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The question guide says that Mathoverflow is for research level mathematics. While I do not perform research in mathematics (I study quantum chemistry), I believe this question is research-level since I cannot find an answer to it using my small knowledge of group theory to search Google Scholar. If my question is too simplistic, let me know and I'll move it elsewhere.

Consider a collection of sets $\{A_i\}$, where each $A_i$ contains $n$ vectors $\{x_1, x_2, \ldots, x_n\}_i$ and each $x_j \in \mathbb{R}^d$. I want to represent each set as a normal form that removes all rotational and permutational degrees of freedom. I would also prefer each normal form to be complete (contains no redundant information), but that's not strictly necessary.

My own attempt at this is not going too well. I start with an overcomplete representation of $A_i$ that enforces rotational invariance, namely the Gramian matrix $G_i = A_i^{T}A_i$. I then take the Cholesky decomposition $G_i = L_i L_i^{T}$, and since each $L_i$ is a matrix with $nd - \frac{1}{2}d(d-1)$ non-zero elements (assuming the vectors in $A_i$ are linearly independent), this specifies a complete representation of $A_i$ minus the $\frac{1}{2}d(d-1)$ rotational degrees of freedom.

But for permutational invariance, I have no clue. I need a symmetric function $F$ such that $$F(P_\alpha(A_i)) = F(P_\beta(A_i)) \neq F(P_\gamma(A_j))$$

for all

$$\alpha, \beta, \gamma \in [1, n!]$$ $$P_\alpha, P_\beta, P_\gamma \in S_n$$ $$i \neq j$$

where $S_n$ is the symmetric group of degree $n$.

$F$ must preserve all non-rotational degrees of freedom, much like the Cholesky decomposition does.

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