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 => 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.