Lifting Cubic Realizations of Weak Orders in Types A and B

TL;DR

This paper explores cubic realizations of posets with projection maps, introducing new graph structures to understand lifts in weak orders of types A and B, leading to unique coordinate representations.

Contribution

It introduces the pre-Reeb and augmented pre-Reeb graphs to analyze compatible cubic lifts in weak orders, providing combinatorial uniqueness results.

Findings

Pre-Reeb graphs are the 1-skeleta of cubes and zonotopes.

Augmented pre-Reeb graphs have total order reachability posets.

Achieves combinatorial uniqueness of compatible cubic coordinates.

Abstract

We study cubic realizations of posets compatible with projection maps, meaning that the projection is represented by deletion of the last coordinate. For cylindrical projections, we introduce the pre-Reeb graph and the augmented pre-Reeb graph, which control compatible cubic lifts and compatible order-embedding cubic lifts, respectively. We apply this construction to the deletion towers in weak order of types A and B. The pre-Reeb graphs are the 1-skeleta of, respectively, cubes and certain zonotopes. In both cases, the augmented pre-Reeb graphs have reachability posets that are total orders, yielding combinatorial uniqueness of the compatible order-embedding cubic coordinates.