reducible to a sum of 2 irreps.
1. SO(2), Young tableau =
reduces to 2 1-d irreps.
2. SO(3), Young tableau =
=> spin=n irrep (dimension 2n+1)
3. SO(2m) Young tableau =
diminsion = 2m choose m, irrep is automatically traceless. irreps = self dual tensor, anti-self dual tensor.