Fundamental polytope for the isometry group of an alcove

Seco L, Garnier A, Neeb KH (2025)

Publication Type: Journal article

Publication year: 2025

Journal

Journal of Algebra Elsevier

Book Volume: 683

Pages Range: 633-671

DOI: 10.1016/j.jalgebra.2025.06.035

Abstract

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 D˜ 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(D˜). 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.

Authors with CRIS profile

Karl Hermann Neeb Lehrstuhl für Mathematik (Lie-Gruppen und Darstellungstheorie)

Involved external institutions

Universidade de Brasília BR Brazil (BR) Laboratoire Amiénois de Mathématique Fondamentale et Appliquée (LAMFA) FR France (FR)

How to cite

APA:

Seco, L., Garnier, A., & Neeb, K.H. (2025). Fundamental polytope for the isometry group of an alcove. Journal of Algebra, 683, 633-671. https://doi.org/10.1016/j.jalgebra.2025.06.035

MLA:

Seco, Lucas, Arthur Garnier, and Karl Hermann Neeb. "Fundamental polytope for the isometry group of an alcove." Journal of Algebra 683 (2025): 633-671.

BibTeX: Download