Chain Polytopes Research Articles

$f$-vector inequalities for order and chain polytopes

The order and chain polytopes are two $0/1$-polytopes constructed from a finite poset. In this paper, we study the $f$-vectors of these polytopes. We investigate how the order and chain polytopes behave under disjoint unions and ordinal sums of posets, and how the $f$-vectors of these polytopes are expressed in terms of $f$-vectors of smaller polytopes. Our focus is on comparing the $f$-vectors of the order and chain polytope built from the same poset. In our main theorem we prove that for a family of posets built inductively by taking disjoint unions and ordinal sums of posets, for any poset $\mathcal {P}$ in this family the $f$-vector of the order polytope of $\mathcal {P}$ is component-wise at most the $f$-vector of the chain polytope of $\mathcal {P}$.

Newton–Okounkov polytopes of flag varieties and marked chain-order polytopes

Marked chain-order polytopes are convex polytopes constructed from a marked poset. They give a discrete family relating a marked order polytope with a marked chain polytope. In this paper, we consider the Gelfand–Tsetlin poset of type A A , and realize the associated marked chain-order polytopes as Newton–Okounkov bodies of the flag variety. Our realization connects previous realizations of Gelfand–Tsetlin polytopes and Feigin–Fourier–Littelmann–Vinberg polytopes as Newton–Okounkov bodies in a uniform way. As an application, we prove that the flag variety degenerates into the irreducible normal projective toric variety corresponding to a marked chain-order polytope. We also construct a specific basis of an irreducible highest weight representation. The basis is naturally parametrized by the set of lattice points in a marked chain-order polytope.

Gröbner fans of Hibi ideals, generalized Hibi ideals and flag varieties

The main goal of this paper is to give explicit descriptions of two maximal cones in the Gröbner fan of the Plücker ideal. These cones correspond to the monomial ideals given by semistandard and PBW-semistandard Young tableaux. For the first cone, as an intermediate result we obtain the description of a maximal cone in the Gröbner fan of any Hibi ideal. For the second, we generalize the notion of Hibi ideals by associating an ideal with every interpolating polytope. This is a family of polytopes that generalizes the order and chain polytopes of a poset (à la Fang–Fourier–Litza–Pegel). We then describe a maximal cone in the Gröbner fan of each of these ideals. We also establish some useful facts concerning PBW-semistandardness, in particular, we prove that it provides a new Hodge algebra structure on the Plücker algebra.

Enriched chain polytopes

Stanley introduced a lattice polytope \(\mathscr{C}_P\) arising from a finite poset P, which is called the chain polytope of P. The geometric structure of \(\mathscr{C}_P\) has good relations with the combinatorial structure of P. In particular, the Ehrhart polynomial of \(\mathscr{C}_P\) 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 Gröbner 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.

Enriched order polytopes and enriched Hibi rings

Stanley introduced two classes of lattice polytopes associated to posets, which are called the order polytope {{mathscr {O}}}_P and the chain polytope {{mathscr {C}}}_P of a poset P. It is known that, given a poset P, the Ehrhart polynomials of {{mathscr {O}}}_P and {{mathscr {C}}}_P are equal to the order polynomial of P that counts the P-partitions. In this paper, we introduce the enriched order polytope of a poset P and show that it is a reflexive polytope whose Ehrhart polynomial is equal to that of the enriched chain polytope of P and the left enriched order polynomial of P that counts the left enriched P-partitions, by using the theory of Gröbner bases. The toric rings of enriched order polytopes are called enriched Hibi rings. It turns out that enriched Hibi rings are normal, Gorenstein, and Koszul. The above result implies the existence of a bijection between the lattice points in the dilations of and . Towards such a bijection, we give the facet representations of enriched order and chain polytopes.

A Continuous Family of Marked Poset Polytopes

For any marked poset we define a continuous family of polytopes, parametrized by a hypercube, generalizing the notions of marked order and marked chain polytopes. By providing transfer maps, we show that the vertices of the hypercube parametrize an Ehrhart equivalent family of lattice polytopes. The combinatorial type of the polytopes is constant when the parameters vary in the relative interior of each face of the hypercube. Moreover, with the help of a subdivision arising from a tropical hyperplane arrangement associated to the marked poset, we give an explicit description of the vertices of the polytope for generic parameters.

Faces of 2-Dimensional Simplex of Order and Chain Polytopes

Each of the descriptions of vertices, edges, and facets of the order and chain polytope of a finite partially ordered set are well known. In this paper, we give an explicit description of faces of 2-dimensional simplex in terms of vertices. Namely, it will be proved that an arbitrary triangle in 1-skeleton of the order or chain polytope forms the face of 2-dimensional simplex of each polytope. These results mean a generalization in the case of 2-faces of the characterization known in the case of edges.

