Tessellations of an affine apartment by affine weight polytopes

TL;DR

This paper proves that affine weight polytopes associated with a root system tessellate a finite-dimensional vector space, providing a foundation for constructing higher affine buildings with complex tessellations.

Contribution

It introduces and analyzes affine weight polytopes, demonstrating their tessellation properties and their role in mixed tessellations related to affine buildings.

Findings

Affine weight polytopes tessellate the vector space.

Established a mixed tessellation involving these polytopes.

Provides groundwork for future higher building constructions.

Abstract

Let $\A$ be a finite dimensional vector space and $\Phi$ be a finite root system in $\A$. To this data is associated an affine poly-simplicial complex. Motivated by a forthcoming construction of connectified higher buildings, we study "affine weight polytopes" associated to these data. We prove that these polytopes tesselate $\A$. We also prove a kind of "mixed" tessellation, involving the affine weight polytopes and the poly-simplical structure on $\A$.