I am looking for literature on a specific topic, but I do not know the terminology. So I have a hard time finding anything relevant. Maybe someone can hint me to the right keywords here.
Tl;dr. Where to read about "symmetric point arrangements", how to construct and classify them, and how to deform them without loosing symmetry.
I am interested in (let me call them) symmetric point arrangements. By that I mean a finite set of points $p_i\in\Bbb R^d,i\in\{1,...,n\}$, that has certain prescribed symmetries. The symmetries are given by a permutation group $\Gamma\subseteq\mathrm{Sym}(n)$. For example, if there is a permutatation $\phi\in \Gamma$ that exchanges $1$ and $2$, then I require the existence of an orthogonal transformation of $\Bbb R^d$ that also exchanges $p_1$ and $p_2$.
I usually define this in the language of representation theory: a point arrangment is symmetric w.r.t. $\Gamma$, if there is an (orthogonal) representation $T\colon\Gamma\to\mathrm O(\Bbb R^d)$ with
$$T_\phi p_i = p_{\phi(i)},\quad\text{for all $\phi\in\Gamma$ and $i\in\{1,...,n\}$}.$$
I am now interested in questions like:
- how to study such arrangements for their metric properties, in particular, only up to orthogonal transformations (does there exists a convenient language already)?
- how to construct and cassify such arrangements?
- when and how can one such arrangement be deformed in another one without loosing symmetry in the process?
- what connections are there between group theoretic properties of $\Gamma$ and metric properties of the arrangement?
- ...
Until now, I used the following search terms, each of which either has not brought up anything, or is different in its meaning to what I am doing:
- symmetric point sets/arrangements/configurations/constellations/families/clouds (essentially no relevant results).
- orbit polytopes (basically the orbit of a point under a matrix group. Biggest difference is that I do not start with a given matrix group, but with a permutation group. Also, questions about deformation seem to be asked less here).
- symmetric polytopes in general, since a polytope is not much more than the set of its vertices (but not all point arrangements are the vertex set of a polytope and, again, deformation is studied less here).
- configuration spaces (seems to be relevant for the topological questions about deformation, but makes the assumption that the points are distinct, which is not relevant for me. Also not necessarily about symmetric configurations).
- geometric representation theory (seems to be something completely different) or "representation theory of point sets" (no relevant results).
- geometric group theory (also something completely different).
- orbifolds (I am not completely certain whether and how this is related to my questions).
- symmetry-forced rigidity of bar-joint frameworks (these use underlying graphs, and have distance constraint that I do not require. I could use the edge-less graph of course, but the question here is mostly for infinitesimal rigidity, and less for what can be brought into what other form by continuous deformation).