Combinatorics Seminar 2018

Speaker: Zeying Xu (徐泽瀛), Shanghai Jiao Tong University

Date: Wed, Apr 25, 2018

Time: 13:30 - 15:00

Venue: 3323A

Title: Tropical hyperplane arrangements and zonotopal tilings


In tropical geometry, people study the tropical semiring consisting of the real numbers with operations of $\min$ and $+$. It has received considerable attention recently due to the strong relationship between several classical questions and their tropical counterparts. Tropical polytopes and tropical hyperplane arrangements have been widely studied. Zonotopes are fundamental objects in combinatorial geometry that are closely related to hyperplane arrangements, oriented matroids, and tilings. We want to study the zonotopal structure arising from tropical hyperplane arrangements. It turns out that zonotopal tropical complexes have nice connections with tree metrics and tropical lines.

For each element $v\in {\mathbb R}^n$, let $[v]$ denote the equivalence class of $v$ in the tropical projective torus $TT^{n-1}={\mathbb R}^n/ {\mathbb R}{\bf 1}$ where ${\bf 1}$ denotes the ``all-ones'' vector in ${\mathbb R}^n$. These equivalence classes will also be called the "points" of $TT^{n-1}$. The tropical hyperplane in $TT^{n-1}$ with apex $[v]$ is the set of elements $[x]\in TT^{n-1}$ for which the maximum in $\{x_1-v_1, \ldots , x_n-v_n \}$ is achieved at least twice. Each tropical hyperplane induces a subdivision of $TT^{n-1}$. Given a discrete set $V$ in $TT^{n-1}$, we have a tropical hyperplane arrangement whose apices are exactly the points in $V$. It induces a subdivision of $TT^{n-1}$ that is defined as the common refinement of the subdivisions of $TT^{n-1}$ induced by the tropical hyperplanes in that arrangement, and we denote the subcomplex of bounded faces of this subdivision by ${\mathcal C}_V$. Develin and Sturmfels call ${\mathcal C}_V$ the tropical complex generated by $V$ and observe that the union of faces in ${\mathcal C}_V$ coincides with the tropical convex set generated by $V$.

A zonotope is an affine projection of a hypercube. A zonotopal tiling is a subdivision of a polytope by zonotopes and a zonotopal filling is a subdivision of the whole Euclidean space by zonotopes.

Tropical linear spaces are those polyhedral complexes that satisfy the balance condition (also called the zero-tension condition). We call one-dimensional tropical linear spaces tropical lines. Each tropical line has the shape of an infinite tree.

Theorem 1 Let $V$ be a finite subset of $TT^{n-1}$. Then ${\mathcal C}_{-V}$ is a zonotopal tiling if and only if $V$ is on a tropical line $T$ and each ray of $T$ contains at least one point from $V$.

Theorem 2 Let $V$ be an infinite discrete subset of $TT^{n-1}$. Then ${\mathcal C}_{-V}$ is a zonotopal filling of $TT^{n-1}$ if and only if $V$ is on a tropical line $T$ and each ray of $T$ contains infinitely many points from $V$.

Suppose $V$ is a finite subset of a tropical line $T$ such that each ray of $T$ contains at least one point from $V$. For each ray $R$ of $T$, we let $v_R$ be the outmost point of $R$ that is in $V$. We call $Ext(V):= \{v_R\in V: R \text{ is a ray of } T\}$ the extremal point set of $V$. Let $conv_T(V)$ be the minimum subtree of $T$ containing $V$. Define $T_V$ to be the subdivision of $conv_T(V)$ by regarding points in $V$ as vertices (dimension zero faces). A partial edge orientation of $T_V$ is called a cyclic orientation if all sink and source vertices are in $V$ and every vertex in $V\setminus Ext(V)$ is on an outgoing arc. Define $CO(T_V)$ to be the poset on all cyclic orientation of $T_V$ ordered under inclusion.

Theorem 3 Let $V$ be a finite set in $TT^{n-1}$. If ${\mathcal C}_{-V}$ is a zonotopal tiling, then the face poset of ${\mathcal C}_{-V}$ is anti-isomorphic with $CO(T_V)$, namely there is a bijection $\alpha$ from the face poset of ${\mathcal C}_{-V}$ to $CO(T_V)$ such that $A$ is contained in $B$ if and only if $\alpha (B)$ is contained in $A$ for all faces $A$ and $B$ of ${\mathcal C}_{-V}$.

This is joint work with Yaokun Wu.