The canonical complex of the weak order

TL;DR

This paper introduces the canonical complex for finite semidistributive lattices, explores its properties under lattice quotients, and provides combinatorial descriptions and algorithms related to the weak order on permutations.

Contribution

It defines the canonical complex for semidistributive lattices, describes its behavior under quotients, and offers combinatorial models and algorithms for the weak order on permutations.

Findings

The canonical complex encodes interval representations in semidistributive lattices.

The complex behaves well under lattice quotients, preserving subcomplex structures.

Explicit bijections and algorithms are provided for the weak order on permutations.

Abstract

We define and study the canonical complex of a finite semidistributive lattice $L$. It is the simplicial complex on the join or meet irreducible elements of $L$ which encodes each interval of $L$ by recording the canonical join representation of its bottom element and the canonical meet representation of its top element. This complex behaves properly with respect to lattice quotients of $L$, in the sense that the canonical complex of a quotient of $L$ is the subcomplex of the canonical complex of $L$ induced by the join or meet irreducibles of $L$ uncontracted in the quotient. We then describe combinatorially the canonical complex of the weak order on permutations in terms of semi-crossing arc bidiagrams, formed by the superimposition of two non-crossing arc diagrams of N. Reading. We provide explicit direct bijections between the semi-crossing arc bidiagrams and the weak order interval posets of G. Ch\^atel, V. Pilaud and V. Pons. Finally, we provide an algorithm to describe the Kreweras maps in any lattice quotient of the weak order in terms of semi-crossing arc bidiagrams.