Subdivisions of lower Eulerian posets

TL;DR

This paper introduces a canonical bijection linking strong formal subdivisions of lower Eulerian posets to specific triples involving the poset, a rank function, and a join condition, with applications to computing the $cd$-index.

Contribution

It establishes a new bijection between strong formal subdivisions and structured triples of lower Eulerian posets, advancing the understanding of their combinatorial properties.

Findings

Bijection between strong formal subdivisions and triples of posets and functions

Application to computing the $cd$-index of Eulerian posets

Extension to $CW$-posets and connections to fan morphisms

Abstract

There is a natural notion of a subdivision of a lower Eulerian poset called a strong formal subdivision, which abstracts the notion of a polyhedral subdivision of a polytope, or a proper, surjective morphism of fans. We show that there is a canonical bijection between strong formal subdivisions and triples consisting of a lower Eulerian poset, a corresponding rank function, and a non-minimal element such that the join with any other element exists. The bijection uses the non-Hausdorff mapping cylinder construction introduced by Barmak and Minian. A corresponding bijection for $CW$-posets is given, as well as an application to computing the $cd$-index of an Eulerian poset. A companion paper explores applications to Kazhdan-Lusztig-Stanley theory.