Isocanted alcoved polytopes

TL;DR

This paper introduces isocanted alcoved polytopes via tropical matrices, explores their combinatorial properties, computes their f-vectors, and verifies several conjectures related to their structure and symmetry.

Contribution

It defines a new class of polytopes, analyzes their combinatorial and geometric properties, and confirms the validity of multiple longstanding conjectures for these polytopes.

Findings

Isocanted alcoved polytopes are centrally symmetric, cubical, and zonotopes.

They have a unique combinatorial type in each dimension.

Their f-vector attains maximum at a specific dimension-dependent point.

Abstract

Through tropical normal idempotent matrices, we introduce isocanted alcoved polytopes, computing their $f$--vectors and checking the validity of the following five conjectures: B\'{a}r\'{a}ny, unimodality, $3^d$, flag and cubical lower bound (CLBC). Isocanted alcoved polytopes are centrally symmetric, almost simple cubical polytopes. They are zonotopes. We show that, for each dimension, there is a unique combinatorial type. In dimension $d$, an isocanted alcoved polytope has $2^{d+1}-2$ vertices, its face lattice is the lattice of proper subsets of $[d+1]$ and its diameter is $d+1$. They are realizations of $d$--elementary cubical polytopes. The $f$--vector of a $d$--dimensional isocanted alcoved polytope attains its maximum at the integer $\lfloor d/3\rfloor$.