Double homotopy Cohen-Macaulayness for the poset of injective words and the classical NC-partition lattice

Myrto Kallipoliti,Martina Kubitzke

doi:10.46298/dmtcs.2935

Abstract

In this paper we study topological properties of the poset of injective words and the lattice of classical non-crossing partitions. Specifically, it is shown that after the removal of the bottom and top elements (if existent) these posets are doubly Cohen-Macaulay. This extends the well-known result that those posets are shellable. Both results rely on a new poset fiber theorem, for doubly homotopy Cohen-Macaulay posets, which can be considered as an extension of the classical poset fiber theorem for homotopy Cohen-Macaulay posets. Dans cet article, nous étudions certaines propriétés topologiques du poset des mots injectifs et du treillis des partitions non-croisées classiques. Plus précisément, nous montrons qu'après suppression des plus petit et plus grand élément (s'ils existent), ces posets sont doublement Cohen-Macaulay. C'est une extension du fait bien connu que ces deux posets sont épluchables ("shellable''). Ces deux résultats reposent sur un nouveau théorème poset-fibre pour les posets doublement homotopiquement Cohen-Macaulay, que l'on peut voir comme extension du théorème poset-fibre classique pour les posets homotopiquement Cohen-Macaulay.

Highlights

Introduction and resultsThis paper focuses on the study of the topology of two different posets – the poset of injective words on n letters and the lattice of non-crossing partitions for the symmetric group (denoted by NC(Sn)).The results we obtain for those two posets rely on a new poset fiber theorem for doubly homotopy Cohen-Macaulay posets and intervals
This theorem can be seen as an extension of the classical poset fiber theorem for homotopy Cohen-Macaulay posets by Quillen [19]
The poset of injective words as well as the one of classical non-crossing partitions have attracted the attention of a lot of different researchers and are well-studied objects

Summary

Introduction and results

This paper focuses on the study of the topology of two different posets – the poset of injective words on n letters (denoted by In) and the lattice of non-crossing partitions for the symmetric group (denoted by NC(Sn)). The poset of injective words as well as the one of classical non-crossing partitions have attracted the attention of a lot of different researchers and are well-studied objects. It was shown by Farmer [13] in 1978 that the regular CW-complex Kn, whose face poset is In, is homotopy equivalent to a wedge of spheres of top dimension. Using Theorem 1.1 we show that this poset, i.e., its order complex, is homotopy Cohen-Macaulay Theorem 1.3 The proper part of the lattice of non-crossing partitions NC(Sn) is doubly homotopy Cohen-Macaulay for n ≥ 3.

Partial orders and simplicial complexes

The poset of injective words

Non-crossing partitions

A poset fiber theorem for doubly homotopy Cohen-Macaulay posets