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

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.

Goldstein, H. Classical Mechanics, 2nd ed. Reading, MA: Addison-Wesley, 1980.Jeffreys,
H. and Jeffreys, B. S. "Isotropic Tensors." §3.03 in Methods
of Mathematical Physics, 3rd ed. Cambridge, England: Cambridge University
Press, pp. 87-89, 1988.Kearsley, E. A. and Fong, J. T.
""Linearly Independent Sets of Isotropic Cartesian Tensors of Ranks up
to Eight." J. Res. Nat. Bureau Standards79B, 49-58, 1975.Sloane,
N. J. A. Sequence A005043/M2587
in "The On-Line Encyclopedia of Integer Sequences."Smith G. F.
"On Isotropic Tensors and Rotation Tensors of Dimension and Order ." Tensor, N. S.19, 79-88, 1968.