Isotropic Tensor

An isotropic tensor is a tensor whose components are unchanged by rotations of the coordinate system. The following statements concern Cartesian tensors in three-dimensional Euclidean space under proper rotations, the special orthogonal group . All rank-0 tensors (scalars) are isotropic. The only isotropic rank-1 tensor (vector) is zero, while every isotropic rank-2 tensor is a scalar multiple of the Kronecker delta and every isotropic rank-3 tensor is a scalar multiple of the permutation tensor (Kearsley and Fong 1975). The latter changes sign under reflection and is therefore not invariant under the full orthogonal group .

The dimensions of the spaces of isotropic tensors of rank 0, 1, 2, ... are 1, 0, 1, 1, 3, 6, 15, 36, 91, 232, ... (OEIS A005043). These are the Riordan numbers (Bernhart 1999), also called the Motzkin sum numbers, and are given by the recurrence relation

(1)

for , with and . Closed forms for are given by

			(2)
			(3)

The terms have the generating function

			(4)
			(5)
			(6)

Products of Kronecker deltas, with a permutation tensor for odd rank, span the isotropic tensors. Linear relations among these products, called syzygies, occur at ranks 5, 7, 8, and all higher ranks (Kearsley and Fong 1975).