Can all 3 vector products $\left(\sum_i v_i u_i w_i\right)$ define linear subspace modulo permutation of coordinates?

Question

To determine a set of $n$ numbers modulo permutation, we can use permutation invariants e.g. some symmetric polynomials like $(\sum_i(x_i)^k)_{k=1..n}$.

The big question is how to generalize it to matrices: define matrix modulo permutation of coordinates $(P M P^T$) by using invariants - e.g. to solve the graph isomorphism problem from algebraic perspective. Constructing such invariants is relatively simple, however, the difficulty is showing their completeness - that they can restrict to only a single matrix modulo permutation. We can cheaply produce lots of them, the question is if they are independent enough?

Here is the simplest problem I was able to get:

There is a $d\leq n$ dimensional linear subspace of $\mathbb{R}^n$ with orthonormal basis $(v^i)_{i=1..d}$

(of fixed vector order, satisfying $\forall_{ij} \sum_{\alpha=1}^n v_{\alpha}^i v_{\alpha}^j = \delta_{ij}$ ).

Does knowing all products of triples $(s_{ijk})_{i\leq j \leq k}$: $$s_{ijk}=\sum_{\alpha=1}^n v_{\alpha}^i v_{\alpha}^j v_{\alpha}^k$$ determine $(v^i)_{i=1..d}$ modulo permutation of $n$ coordinates?

This way we get $d(d+1)(d+2)/6$ invariants (of permutation of coordinates) for $d(2n-d+1)$ values - it has a chance to be sufficient only for $d$ large enough: $d(d+1)(d+2)/6\geq d(2n-d)$. But this is not necessarily sufficient as constraints can be dependent.

Is there some minimal $d$ for given $n$ for which we can generally prove uniqueness?

How to generally attack a question of existence of large enough subset of independent conditions in a larger set of conditions?

Is there some literature about such "products of triples"?

Answer 1

So I'm not sure if you are talking about $\textsf{Permutational Code Equivalence}$ or $\textsf{Tensor Isomorphism}$.

The $\textsf{Permutational Code Equivalence}$ problem takes as input two linear codes $C_{1}$ and $C_{2}$ given be generator matrices $M_{1}$ and $M_{2}$. We ask for a permutation $\pi$ such that $C_{1}^{\pi} = C_{2}$. It is known that $\textsf{Permutational Code Equivalence}$ is $\textsf{Graph Isomorphism}$-hard.

$\textsf{Tensor Isomorphism}$ is a generalization of isomorphism testing to the multilinear algebraic setting. I'm certainly not an expert. The Grochow--Qiao paper is the reference here (https://arxiv.org/abs/1907.00309).