Stanley introduced a lattice polytope $$\mathscr{C}_P$$ C P arising from a finite poset P , which is called the chain polytope of P . The geometric structure of $$\mathscr{C}_P$$ C P has good relations with the combinatorial structure of P . In particular, the Ehrhart polynomial of $$\mathscr{C}_P$$ C P is given by the order polynomial of P . In the present paper, associated to P , we introduce a lattice polytope ℰ P , 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 Gröbner bases, we see that ℰ P is a reflexive polytope with a flag regular unimodular triangulation. Moreover, the h *-polynomial of ℰ P is equal to the h -polynomial of a flag triangulation of a sphere. On the other hand, by showing that the Ehrhart polynomial of ℰ P coincides with the left enriched order polynomial of P , it follows from works of Stembridge and Petersen that the h *-polynomial of ℰ P is γ -positive. Stronger, we prove that the γ -polynomial of ℰ P is equal to the f -polynomial of a flag simplicial complex.

中文翻译：

---

富链多胞体

Stanley 引入了由有限偏序 P 产生的格多胞体 $$\mathscr{C}_P$$ CP ，称为 P 的链多胞体。$$\mathscr{C}_P$$ CP 的几何结构与 P 的组合结构有很好的关系。特别是，$$\mathscr{C}_P$$ CP 的 Ehrhart 多项式由 P 的阶多项式给出。在本文中，与 P 相关联，我们介绍了一个晶格多面体 ℰ P ，它被称为 P 的富集链多面体，并研究了这个多面体的几何和组合特性。凭借 Gröbner 基上的代数技术，我们看到 ℰ P 是具有标志规则单模三角剖分的自反多胞体。此外，ℰ P 的 h * 多项式等于球体标志三角剖分的 h 多项式。另一方面，通过证明 ℰ P 的 Ehrhart 多项式与 P 的左富集阶多项式重合，从 Stembridge 和 Petersen 的工作中可以得出，ℰ P 的 h * 多项式是 γ 正的。更强的是，我们证明 ℰ P 的 γ 多项式等于标志单纯复形的 f 多项式。