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On an extension problem on the moment curve

Seunghun Lee, Eran Nevo

TL;DR

This work analyzes when a finite geometric simplicial complex on the moment curve $\gamma_d$ can be extended to a triangulation of conv$(A)$ without new vertices. It shows a sharp dimension threshold: extendability holds for $2\le d\le 4$, while for $d\ge 5$ there are non-extendable configurations on $n\ge d+3$ vertices, with the $d=4$ case yielding a striking algebraic application via a correspondence of Oppermann–Thomas that all maximal rigid objects in $\mathcal{O}_{A_n^{2}}$ are cluster tilting. The approach blends height-based orderings, interlacing criteria, and the lattice structure of the higher Stasheff–Tamari posets to build extensions in low dimensions and to construct explicit non-extendable examples in high dimensions. The results deepen connections between discrete geometry of cyclic polytopes and higher Auslander–Reiten theory, and they suggest both algorithmic implications and avenues for further algebraic proofs and combinatorial refinements.

Abstract

We show that for $2\le d\le 4$, every finite geometric simplicial complex $Δ$ in $\mathbb{R}^d$ with vertices on the moment curve can be extended to a triangulation $T$ of the cyclic polytope $C$ where $Δ, T$ and $C$ all have the same vertex set. Further, for $d\ge 5$ we construct for every $n\ge d+3$ complexes $Δ$ on $n$ vertices for which no such triangulations $T$ exist. Our result for $d=4$ has the following novel algebraic application, due to a correspondence by Oppermann and Thomas (JEMS, 2012): every maximal rigid object in $\mathcal{O}_{A_n^{2}}$ is cluster tilting, where $\mathcal{O}_{A_n^δ}$ denotes a higher dimensional cluster category introduced by Oppermann and Thomas for $A_n^δ$, where $A_n^δ$ denotes a higher Auslander algebra of linearly oriented type $A$.

On an extension problem on the moment curve

TL;DR

This work analyzes when a finite geometric simplicial complex on the moment curve can be extended to a triangulation of conv without new vertices. It shows a sharp dimension threshold: extendability holds for , while for there are non-extendable configurations on vertices, with the case yielding a striking algebraic application via a correspondence of Oppermann–Thomas that all maximal rigid objects in are cluster tilting. The approach blends height-based orderings, interlacing criteria, and the lattice structure of the higher Stasheff–Tamari posets to build extensions in low dimensions and to construct explicit non-extendable examples in high dimensions. The results deepen connections between discrete geometry of cyclic polytopes and higher Auslander–Reiten theory, and they suggest both algorithmic implications and avenues for further algebraic proofs and combinatorial refinements.

Abstract

We show that for , every finite geometric simplicial complex in with vertices on the moment curve can be extended to a triangulation of the cyclic polytope where and all have the same vertex set. Further, for we construct for every complexes on vertices for which no such triangulations exist. Our result for has the following novel algebraic application, due to a correspondence by Oppermann and Thomas (JEMS, 2012): every maximal rigid object in is cluster tilting, where denotes a higher dimensional cluster category introduced by Oppermann and Thomas for , where denotes a higher Auslander algebra of linearly oriented type .

Paper Structure

This paper contains 12 sections, 30 theorems, 13 equations, 9 figures, 1 table.

Table of Contents

  1. Introduction
  2. Ordering simplices and triangulations by height
  3. Ordering simplices by height
  4. Height of a triangulation and Higher Stasheff-Tamari posets
  5. Submersion set of a triangulation and lattice property
  6. Extending simplices into a triangulation
  7. Proof of Theorem \ref{['thm_main']}(i) modulo the existence of suitable triangulations
  8. Proof of Theorem \ref{['thm_level_d=2']}
  9. Proof of Theorem \ref{['thm_level_d=3']}
  10. Finding a 2-dimensional triangulation avoiding certain interlacing patterns
  11. Proof of Theorem \ref{['thm_main']}(ii)
  12. Final remarks

Key Result

Theorem 1.2

(i) For every $D\le 4$ and every finite collection $\mathcal{F}$ of pairwise non-overlapping simplices in $\mathbb{R}^D$ on $A\subseteq \gamma_D$, $\mathcal{F}$ can be extended into a triangulation $T$ of the cyclic polytope $\mathop{\mathrm{conv}}\nolimits(A)$ such that $A$ is exactly the vertex se

Figures (9)

  • Figure 1: Illustration of the mutually exclusive cases (A)-(D).
  • Figure 2: Extension of $\sigma$ into a triangulation $T(\sigma) \in S(n,2)$.
  • Figure 3: Illustrations of $\sigma=\{v_1, v_2, v_3\}$ and the intervals $J_L$, $J_R$ and $I_M\subset J_M$.
  • Figure 4: Left: case when $w_2'$ is at the same side of $w_1'$ or $w_3'$ against (Ms). In this case the triangle $t'=w_1'w_2'w_3'$ can be ignored. Middle: case when $w_2' \notin V$ against (MV). In this case $e_s$ violates (Me). Right: case against (RV). In this case $e_s$ violates (Re) unless $s_1\leq r_1$, in which case we can ignore $e_R=r_1r_2$.
  • Figure 5: Left: when neither (i) nor (ii) holds. Not to violate (LMR), we have $r_1\leq w_1$ and $w_3\leq l_2$. Then (LR) is violated. Middle: when $w_1<\min(V)$ and $\max(V)<w_3$. Then we violate (Ms) or (Me). Right: when $w_3>\max(V)$ and $\min(V)=m\leq w_1$, but (ii) is violated. Here depicted the case $l_2<w_3$ hence (LMR) is violated. Else $w_3 \leq l_2$ and then $e_s$ violates (Le).
  • ...and 4 more figures

Theorems & Definitions (62)

  • Theorem 1.2
  • Theorem 1.3: opperman_thomas
  • Corollary 1.4
  • Proposition 2.1
  • Proposition 2.2
  • Proposition 2.3
  • proof
  • Proposition 2.4
  • proof
  • Corollary 2.5
  • ...and 52 more