Séminaire Lotharingien de Combinatoire, 95B.103 (2026), 12 pp.
Timothy Blanton, Jesús A. De Loera and Melissa Sherman-Bennett
On Orbit Realizations of Coxeter Permutahedra
Abstract.
The permutahedron is a fundamental polytope associated with the symmetric group and serves as a central example of the interaction between geometry, combinatorics, and algebra. More generally, for W a finite reflection group, one can define the W-permutahedron. In this work, we study the \emph{orbit realizations} of the W-permutahedron, which are obtained by taking the convex hull of the W-orbit of any point in the fundamental chamber. Although all orbit realizations have the same normal fan, they do not have the same geometric structure. We show that the orbit realization space of the W-permutahedron is contractible. We also explore whether various geometric and combinatorial features persist among all orbit realizations. In particular, we show that certain subsets of vertices always form simplices, and some triangulations similarly persist.
We experimentally explore the cases of S3
and S4 and we uncover some non-persistent phenomena.
Our results suggest a rich internal structure within orbit realization spaces, motivating further study of their topology and combinatorial geometry.
Received: November 15, 2025.
Accepted: February 15, 2026.
Final version: April 1, 2026.
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