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

TL;DR

This paper generalizes the weak order on permutations to an $s$-weak order on $s$-trees, describing its combinatorial and geometric structures, including lattice quotients and polyhedral realizations.

Contribution

It introduces a comprehensive combinatorial and geometric framework for the $s$-weak order and its quotients, extending existing theories from permutations to $s$-trees.

Findings

Characterization of join irreducible elements and canonical join representations.

Construction of geometric realizations as polyhedral complexes.

Extension of shards and shard polytopes to 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.