Enriched chain polytopes
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ENRICHED CHAIN POLYTOPES
BY
Hidefumi Ohsugi Department of Mathematical Sciences, School of Science and Technology Kwansei Gakuin University, Sanda, Hyogo 669-1337, Japan e-mail: [email protected]
AND
Akiyoshi Tsuchiya Graduate School of Mathematical Sciences, University of Tokyo Komaba, Meguro-ku, Tokyo 153-8914, Japan e-mail: [email protected]
ABSTRACT
Stanley introduced a lattice polytope CP arising from a finite poset P , which is called the chain polytope of P . The geometric structure of CP has good relations with the combinatorial structure of P . In particular, the Ehrhart polynomial of CP is given by the order polynomial of P . In the present paper, associated to P , we introduce a lattice polytope EP , which is called the enriched chain polytope of P , and investigate geometric and combinatorial properties of this polytope. By virtue of the algebraic technique on Gr¨ obner bases, we see that EP is a reflexive polytope with a flag regular unimodular triangulation. Moreover, the h∗ -polynomial of EP is equal to the h-polynomial of a flag triangulation of a sphere. On the other hand, by showing that the Ehrhart polynomial of EP coincides with the left enriched order polynomial of P , it follows from works of Stembridge and Petersen that the h∗ -polynomial of EP is γ-positive. Stronger, we prove that the γ-polynomial of EP is equal to the f -polynomial of a flag simplicial complex.
Received December 7, 2018 and in revised form July 1, 2019
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H. OHSUGI AND A. TSUCHIYA
Isr. J. Math.
Introduction A lattice polytope P ⊂ Rn of dimension n is a convex polytope all of whose vertices have integer coordinates. Given a positive integer m, we define LP (m) = |mP ∩ Zn |. The study on LP (m) originated in Ehrhart [5] who proved that LP (m) is a polynomial in m of degree n with the constant term 1. We say that LP (m) is the Ehrhart polynomial of P. The generating function of the lattice point enumerator, i.e., the formal power series EhrP (x) = 1 +
∞
LP (k)xk ,
k=1
is called the Ehrhart series of P. It is well known that it can be expressed as a rational function of the form h∗ (P, x) EhrP (x) = . (1 − x)n+1 The polynomial h∗ (P, x) is a polynomial in x of degree at most n with nonnegative integer coefficients ([18]) and it is called the h∗ -polynomial (or the δ-polynomial) of P. Moreover, one has Vol(P) = h∗ (P, 1), where Vol(P) is the normalized volume of P. In [19], Stanley introduced a class of lattice polytopes associated with finite partially ordered sets. Let P = [n] := {1, 2, . . . , n} be a partially ordered set (poset, for short). An antichain of P is a subset of P consisting of pairwise incomparable elements of P . Note that the empty set ∅ is an antichain of P . The chain polytope CP of P is the convex hull of {ei1 + · · · + eik : {i1 , . . . , ik } is an antichain of P }, where ei is the i-th unit coordinate vector of Rn and the empty set ∅ corresponds to the origin 0 of Rn . Then CP is a lattice polytope of dimension n. There is a close interplay between the combinatorial structur
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