# A theory of (or reference for) symmetric point arrangements

I wonder where I can find something written on symmetric point arrangements (see definition below). I am interested in general references, preferably books that introduce (or papers that use) some notation/terminology which is accepted as standard. I would also be happy if you could provide helpful terminology, as my search terms have not brought up much. Terms I used: "point arrangements/constellations/configurations".

A point arrangement is a (finite) family of points $$p_i\in\Bbb R^d,i\in N:=\{1,...,n\}$$. It is said to be symmetric w.r.t. a group $$\Gamma\subseteq\mathrm{Sym}(N)$$, if there is a representation $$\smash{\rho :\Gamma\to\mathrm{O}(\Bbb R^d)}$$ with

$$\rho(\phi)p_i=p_{\phi(i)},\qquad\text{for all \phi\in \Gamma and i\in N}.$$

One prototypical question that I might ask is the following:

Given $$n\in\Bbb N$$ and a group $$\Gamma\subseteq\mathrm{Sym}(N)$$ acting as permutation group on $$N:=\{1,...,n\}$$. What are the symmetric point arrangements of $$n$$ points w.r.t. group $$\Gamma$$?

One can then go about asking questions about some geometric properties of these point arrangements depending on some properties of the group.

I have answers to some of these question, basically using (real) representation theory of finite groups. The connection to representation theory is not deep, and this is why I think someone should have come up with this before. I am currently writing a paper, and struggle with myself whether to include these "basics" when I can possible reference them somewhere. I am also worried about using a non-standard language, which then has to be corrected as soon as a referee hints me to suitable literature.

Note, that my interests are different from classifying point groups or orbit polytopes. I start with a permutation group on a fixed number of points instead of a given representation in $$\smash{\Bbb R^d}$$.

• This is called point group in chemistry, but chemists are only interested in cases where $d\leq3$. – Bullet51 Apr 15 at 14:22
• @Bullet51 Thanks for the reply. I know point groups as the finite subgroups of $\mathrm O(\Bbb R^d)$. Does it have a different meaning in chemistry? – M. Winter Apr 15 at 14:30
• Surely not. The classification of point groups follows from the classification of Coxeter groups, so one can check the structure of $Γ$ and fit it in a suitable Coxeter group. – Bullet51 Apr 15 at 14:52
• @Bullet51 Is is known that every point group is a Coxete groups, or a simple modification of one? (e.g. the symmetry group of the Grand antiprism is not a coxeter group). – M. Winter Apr 15 at 14:56
• The symmetry group of the Grand antiprism could be represented as a product of two Coxeter groups, namely, $D_{10}$ and $D_{10}$. – Bullet51 Apr 15 at 15:29