1 Introduction

Two double poset polytopes

Abstract.

To every poset , Stanley (1986) associated two polytopes, the order polytope and the chain polytope, whose geometric properties reflect the combinatorial qualities of . This construction allows for deep insights into combinatorics by way of geometry and vice versa. Malvenuto and Reutenauer (2011) introduced double posets, that is, (finite) sets equipped with two partial orders, as a generalization of Stanley’s labelled 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 characterize -level polytopes among our double poset polytopes.

Fulkerson’s anti-blocking polytopes from combinatorial optimization subsume stable set polytopes of graphs and chain polytopes of posets. We determine the geometry of Minkowski- and Cayley sums of anti-blocking polytopes. In particular, we describe a canonical subdivision of Minkowski sums of anti-blocking polytopes that facilitates the computation of Ehrhart (quasi-)polynomials and volumes. This also yields canonical triangulations of double poset polytopes.

Finally, we investigate the affine semigroup rings associated to double poset polytopes. We show that they have quadratic Gröbner bases, which gives an algebraic description of the unimodular flag triangulations described in the first part.

Key words and phrases:
double posets, double order polytope, double chain polytope, Birkhoff lattice, Ehrhart polynomials, volumes, anti-blocking polytopes, Gröbner bases
2010 Mathematics Subject Classification:
06A07, 06A11, 52B12, 52B20

1. Introduction

A (finite) partially ordered set (or poset, for short) is a finite set together with a reflexive, transitive, and anti-symmetric relation . The notion of partial order pervades all of mathematics and the enumerative and algebraic combinatorics of posets is underlying in computations in virtually all areas. In 1986, Stanley [34] defined two convex polytopes for every poset that, in quite different ways, geometrically capture combinatorial properties of . The order polytope is set of all order preserving functions into the interval . That is, all functions such that

for all with . Hence, parametrizes functions on and many properties of are encoded in the boundary structure of : faces of are in correspondence with quotients of . In particular, the vertices of are in bijection to filters of . But also metric and arithmetic properties of can be determined from . The order polytope naturally has vertices in the lattice and its Ehrhart polynomial , up to a shift, coincides with the order polynomial ; see Section 4.2 for details. A full-dimensional simplex with vertices in a lattice is unimodular with respect to if it has minimal volume. The normalized volume relative to is the Euclidean volume scaled such that the volume of a unimodular simplex is . If the lattice is clear from the context, we denote the normalized volume by . By describing a canonical triangulation of into unimodular simplices, Stanley showed that is exactly the number of linear extensions of , that is, the number of refinements of to a total order. We will review these results in more detail in Section 4.2. This bridge between geometry and combinatorics can, for example, be used to show that computing volume is hard (cf. [4]) and, conversely, geometric inequalities can be used on partially ordered sets; see [26, 34].

The chain polytope is the collection of functions such that

(1)

for all chains in . In contrast to the order polytope, does not determine . In fact, is defined by the comparability graph of and bears strong relations to so-called stable set polytopes of perfect graphs; see Section 3.2. Surprisingly, it is shown in [34] that the chain polytope and the order polytope have the same Ehrhart polynomial and hence , which shows that the number of linear extensions only depends on the comparability relation. Stanley’s poset polytopes are very natural objects that appear in a variety of contexts in combinatorics and beyond; see [1, 25, 32, 11].

Inspired by Stanley’s labelled posets, Malvenuto and Reutenauer [28] introduced double poset in the context of combinatorial Hopf algebras. A double poset is a triple consisting of a finite ground set and two partial order relations and on . We will write and to refer to the two underlying posets. If is a total order, then this corresponds to labelled poset in the sense of Stanley [33], which is the basis for the rich theory of -partitions. The combinatorial study of general double posets gained momentum in recent years with a focus on algebraic aspects; see, for example, [9, 10]. The goal of this paper is to build a bridge to geometry by introducing two double poset polytopes that, like the chain- and the order polytope, geometrically reflect the combinatorial properties of double posets and, in particular, the interaction between the two partial orders.

