Isotropic Tensor

A tensor which has the same components in all rotated coordinate systems. All rank-0 tensors (scalars) are isotropic, but no rank-1 tensors (vectors) are. The unique rank-2 isotropic tensor is the Kronecker delta, and the unique rank-3 isotropic tensor is the permutation symbol (Goldstein 1980, p. 172).

The number of isotropic tensors of rank 0, 1, 2, ... are 1, 0, 1, 1, 3, 6, 15, 36, 91, 232, ... (OEIS A005043). These numbers are called the Motzkin sum numbers and are given by the recurrence relation

(1)

with and . Closed forms for are given by

			(2)
			(3)

The terms have the generating function

			(4)
			(5)
			(6)

Starting at rank 5, syzygies play a role in restricting the number of isotropic tensors. In particular, syzygies occur at rank 5, 7, 8, and all higher ranks.