Definition of a tensor:
LJunL^=Luu'Lvv'Ju'v'

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).

Theorem:
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
Let P=

12
3

then PTijk=1/3(Tijk+Tjik-Tkji-Tkij)

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.

Examples:
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.

source
jl@crush.caltech.edu index
lie_algebra
dual
SO