On the canonical ideal of the Ehrhart ring of the chain polytope of a poset

Let P be a poset, O(P) the order polytope of P and C(P) the chain polytope of P. In this paper, we study the canonical ideal of the Ehrhart ring K[C(P)] of C(P) over a field K and characterize the level (resp. anticanonical level) property of K[C(P)] by a combinatorial structure of P. In particular, we show that if K[C(P)] is level (resp. anticanonical level), then so is K[O(P)]. We exhibit examples which show the converse does not hold.Moreover, we show that the symbolic powers of the canonical ideal of K[C(P)] are identical with ordinary ones and degrees of the generators of the canonical and anticanonical ideals are consecutive integers.

Cutting Convex Polytopes by Hyperplanes

Cutting a polytope is a very natural way to produce new classes of interesting polytopes. Moreover, it has been very enlightening to explore which algebraic and combinatorial properties of the original polytope are hereditary to its subpolytopes obtained by a cut. In this work, we devote our attention to all the separating hyperplanes for some given polytope (integral and convex) and study the existence and classification of such hyperplanes. We prove the existence of separating hyperplanes for the order and chain polytopes for any finite posets that are not a single chain, and prove there are no such hyperplanes for any Birkhoff polytopes. Moreover, we give a complete separating hyperplane classification for the unit cube and its subpolytopes obtained by one cut, together with some partial classification results for order and chain polytopes.

Antichain Toggling and Rowmotion

In this paper, we analyze the toggle group on the set of antichains of a poset. Toggle groups, generated by simple involutions, were first introduced by Cameron and Fon-Der-Flaass for order ideals of posets. Recently Striker has motivated the study of toggle groups on general families of subsets, including antichains. This paper expands on this work by examining the relationship between the toggle groups of antichains and order ideals, constructing an explicit isomorphism between the two groups (for a finite poset). We also focus on the rowmotion action on antichains of a poset that has been well-studied in dynamical algebraic combinatorics, describing it as the composition of antichain toggles. We also describe a piecewise-linear analogue of toggling to the Stanley’s chain polytope. We examine the connections with the piecewise-linear toggling Einstein and Propp introduced for order polytopes and prove that almost all of our results for antichain toggles extend to the piecewise-linear setting.

Order-chain polytopes

Given two families X and Y of integral polytopes with nice combinatorial and algebraic properties, a natural way to generate a new class of polytopes is to take the intersection P = P 1 ∩ P 2 , where P 1 ∈ X , P 2 ∈ Y . Two basic questions then arise: 1) when P is integral and 2) whether P inherits the “old type” from P 1 , P 2 or has a “new type”, that is, whether P is unimodularly equivalent to a polytope in X ∪ Y or not. In this paper, we focus on the families of order polytopes and chain polytopes. Following the above framework, we create a new class of polytopes which are named order-chain polytopes. When studying their volumes, we discover a natural relation with Ehrenborg and Mahajan’s results on maximizing descent statistics.

Volume, Facets and Dual Polytopes of Twinned Chain Polytopes

Let $(P,\leq_P)$ and $(Q,\leq_Q)$ be finite partially ordered sets with $|P|=|Q|=d$, and $\mathcal{C}(P) \subset \mathbb{R}^d$ and $\mathcal{C}(Q) \subset \mathbb{R}^d$ their chain polytopes. The twinned chain polytope of $P$ and $Q$ is the lattice polytope $\Gamma(\mathcal{C}(P),\mathcal{C}(Q)) \subset \mathbb{R}^d$ which is the convex hull of $\mathcal{C}(P) \cup (-\mathcal{C}(Q))$. It is known that twinned chain polytopes are Gorenstein Fano polytopes with the integer decomposition property. In the present paper, we study combinatorial properties of twinned chain polytopes. First, we will give the formula of the volume of twinned chain polytopes in terms of the underlying partially ordered sets. Second, we will identify the facet-supporting hyperplanes of twinned chain polytopes in terms of the underlying partially ordered sets. Finally, we will provide the vertex representations of the dual polytopes of twinned chain polytopes.

Reflexive polytopes arising from partially ordered sets and perfect graphs

Reflexive polytopes which have the integer decomposition property are of interest. Recently, some large classes of reflexive polytopes with integer decomposition property coming from the order polytopes and the chain polytopes of finite partially ordered sets are known. In the present paper, we will generalize this result. In fact, by virtue of the algebraic technique on Gröbner bases, new classes of reflexive polytopes with the integer decomposition property coming from the order polytopes of finite partially ordered sets and the stable set polytopes of perfect graphs will be introduced. Furthermore, the result will give a polyhedral characterization of perfect graphs. Finally, we will investigate the Ehrhart $$\delta $$ -polynomials of these reflexive polytopes.

