This question already has an answer here:
The definition of symmetric second order tensors (e.g. linear operators, bilinear forms) corresponds essentially to their matrix representations being symmetric matrices.
Question: Is there a notion of transpose for higher order multidimensional arrays (e.g. three- or four-dimensional arrays)?
A natural generalization of symmetric tensor for orders $>2$ would be to define a symmetric higher order tensor to be one whose multidimensional array representation is "symmetric".
However, such a definition would require there to exist some notion of transpose/adjoint for higher-dimensional multidimensional arrays (i.e. $>2$).
For example, can we make a correspondence between "symmetric" three-dimensional arrays and homogeneous polynomials of degree three, the same way we have a correspondence between symmetric matrices and quadratic forms? Or between "symmetric" four-dimensional arrays and homogeneous quartic polynomials?
Wikipedia says that there is such a thing as the symmetric algebra over a vector space, which is a sub-algebra of the tensor algebra, and that the elements of this subalgebra (when the characteristic of the underlying field is zero, which is the only case that interests me) are called symmetric tensors, which do in fact correspond naturally to homogeneous polynomials, e.g. symmetric third order tensors and cubic forms, symmetric fourth order tensors and quartic forms:
The space of symmetric tensors of order r on a finite-dimensional vector space is naturally isomorphic to the dual of the space of homogeneous polynomials of degree r on V.
However, in the same way that symmetric second order tensors naturally have symmetric matrix representations, do higher order symmetric tensors also naturally map to some type of multi-dimensional array which is invariant under some type of generalized transpose operation?