Symmetrizing polytopes and posets

TL;DR

This paper introduces a method to symmetrize polytopes and posets using reflection groups, enabling the analysis of their combinatorics and normal fans, and offers a new approach to realizing symmetric face posets as polytopes.

Contribution

The authors define the $rak{G}$-symmetrization of polytopes and develop a framework to recover their combinatorics and fans from refined fundamental fans, advancing the realization problem for symmetric posets.

Findings

The combinatorics and normal fan of symmetrized polytopes can be derived from their refined fundamental fans.

A new approach to the realization problem of symmetric face posets as polytopes is proposed.

The method simplifies constructing polytopes with prescribed symmetric face posets.

Abstract

Motivated by the authors' work on permuto-associahedra, which can be considered as a symmetrization of the associahedron using the symmetric group, we introduce and study the $\mathfrak{G}$-symmetrization of an arbitrary polytope $P$ for any reflection group $\mathfrak{G}$. We show that the combinatorics, and moreover, the normal fan of such a symmetrization can be recovered from its refined fundamental fan, a decorated poset describing how the normal fan of $P$ subdivides the fundamental chamber associated to the reflection group $\mathfrak{G}$. One important application of our results is providing a way to approach the realization problem of a $\mathfrak{G}$-symmetric poset F, that is, the problem of constructing a polytope whose face poset is F. Instead of working with the original poset F, we look at its dual poset T (which is $\mathfrak{G}$-symmetric as well) and focus on a generating subposet Z of T, and reduce the problem to realizing Z as a refined fundamental fan.