1.1. Double order polytopes

For a double poset , we define the double order polytope as

This is a -dimensional polytope in . Its vertices are trivially in bijection to filters of and . This is a lattice polytope with respect to but we will mostly view as a lattice polytope with respect to the affine lattice . That is, up to a translation by , is the polytope

which is a lattice polytope with respect to . In Section 2.2, we describe the facets of in terms of chains and cycles alternating between and and, for the important case of compatible double posets, we completely determine the facial structure in Section 2.3 in terms of double Birkhoff lattices . The double order polytope automatically has and as facets. The non-trivial combinatorial structure is captured by the reduced double order polytope

which is a lattice polytope with respect to by our choice of embedding.

By placing and at heights and , respectively, we made sure that always contains the origin. Every poset trivially induces a double poset and for an induced double poset, is centrally-symmetric and, up to a (lattice-preserving) shear, is the polytope

where is the poset with the opposite order. Geissinger [13] introduced a polytope associated to valuations on distributive lattices with values in . In Section 2.4, we show a surprising connection between Geissinger’s valuation polytopes and polars of the (reduced) double order polytopes of . We will review notions from the theory of double posets and emphasize their geometric counterparts.

1.2. Double chain-, Hansen-, and anti-blocking polytopes

The double chain polytope associated to a double poset is the polytope

The reduced version is studied in Section 3 in the context of anti-blocking polytopes. According to Fulkerson [12], a full-dimensional polytope is anti-blocking if for any , it contains all points with for . It is obvious from (1) that chain polytopes are anti-blocking. Anti-blocking polytopes are important in combinatorial optimization and, for example, contain stable set polytopes of graphs. For two polytopes , we define the Cayley sum as the polytope

and we abbreviate . Thus,

Section 3 is dedicated to a detailed study of the polytopes as well as their sections for anti-blocking polytopes . We completely determine the facets of in terms of in Section 3.1, which yields the combinatorics of . In Section 3.3, we describe a canonical subdivision of and for anti-blocking blocking polytopes . Moreover, if have regular, unimodular, or flag triangulations, then so has (for an appropriately chosen affine lattice). The canonical subdivision enables us to give explicit formulas for the volume and the Ehrhart (quasi-)polynomials of these classes of polytopes.

The chain polytope only depends on the comparability graph of and, more precisely, is the stable set polytope of . Thus, only depends on the double graph . For a graph , let be its stable set polytope; see Section 3.1 for precise definitions. Lovász [27] characterized perfect graphs in terms of and Hansen [17] studied the polytopes . If is perfect, then Hansen showed that the polar is linearly isomorphic to where is the complement graph of . In Section 3.2, we generalize this result to all Cayley sums of anti-blocking polytopes.

1.3. -level polytopes and volume

A full-dimensional polytope is called -level if for any facet-defining hyperplane there is such that contains all vertices of . The class of -level polytopes plays an important role in, for example, the study of centrally-symmetric polytopes [30, 17], polynomial optimization [14, 15], statistics [37], and combinatorial optimization [31]. For example, Lovász [27] characterizes perfect graphs by the -levelness of their stable set polytopes and Hansen showed that is -level if is perfect. In fact, we extend this to yet another characterization of perfect graphs in Corollary 3.11. This result implies that is -level for double posets induced by posets. However, it is in general not true that is -level if and are. A counterexample is the polytope , where is the -hypersimplex. The starting point for this paper was the question for which double posets the polytopes and are -level. Answers are given in Corollary 2.9, Proposition 2.10, and Corollary 3.11. A new class of -level polytopes comes from valuation polytopes; see Corollary 2.19. Sullivant [37, Thm. 2.4] showed that -level lattice polytopes have the interesting property that any pulling triangulation that uses all lattice points in is unimodular. Hence, for -level lattice polytopes, the normalized volume is the number of simplices. In particular, is -level and Stanley’s canonical triangulation is a pulling triangulation. Stanley defined a piecewise linear homeomorphism between and whose domains of linearity are exactly the simplices of the canonical triangulation. Since this transfer map is lattice preserving, it follows that , which also implies the volume result. In Section 4, we generalize this transfer map to a lattice preserving PL homeomorphism for any compatible double poset . In particular, is mapped to . This also transfers the canonical flag triangulation of to a canonical flag triangulation of . Abstractly, the triangulation can be described in terms of a suitable subcomplex of the order complex of the double Birkhoff lattice . In Section 4.2, we give explicit formulas for the Ehrhart polynomial and the volume of if is compatible and for in general.

