Sometimes a point set in Euclidean space may have a shadow with an unexpected symmetry. The purpose here is to ask when this happens or when it doesn't happen (in some generality). This requires a few definitions.
First, suppose $V$ is a vector space and $\Omega\subset V$ is a subset. Define a "shadow" of $\Omega$ as the image $\pi(\Omega)$, where $\pi:V\rightarrow W$ is a linear transformation. In this context, I am thinking of $\pi$ as an orthogonal projection, where $W$ is a subspace of $V$, although it is not required to make this identification.
Next, suppose $V$ is a vector space and $\Omega\subset V$ is a subset. Define a "permutation symmetry" of $\Omega$ as a permutation of $\Omega$ which extends to a linear operator on $V$. For example, suppose $V=\mathbb{R}^d$ and $\Omega\subset V$ has $n$ points. Then one may regard $\Omega$ as a $d\times n$ matrix and a permutation $\sigma\in\mathrm{Perm}(\Omega)$ is a permutation symmetry if there is a $d\times d$ matrix $M$ such that $M\Omega=\Omega P_\sigma$, where $P_\sigma$ is the permutation matrix corresponding to $\sigma$.
Next, suppose $V$ is a vector space and $\Omega\subset V$ is a subset. Call a linear transformation $\pi:V\rightarrow W$ "generic" with respect to $\Omega$ if the restriction of $\pi$ to $\Omega$ is a bijection. For simplicity, regard $\Omega$ as finite so that a projection $\pi$ is generic if $|\pi(\Omega)| = | \Omega|$.
Write $\Omega_1\mapsto\Omega_2$ if $\Omega_1\subset V_1$, $\Omega_2\subset V_2$ and there is a linear transformation $\pi:V_1\rightarrow V_2$ which is generic with respect to $\Omega_1$ and such that $\pi(\Omega_1)=\Omega_2$. If $\Omega_1\mapsto\Omega_2$, then one has a natural way to identify permutation symmetries of $\Omega_2$ with permutations of $\Omega_1$. However, such a permutation may not be a permutation symmetry of $\Omega_1$. Thus, if $\Omega_1\mapsto\Omega_2$, then call a permutation symmetry $\sigma$ of $\Omega_2$ "inherited" if it is also a permutation symmetry of $\Omega_1$.
Here is an example of a non-inherited permutation symmetry of a shadow. Define a point configuration by $$\Omega_1=\left[\begin{matrix} 1 & 1 & -1 & -1 & 0 & 0 & 0 & 0 \\ 1 & -1 & 1 & -1 & 0 & 0 & 0 & 0 \\ 1 & -1 & -1 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 1 & -1 & -1 \\ 0 & 0 & 0 & 0 & 1 & -1 & 1 & -1 \\ 0 & 0 & 0 & 0 & 1 & -1 & -1 & 1 \end{matrix}\right].$$ Then $\Omega_1$ has a shadow $$\Omega_2=\left[\begin{matrix} 1 & 1 & -1 & -1 & 0 & 0 & 0 & 0 \\ 1 & -1 & 1 & -1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 1 & -1 & -1 \\ 0 & 0 & 0 & 0 & 1 & -1 & 1 & -1 \\ \end{matrix}\right].$$ The latter is obtained from the former by deleting rows 3 and 6, and one may see that this projection is generic. Notice that $\Omega_1$ consists of the vertices of two regular tetrahedra in complementary subspaces in $\mathbb{R}^6$ and $\Omega_2$ is the set of vertices of a 4-dimensional cross polytope. One may check that there are permutation symmetries of $\Omega_2$ which do not extend to permutation symmetries of $\Omega_1$.
Here are some general questions: Which objects have generic shadows with non-inherited permutation symmetries? The set $\Omega_1$ above provides an example of this. Which objects have the property that every permutation symmetry of every generic shadow is inherited? The vertex set of the simplex, for example, has this property since the permutation symmetry group of this set is the entire symmetric group. What other families of point configurations have this property? I am most interested in point sets possessing a transitive permutation symmetry group.
