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Subdivisions of Hypersimplices: with a View Toward Finite Metric Spaces

Laura Casabella, Michael Joswig, Lars Kastner

Abstract

The secondary fan $Σ(k,n)$ is a polyhedral fan which stratifies the regular subdivisions of the hypersimplices $Δ(k,n)$. We find new infinite families of rays of $Σ(k,n)$, and we compute the fans $Σ(2,7)$ and $Σ(3,6)$. In the special case $k=2$ the fan $Σ(2,n)$ is closely related to the metric fan $\mathop{MF}(n)$, which forms a natural parameter space for the metric spaces on $n$ points. So our results yield a classification of the finite metric spaces on seven points.

Subdivisions of Hypersimplices: with a View Toward Finite Metric Spaces

Abstract

The secondary fan is a polyhedral fan which stratifies the regular subdivisions of the hypersimplices . We find new infinite families of rays of , and we compute the fans and . In the special case the fan is closely related to the metric fan , which forms a natural parameter space for the metric spaces on points. So our results yield a classification of the finite metric spaces on seven points.
Paper Structure (7 sections, 21 theorems, 23 equations, 2 figures, 3 tables)

This paper contains 7 sections, 21 theorems, 23 equations, 2 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Regular Subdivisions and Their Tight Spans
  3. Subdivisions of Hypersimplices
  4. Volumes and Eulerian Numbers
  5. Finite Metric Spaces
  6. Further Computations
  7. Conclusion

Key Result

Theorem 2

Let $\omega$ be a lifting function of a point set $A$. Then there is a coherent decomposition where $\omega_0$ is split prime, and this decomposition is unique among all coherent decompositions of $\omega$.

Figures (2)

  • Figure 1: Tight spans of the non-split coarsest subdivisions of the hypersimplex $\Delta(2,6)$. The top six are $2$-dimensional, while the bottom four are $3$-dimensional. The $3$-dimensional cases feature a unique $3$-cell, whose edges are marked red
  • Figure 2: Dressian $\mathop{\mathrm{Dr}}\nolimits(2,5)$ embedded into (the graph of) $\Sigma(2,5)$

Theorems & Definitions (44)

  • Remark 1
  • Theorem 2: Hirai Hirai:2006a; Herrmann and Joswig HerrmannJoswig:2008
  • Lemma 3
  • proof
  • Proposition 4
  • proof
  • Corollary 5
  • proof
  • Example 6
  • Theorem 7: Speyer Speyer:2008
  • ...and 34 more