Cubical realizations of flag nestohedra and Gal's conjecture

TL;DR

This paper demonstrates that all flag nestohedra can be constructed from cubes through face shavings, provides new geometric realizations, and confirms Gal's conjecture for these polytopes, including bounds on their gamma-vectors.

Contribution

It introduces a new geometric construction method for flag nestohedra and proves Gal's conjecture for all such polytopes, with explicit gamma-vector bounds.

Findings

Flag nestohedra can be obtained from cubes by face shavings.

New Delzant geometric realizations of flag nestohedra are provided.

Gal's conjecture is confirmed for all flag nestohedra, with gamma-vector bounds.

Abstract

We study nestohedra $P_B$ corresponding to building sets $B$. It is shown that every flag nestohedron can be obtained from a cube by successive shavings faces of codimension 2. We receive new Delzant geometric realization of flag nestohedra. The main result of the paper is that Gal's conjecture holds for every flag nestohedron. Moreover, we get the exact estimation of $\gamma$-vectors of $n$-dimensional flag nestohedra: $0\leq\gamma_i(P_B)\leq\gamma_i(Pe^n)$.