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