Fundamental polytope for the isometry group of an alcove

TL;DR

The paper characterizes the full isometry group $\mathrm{Aut}(\mathcal{A})$ of a fundamental alcove for an irreducible affine reflection group, showing it is isomorphic to the affine Dynkin diagram automorphism group $\mathrm{Aut}(\widetilde{D})$ and that $\mathrm{Aut}(\mathcal{A})$ is an abstract Coxeter group generated by affine involutions. It demonstrates a concrete link between alcove geometry and diagram automorphisms via a semidirect product $\mathrm{Aut}(\mathcal{A}) = \Omega \rtimes \mathrm{Aut}(D)$ with an isomorphism $\pi: \mathrm{Aut}(\mathcal{A}) \to \mathrm{Aut}(\widetilde{D})$, and provides an explicit inverse $\theta$. The article then constructs a fundamental polytope $\mathcal{L}$ for the action of $\mathrm{Aut}(\mathcal{A})$ on $\mathcal{A}$ by slicing the Komrakov–Premet polytope $\mathcal{K}$ with hyperplanes determined by balanced minuscule roots, yielding a vertex description controlled by these roots. Detailed realizations are given for types $A_n$, $D_n$, and $E_6$, illustrating how affine diagram automorphisms and minuscule data govern the symmetry and polyhedral structure, with discussions on related stratified centralizers in the appendix.

Abstract

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 $Ω$. 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 $Ω$, 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.

Figures (3)

• Figure 1: Group structure of $\widetilde{W}$ with permutations of roots $v_j$ in cycle notation and indices on the corresponding affine Dynkin diagrams in Table \ref{['table:thm1']}.
• Figure 2: Generators of $\widetilde{W}$ as product of generators of $\mathop{\mathrm{Aut}}\nolimits(\widetilde{D})$
• Figure :

Theorems & Definitions (80)

• Theorem A
• Theorem B
• Theorem 2.1
• proof
• Corollary 2.2
• proof
• Corollary 2.3
• proof
• Corollary 2.4
• proof