Bruhat order on plane posets and applications

TL;DR

This paper introduces a Bruhat order on plane posets, demonstrating its compatibility with algebraic structures and its role in defining a Hopf pairing, thus connecting combinatorial and algebraic frameworks.

Contribution

It defines a Bruhat order on plane posets and shows its compatibility with their algebraic operations, linking combinatorics with Hopf algebra theory.

Findings

Bruhat order on plane posets is isomorphic to the weak order on partitions.

Order compatibility with algebraic products is established.

A non-degenerate Hopf pairing is encoded by this order.

Abstract

A plane poset is a finite set with two partial orders, satisfying a certain incompatibility condition. The set PP of isoclasses of plane posets owns two products, and an infinitesimal Hopf algebra structure is defined on the vector space H_PP generated by PP, using the notion of biideals of plane posets. We here define a partial order on PP, making it isomorphic to the set of partitions with the weak Bruhat order. We prove that this order is compatible with both products of PP; moreover, it encodes a non degenerate Hopf pairing on the infinitesimal Hopf algebra H_PP.