Geometric realizations of the $s$-weak order and its lattice quotients

TL;DR

This work generalizes the weak order on permutations to the s-weak order on s-trees via an insertion framework into s-bushes, yielding a rich polyhedral and combinatorial structure. The authors develop s-arcs, s-shards, and non-crossing s-arc diagrams to encode canonical representations, join-irreducibles, and forcing relations, proving the s-weak order is a congruence-uniform lattice and extending facial notions. They construct geometric realizations of the s-weak order and all its lattice quotients through s-foams, shard polytopes, shardoplexes, and quotientoplexes, using tropical geometry to realize and relate these objects to quotient fans and quotientsopes. The framework recovers classical objects (Tamari, Cambrian, permutree lattices) as special cases and provides a unified tropical-geometric machinery for understanding lattice quotients of the s-weak order.

Abstract

For an $n$-tuple $s$ of non-negative integers, the $s$-weak order is a lattice structure on $s$-trees, generalizing the weak order on permutations. We first describe the join irreducible elements, the canonical join representations, and the forcing order of the $s$-weak order in terms of combinatorial objects, generalizing the arcs, the non-crossing arc diagrams, and the subarc order for the weak order. We then extend the theory of shards and shard polytopes to construct geometric realizations of the $s$-weak order and all its lattice quotients as polyhedral complexes, generalizing the quotient fans and quotientopes of the weak order.

Figures (33)

• Figure 1: The $(1,2,0)$-weak order (left) and the $(1,2,0)$-foam (right).
• Figure 2: The congruence lattice of the $(1,2,0)$-weak order, where each congruence $\equiv$ is replaced by its ${\boldsymbol{s}}$-arc down set (left), its quotient foam $\quotientFoam$ (middle) and its quotientoplex $\quotientoplex$ (right).
• Figure 3: The braid fan for $n = 3$ with cones labeled by ordered set partitions and bushes (left) and for $n = 4$ with maximal cones labeled by permutations and increasing binary trees (right).
• Figure 4: Flow of the ${\boldsymbol{s}}$-insertion algorithm to compute $\mathop{\mathrm{\mathsf{B}}}\nolimits({\boldsymbol{s}}, {\boldsymbol{x}})$ for ${\boldsymbol{s}} = (1, 2, 2, 0, 2, 2, 1, 2, 1)$ and ${\boldsymbol{x}} = (5, 6, 3, 5, 4, 4, 5.5, 1.5, .25)$. We indicate each gap label $(u,\rho)$ by writing $\rho$ in red above the node $u$. The equalities or inequalities that led to this ${\boldsymbol{s}}$-tree are indicated at each step of the algorithm.
• Figure 5: The ${\boldsymbol{s}}$-foam for ${\boldsymbol{s}} = (1,2,0)$ (left) and ${\boldsymbol{s}} = (2,1,0)$ (right).

Theorems & Definitions (148)

• Theorem A
• Theorem B
• Theorem C
• Theorem D
• Corollary
• Definition 1
• Remark 2
• Remark 3
• Definition 4
• Remark 5