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Permutation ensembles and acyclicity on product of simplices

SuHo Oh

TL;DR

This work addresses the extendability of boundary data for tilings of $nΔ_{n-1}$ via $S_n$-ensembles, introducing a sharp obstruction in the $n=4$ case: a $33$-cyclic pattern. The main theorem proves that an $S_4$-ensemble admits a permutation appearing in every row and column (hence enabling a complete tiling of $4Δ_3$) if and only if the ensemble avoids any $33$-cyclic pattern. The approach combines permutation-ensemble encoding with matching-ensemble techniques and case analyses to connect boundary permutations to fine mixed subdivisions through the Cayley trick, yielding a precise criterion for extendability. These results inform broader questions about extendability in higher dimensions ($nΔ_{n-1}$) and invite geometric interpretations within tropical geometry and triangulations of product of simplices.

Abstract

We propose the study of $S_n$-ensembles: $n \times n$ arrays of permutations of $[n]$ that encode the boundary data of $nΔ_{n-1}$. We characterize precisely when an $S_4$-ensemble contains a permutation appearing exactly four times, leading to a complete description of the types of boundary of $4Δ_3$ that can be extended to fine mixed subdivisions.

Permutation ensembles and acyclicity on product of simplices

TL;DR

This work addresses the extendability of boundary data for tilings of via -ensembles, introducing a sharp obstruction in the case: a -cyclic pattern. The main theorem proves that an -ensemble admits a permutation appearing in every row and column (hence enabling a complete tiling of ) if and only if the ensemble avoids any -cyclic pattern. The approach combines permutation-ensemble encoding with matching-ensemble techniques and case analyses to connect boundary permutations to fine mixed subdivisions through the Cayley trick, yielding a precise criterion for extendability. These results inform broader questions about extendability in higher dimensions () and invite geometric interpretations within tropical geometry and triangulations of product of simplices.

Abstract

We propose the study of -ensembles: arrays of permutations of that encode the boundary data of . We characterize precisely when an -ensemble contains a permutation appearing exactly four times, leading to a complete description of the types of boundary of that can be extended to fine mixed subdivisions.
Paper Structure (10 sections, 14 theorems, 15 equations, 3 figures, 14 tables)

This paper contains 10 sections, 14 theorems, 15 equations, 3 figures, 14 tables.

Table of Contents

  1. Introduction
  2. Connections to product of simplices
  3. Basics on product of simplices
  4. Matching ensembles
  5. Permutation ensembles and fine mixed subdivisions of $n\Delta_{n-1}$.
  6. Main result
  7. Definition of the 33-cyclic pattern.
  8. Generating entries
  9. Proof of the main result.
  10. Further directions

Key Result

Theorem 2.1

(Spread out simplices) zbMATH05192027 A collection of $n$ triangles in $n\Delta_2$ can be extended to a tiling if and only if it is spread out.

Figures (3)

  • Figure 1: A fine mixed subdivision of $3\Delta_2$ corresponding to \ref{['tab:s3ensemble']}.
  • Figure 2: How the triangulation relates to the mixed subdivisions of $2\Delta_2$.
  • Figure 3: Spanning trees describing the tiles of a mixed subdivision of $3\Delta_2$, and the collection of matchings obtained from them.

Theorems & Definitions (37)

  • Definition 1
  • Theorem 2.1
  • Theorem 2.2
  • Conjecture 1
  • Conjecture 2
  • Remark 1
  • Definition 2
  • Theorem 2.3: OYoo15
  • Example 1
  • Proposition 1
  • ...and 27 more