Séminaire Lotharingien de Combinatoire, 91B.60 (2024), 12 pp.
Rafael S. González D'León, Alejandro H. Morales, Eva Philippe,
Daniel Tamayo Jiménez and Martha Yip
Realizing the s-Permutahedron via Flow Polytopes
Abstract.
In 2019, Ceballos and Pons introduced the s-weak order on
s-decreasing trees, for any weak composition s. They proved its
lattice structure and conjectured that it could be realized as the
1-skeleton of a polyhedral subdivision of a zonotope of
dimension n-1. We answer their conjecture in the case where s is a
(strict) composition by providing three geometric realizations of the
s-permutahedron.
The first one is the dual graph of a triangulation of a flow polytope
of high dimension. The second, obtained using the Cayley trick, is the
dual graph of a fine mixed subdivision of a sum of hypercubes that has
the conjectured dimension. The third, obtained using tropical
geometry, is the 1-skeleton of a polyhedral complex for which we can
provide explicit coordinates of the vertices and whose support is a
permutahedron as conjectured.
Received: November 15, 2023.
Accepted: February 15, 2024.
Final version: April 1, 2024.
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