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Consider a set of vectors $M$ from an inner product space $V$. The ordered set of inner products of all pairs of elements in $M$ uniquely characterizes $M$ up to isomorphism.

Suppose now that $V$ is $\mathbb{R}^n$ and $\{M_i\}$ is a collection of vector sets (that all have the same cardinality). The $M_i \in \{M_i\}$ can be represented as Gramian matrices — these are invariant to rotation but not permutation (in $n$-dimensional space).

On the other hand, the ordered set of inner products for a given $M_i$ is invariant to both rotation and permutation. So it seems to me that if one wishes to compare the elements of $\{M_i\}$ in a way that is invariant to both the ordering and collective rotation of each set of vectors in $\{M_i\}$, then using ordered sets of inner products would be the most general way to represent each $M_i$ (and this could perhaps be done in some approximate fashion by using a kernel density estimate of the inner product distribution function).

I'm sure there has been plenty of research on inner product distribution functions in the mathematical community; however, I cannot seem to find the correct terminology that is used to describe this type of research. I've searched the literature using various combinations and variants of the above terms, but I can't really find much.

What is the correct terminology for this mathematical concept?

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