2025/01/03 by Seco, Lucas, Garnier, Arthur, Neeb, Karl-Hermann
#20F55 #51F15 #57Q15 #Combinatorics (math.CO) #FOS: Mathematics #Group Theory (math.GR) #Primary 17B22 #Secondary 57R91
paper · doi:10.48550/arxiv.2501.01654
A fundamental alcove A is a tile in a paving of a vector space V by an affine reflection group Waff. Its geometry encodes essential features of Waff, such as its affine Dynkin diagram \widetildeD and fundamental group Ω. In this article we investigate its full isometry group Aut(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 A is the simplest of polyhedra, nevertheless it is seldom a regular one. In our first main result we show that Aut(A) is isomorphic to Aut(\widetildeD). Building on this connection, we establish that Aut(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 K for the action of Ω, a family of fundamental polytopes for the action of Aut(A) on A, whose vertices are contained in the vertices of 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.