Lifting Cubic Realizations of Weak Orders in Types A and B

Daria Poliakova

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arXiv · 2604.20708

Lifting Cubic Realizations of Weak Orders in Types A and B

Daria Poliakova

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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.

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Daria Poliakova. 2026-04-22. Lifting Cubic Realizations of Weak Orders in Types A and B. https://arxiv.org/abs/2604.20708

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