Preorders on maximal chains: hyperplane arrangements, Cambrian lattices, and maximal green sequences
Mikhail Gorsky, Nicholas J. Williams

TL;DR
This paper explores the structure of preorders on maximal chains in polygonal lattices, with applications to hyperplane arrangements, Coxeter groups, Cambrian lattices, and maximal green sequences, revealing new connections and properties.
Contribution
It introduces a general framework for preorders on maximal chains in polygonal lattices, extending to hyperplane arrangements and Cambrian lattices, and relates these to algebraic structures like torsion-free classes.
Findings
Preorders on maximal chains can be descended to quotients with order-preserving maps.
Induced maps on maximal chains are always connected, even if not intervals.
Descriptions of these maps relate to orientations of root subsystems and stability conditions.
Abstract
We study preorders on (equivalence classes of) maximal chains in the general context of polygonal lattices endowed with suitably nice edge labellings. We show that, given a quotient of polygonal lattices, such edge labellings descend to the quotient, and that there is an induced order-preserving surjective map on the preordered sets of equivalence classes of maximal chains. Under a natural condition ensuring that the domain is a poset, the map is a contraction of preordered sets. We apply this to lattices of regions of simplicial hyperplane arrangements, where the preorders are partial orders, in particular to finite Coxeter arrangements. For the latter, each choice of Coxeter element gives us a different partial order on the set of equivalence classes of maximal chains; these generalise certain reoriented higher Bruhat orders in dimension two. The maps of posets of maximal chains…
Peer Reviews
No public reviews on file for this paper yet. If you reviewed it on a platform where reviews are public (OpenReview, ICLR, NeurIPS, ICML), you can paste yours below so the community can read it here.
Videos
No videos yet. Explain this paper in a talk, walkthrough, or lecture? Add one.
Taxonomy
TopicsAdvanced Combinatorial Mathematics · Advanced Algebra and Logic · Quasicrystal Structures and Properties
