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 $\smash{\Bbb R^d}$.

  • 1
    $\begingroup$ This is called point group in chemistry, but chemists are only interested in cases where $d\leq3$. $\endgroup$ – Bullet51 Apr 15 at 14:22
  • $\begingroup$ @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? $\endgroup$ – M. Winter Apr 15 at 14:30
  • $\begingroup$ 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. $\endgroup$ – Bullet51 Apr 15 at 14:52
  • $\begingroup$ @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). $\endgroup$ – M. Winter Apr 15 at 14:56
  • $\begingroup$ The symmetry group of the Grand antiprism could be represented as a product of two Coxeter groups, namely, $D_{10}$ and $D_{10}$. $\endgroup$ – Bullet51 Apr 15 at 15:29

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.