A poset fiber theorem for doubly Cohen-Macaulay posets and its applications to non-crossing partitions and injective words

TL;DR

This paper introduces a new poset fiber theorem and demonstrates that certain lattices of non-crossing partitions and injective words are doubly Cohen-Macaulay, enhancing understanding of their topological and combinatorial properties.

Contribution

It presents a novel poset fiber theorem and applies it to prove doubly Cohen-Macaulay properties of specific posets, extending prior Cohen-Macaulay results.

Findings

Non-crossing partition lattices are doubly Cohen-Macaulay after removing extremal elements.

Poset of injective words also exhibits double Cohen-Macaulayness.

The new fiber theorem is a versatile tool for topological poset analysis.

Abstract

This paper studies topological properties of the lattices of non-crossing partitions of types A and B and of the poset of injective words. Specifically, it is shown that after the removal of the bottom and top elements (if existent) these posets are doubly Cohen-Macaulay. This strengthens the well-known facts that these posets are Cohen-Macaulay. Our results rely on a new poset fiber theorem which turns out to be a useful tool to prove double (homotopy) Cohen-Macaulayness of a poset. Applications to complexes of injective words are also included.