Vertex maps between simplices, cubes, and crosspolytopes

TL;DR

This paper investigates the vertices of hom-polytopes between various high-dimensional convex polytopes, revealing patterns and solving specific cases that contribute to understanding a conjectural homological theory of convex polytopes.

Contribution

It describes the vertex sets of hom-polytopes between boxes, diamonds, simplices, and cubes, advancing the systematic understanding of these geometric mappings.

Findings

Vertex sets of Hom(box_m,simplex_n), Hom(diamond_m,simplex_n), and Hom(diamond_m,diamond_n) are characterized.

Patterns in vertex structures suggest connections to homological algebra.

Open problems remain for simple source and target polytopes.

Abstract

We study the vertices of the polytopes of all affine maps (a.k.a. hom-polytopes) between higher dimensional simplices, cubes, and crosspolytopes. Systematic study of general hom-polytopes was initiated in [3]. The study of such vertices is the classical aspect of a conjectural homological theory of convex polytopes. One quickly runs into open problems even for simple source and target polytopes. The vertices of Hom(simplex_m,-) and Hom(-,cube_n) are easily understood. In this work we describe the vertex sets of Hom(box_m,simplex_n), Hom(diamond_m,simplex_n), and Hom(diamond_m,diamond_n). The emergent pattern in our arguments is reminiscent of diagram chasing in homological algebra.