Quadratic Gröbner bases arising from partially ordered sets

The order polytope $\mathcal {O}(P)$ and the chain polytope $\mathcal {C}(P)$ associated to a partially ordered set $P$ are studied. In this paper, we introduce the convex polytope $\Gamma (\mathcal {O}(P), -\mathcal {C}(Q))$ which is the convex hull of $\mathcal {O}(P) \cup (-\mathcal {C}(Q))$, where both $P$ and $Q$ are partially ordered sets with $|P|=|Q|=d$. It will be shown that $\Gamma (\mathcal {O}(P), -\mathcal {C}(Q))$ is a normal and Gorenstein Fano polytope by using the theory of reverse lexicographic squarefree initial ideals of toric ideals.

Facets and volume of Gorenstein Fano polytopes

AbstractIt is known that every integral convex polytope is unimodularly equivalent to a face of some Gorenstein Fano polytope. It is then reasonable to ask whether every normal polytope is unimodularly equivalent to a face of some normal Gorenstein Fano polytope. In the present paper, it is shown that, by giving new classes of normal Gorenstein Fano polytopes, each order polytope as well as each chain polytope of dimension d is unimodularly equivalent to a facet of some normal Gorenstein Fano polytopes of dimension . Furthermore, investigation on combinatorial properties, especially, Ehrhart polynomials and volume of these new polytopes will be achieved. Finally, some curious examples of Gorenstein Fano polytopes will be discovered.

The numbers of edges of the order polytope and the chain polytope of a finite partially ordered set

Let P be an arbitrary finite partially ordered set. It will be proved that the number of edges of the order polytope 풪(P) is equal to that of the chain polytope C(P). Furthermore, it will be shown that the degree sequence of the finite simple graph which is the 1-skeleton of 풪(P) is equal to that of C(P) if and only if 풪(P) and C(P) are unimodularly equivalent.

Two Double Poset Polytopes

To every poset $P$, Stanley [Discrete Comput. Geom., 1 (1986), pp. 9--23] associated two polytopes, the order polytope and the chain polytope, whose geometric properties reflect the combinatorial qualities of $P$. This construction allows for deep insights into combinatorics by way of geometry and vice versa. Malvenuto and Reutenauer [J. Combin. Theory Ser. A, 118 (2011), pp. 1322--1333] introduced double posets, that is, (finite) sets equipped with two partial orders, as a generalization of Stanley's labeled posets. Many combinatorial constructions can be naturally phrased in terms of double posets. We introduce the double order polytope and the double chain polytope and we amply demonstrate that they geometrically capture double posets, i.e., the interaction between the two partial orders. We describe the facial structures, Ehrhart polynomials, and volumes of these polytopes in terms of the combinatorics of double posets. We also describe a curious connection to Geissinger's valuation polytopes and we c...

Marked chain-order polytopes

We introduce in this paper the marked chain-order polytopes associated to a marked poset, generalizing the marked chain polytopes and marked order polytopes by putting them as extremal cases in an Ehrhart equivalent family. Some combinatorial properties of these polytopes are studied. This work is motivated by the framework of PBW degenerations in representation theory of Lie algebras.

Unimodular Equivalence of Order and Chain Polytopes

Order polytope and chain polytope are two polytopes that arise naturally from a finite partially ordered set. These polytopes have been deeply studied from viewpoints of both combinatorics and commutative algebra. Even though these polytopes possess remarkable combinatorial and algebraic resemblance, they seem to be rarely unimodularly equivalent. In the present paper, we prove the following simple and elegant result: the order polytope and chain polytope for a poset are unimodularly equivallent if and only if that poset avoid the 5-element "X" shape subposet. We also explore a few equivalent statements of the main result.

Marked poset polytopes: Minkowski sums, indecomposables, and unimodular equivalence

We analyze marked poset polytopes and generalize a result due to Hibi and Li, answering whether the marked chain polytope is unimodular equivalent to the marked order polytope. Both polytopes appear naturally in the representation theory of semi-simple Lie algebras, and hence we can give a necessary and sufficient condition on the marked poset such that the associated toric degenerations of the corresponding partial flag variety are isomorphic. We further show that the set of lattice points in such a marked poset polytope is the Minkowski sum of sets of lattice points for 0–1 polytopes. Moreover, we provide a decomposition of the marked poset into indecomposable marked posets, which respects this Minkowski sum decomposition for the marked chain polytopes.