Abstract
A lattice polytope mathscr {P} subset mathbb {R}^d is called a locally anti-blocking polytope if for any closed orthant {mathbb R}^d_{varepsilon } in mathbb {R}^d, mathscr {P} cap mathbb {R}^d_{varepsilon } is unimodularly equivalent to an anti-blocking polytope by reflections of coordinate hyperplanes. We give a formula for the h^*-polynomials of locally anti-blocking lattice polytopes. In particular, we discuss the gamma -positivity of h^*-polynomials of locally anti-blocking reflexive polytopes.
Highlights
A lattice polytope is a convex polytope all of whose vertices have integer coordinates
We investigate the h∗-polynomials of locally anti-blocking lattice polytopes
Enriched chain polytopes are unconditional and reflexive, and their h∗-polynomials are always γ -positive. Combining these facts and Theorem 1.1, we know that, for a locally anti-blocking reflexive polytope P, if every P ∩ Rdε is the intersection of Rdε and either an enriched chain polytope or a symmetric edge reflexive polytope of type B, the h∗-polynomial of P is γ -positive (Corollary 4.2)
Summary
A lattice polytope is a convex polytope all of whose vertices have integer coordinates. Enriched chain polytopes are unconditional and reflexive, and their h∗-polynomials are always γ -positive Combining these facts and Theorem 1.1, we know that, for a locally anti-blocking reflexive polytope P, if every P ∩ Rdε is the intersection of Rdε and either an enriched chain polytope or a symmetric edge reflexive polytope of type B, the h∗-polynomial of P is γ -positive (Corollary 4.2). By using this result, we show that the h∗-polynomials of several classes of reflexive polytopes are γ -positive.
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