Consider a tensor of dimension $[d]\times[d]\times[d]$ which is symmetric with respect to every permutation of the indices. Are there any $\textbf{explicit}$ formulas for notions like determinant-like structures on this ? 


Also do these notions generalize to tensors of dimension $\overbrace{[d]\times[d]\times...\times[d]}^{n}$. 



Determinants of tensors
For a recent study of the determinant of a tensor (symmetric or not), equal to the product of eigenvalues, see [E-Determinants of Tensors](http://arxiv.org/abs/1109.0348) (2011). The history of this [hyperdeterminant](https://en.wikipedia.org/wiki/Hyperdeterminant) goes back to Cayley (1843). The classic textbook is [Discriminants, Resultants, and Multidimensional Determinants](https://books.google.nl/books/about/Discriminants_Resultants_and_Multidimens.html?id=n6ZxAKevSbsC&source=kp_cover&hl=en) by Gelfand, Kapranov, and Zelevinsky. 


Here is an [explicit expression](http://mathworld.wolfram.com/Hyperdeterminant.html) for the $2\times 2\times 2$ case, and [here](http://arxiv.org/abs/1310.3257) is the $3\times 3\times 3$ case (the [tables](http://arxiv.org/src/1112.2949v2/anc/invariant12.txt) needed to write down that explicit formula are quite unwieldy). 


