I am compiling a paper in which I advertise (and use) the following notion of arrangement spaces (I made up the name, as I found no standard name in the literature).
Let $v_i\in\Bbb R^d,i\in N:=\{1,...,n\}$ be a finite family of points. Let's call this an arrangement and write $v$ when referring to all the points at once.
Definition. The arrangement space $U\subseteq\Bbb R^n$ of $v$ is the column span of the matrix $M$ in which the $v_i$'s are the rows.
The definition is motivated by a recurring idea in geometry, and despite its simplicity has some interesting and non-trivial applications. While I have seen numerous computations that could have been formulated with this concept, I have never seen anyone giving a name to this specific idea.
So I wonder where some of you came across this idea, how you have used it, and where you have already seen it under a different name. I will start to name a few:
- The arrangement space determines the arrangement up to invertible linear transformations. It is therefore of interest wherever people study linear, affine and convex dependencies between points (point configurations, oriented matroids, ...)
- The arrangement space determines all properties that are determined up to invertible linear transformations. E.g. the dimension of the span of the points, or whether the points are a linear transformation of a rational arrangement or a 01-arrangement.
- The space of possible $d$-dimensional arrangement spaces in $\Bbb R^n$ is the Grassmannian $\mathrm{Gr}(n,d)$. We therefore can parametrize arrangments via the Grassmannian, where distinct points in $\mathrm{Gr}(n,d)$ describe arrangements that are not related by linear transformations.
- Arrangement spaces give a natural definition of a form of Gale duality, which I have only seen defined in artificial and technical ways. The idea is as follows: $$v\;\;\mapsto \;\;U\;\;\mapsto\;\; U^\bot\;\;\mapsto\;\;\bar v.$$ Two arrangements are Gale duals of each other, if and only if their arrangement spaces are orthogonal complements of each other.
- The linear matroid on $v$ can equivalently be defined by just knowing its arrangement space $U$: the independent sets are $I\subseteq N$ for which $U^\bot$ has trivial intersection with $\mathrm{span}\{e_i\mid i\in I\}$. In this form it is especially easy to prove that the dual matroid is again a linear matroid, namely, it is the linear matroid of the Gale dual of $v$ (as defined above).
- Let $\Gamma\subseteq\mathrm{Sym}(N)$ be a permutation group that acts on $\Bbb R^n$ via permutation matrices. If the arrangement space of $v$ is $\Gamma$-invariant, then the arrangement expresses certain symmetries prescribed by $\Gamma$. The arrangement space can hence be used in the study of symmetric arrangements.
- If $G=(N,E)$ is a graph on $N$, and the arrangement space of $v$ is an eigenspace of $G$, then the points in $v$ are in a certain balanced configuration w.r.t. the edges of $G$. This has then be applied in graph drawings, stable arrangements from strongly regular graphs, eigenpolytopes, etc.
- ...
Note how all these are connected via their common use of the arrangement space. Hence, if you provide an answer, please explain how the arrangment space can be used to simplify what you are doing.
