Fundamental polytope for the isometry group of an alcove
A fundamental alcove $\mathcal{A}$ is a tile in a paving of a vector space $V$ by an affine reflection group $W_{\mathrm{aff}}$. Its geometry encodes essential features of $W_{\mathrm{aff}}$, such as its affine Dynkin diagram $\widetilde{D}$ and fundamental group $\Omega$. In this article we investigate its full isometry group $\mathrm{Aut}(\mathcal{A})$. It is well known that the isometry group of a regular polyhedron is generated by hyperplane reflections on its faces. Being a simplex, an alcove $\mathcal{A}$ is the simplest of polyhedra, nevertheless it is seldom a regular one. In our first main result we show that $\mathrm{Aut}(\mathcal{A})$ is isomorphic to $\mathrm{Aut}(\widetilde{D})$. Building on this connection, we establish that $\mathrm{Aut}(\mathcal{A})$ is an abstract Coxeter group, with generators given by affine isometric involutions of the ambient space. Although these involutions are seldom reflections, our second main result leverages them to construct, by slicing the Komrakov--Premet fundamental polytope $\mathcal{K}$ for the action of $\Omega$, a family of fundamental polytopes for the action of $\mathrm{Aut}(\mathcal{A})$ on $\mathcal{A}$, whose vertices are contained in the vertices of $\mathcal{K}$ and whose faces are parametrized by the so-called balanced minuscule roots, which we introduce here. In an appendix, we discuss some related negative results on stratified centralizers and equivariant triangulations.