The local $h$-polynomial of the edgewise subdivision of the simplex

Abstract

The $r$-fold edgewise subdivision is a well studied flag triangulation of the simplex with interesting algebraic, combinatorial and geometric properties. An important enumerative invariant, namely the local $h$-polynomial, of this triangulation is computed and shown to be $\gamma$-nonnegative by providing explicit combinatorial interpretations to the corresponding coefficients. A construction of a flag triangulation of the seven-dimensional simplex whose local $h$-polynomial is not real-rooted is also described.

arXiv comment11 pages, 3 figures; to appear in the Bulletin of the Hellenic Mathematical Society

References in corpus (5)

• Lectures on 0/1-polytopes
• Real Root Conjecture fails for five and higher dimensional spheres
• Existence of unimodular triangulations - positive results
• Compatible polynomials and edgewise subdivisions
• Edgewise subdivisions, local $h$-polynomials and excedances in the wreath product $\ZZ_r \wr \mathfrak{S}_n$

Cited by in corpus (3)

• Gamma-positivity in combinatorics and geometry
• Pattern Recognition on Oriented Matroids: Symmetric Cycles in the Hypercube Graphs
• Local $h$-vectors of Quasi-Geometric and Barycentric Subdivisions