1.4. Double Hibi rings

Hibi [19] studied rings associated to finite posets that give posets an algebraic incarnation and that are called Hibi rings. In modern language, the Hibi ring associated to a poset is the semigroup ring associated to . Many properties of posses an algebraic counterpart and, in particular, Hibi exhibited a quadratic Gröbner basis for the associated toric ideal. In Section 5, we introduce the double Hibi rings as suitable analogs for double posets, which are the semigroup rings associated to . We construct a quadratic Gröbner basis for the cases of compatible double posets. Using a result by Sturmfels [36, Thm. 8.3], this shows the existence of a unimodular and flag triangulation of which coincides with the triangulation in Section 4. We also construct a quadratic Gröbner basis for the rings for arbitrary double posets and we remark on the algebraic implications for double posets.

Acknowledgements. We would like to thank Christian Stump and Stefan Felsner for many helpful conversations regarding posets and we thank Vic Reiner for pointing out [28]. We would also like to thank the referees for valuable suggestions. T. Chappell was supported by a Phase-I scholarship of the Berlin Mathematical School. T. Friedl and R. Sanyal were supported by the DFG-Collaborative Research Center, TRR 109 “Discretization in Geometry and Dynamics”. T. Friedl received additional support from the Dahlem Research School at Freie Universität Berlin.

2. Double order polytopes

2.1. Order polytopes

Let be a poset. We write for the poset obtained from by adjoining a minimum and a maximum . For an order relation , we define a linear form by

for any . Moreover, for , we define and . With this notation, is contained in if and only if

(2)

Every nonempty face of is of the form

for some linear function . Later, we want to identify with its vector of coefficients and thus we write

Combinatorially, faces can be described using face partitions: To every face is an associated collection of nonempty and pairwise disjoint subsets that partition . According to Stanley [34, Thm 1.2], a partition of is a (closed) face partition if and only if each is a connected poset and for some is a partial order on . Of course, it is sufficient to remember the non-singleton parts and we define the reduced face partition of as . The normal cone of a nonempty face is the polyhedral cone

The following description of follows directly from (2).

Proposition 2.1.

Let be a finite poset and a nonempty face with reduced face partition . Then

We note the following simple but very useful consequence of this description.

Corollary 2.2.

Let be a nonempty face with reduced face partition . Then for every and the following hold:

  1. if for some , then ;

  2. if for some , then ;

  3. if , then .

The vertices of are exactly the indicator functions where is a filter. For a filter , we write for the filter induced in .

Proposition 2.3.

Let be a face with (reduced) face partition and let be a filter. Then if and only if

for all .

That is, belongs to if and only if does not separate any two comparable elements in , for all .

2.2. Facets of double order polytopes

Let be a double poset. The double order polytope is a -dimensional polytope in with coordinates . It is obvious that the vertices of are exactly for filters and , respectively. To get the most out of our notational convention, for we define

By construction, and are facets that are obtained by maximizing the linear function over . We call the remaining facets vertical, as they are of the form , where are certain nonempty proper faces for . The vertical facets are in bijection with the facets of the reduced double order polytope .

More precisely, if is a facet, then there is a linear function such that where and . This linear function is necessarily unique up to scaling and hence the faces are characterized by the property

(3)

We will call a linear function rigid if it satisfies (3) for a pair of faces . Our next goal is to give an explicit description of all rigid linear functions for which then yields a characterization of vertical facets.

An alternating chain of a double poset is a finite sequence of distinct elements

(4)

where . For an alternating chain , we define a linear function by

Here, we severely abuse notation and interpret as . Note that if and we call a proper alternating chain if . An alternating cycle of is a sequence of length of the form

where and for . We similarly define a linear function associated to by

Note that it is possible that a sequence of elements gives rise to two alternating chains, one starting with and one starting with . On the other hand, every alternating cycle of length yields alternating cycles starting with and alternating starting with .

Proposition 2.4.

Let be a double poset. If is a rigid linear function for , then for some alternating chain or alternating cycle and .

Proof.

Let and be the two faces for which (3) holds and let be the corresponding reduced face partitions. We define a directed bipartite graph with nodes and accordingly. If is contained in some part of , then we add a corresponding node to Consistently, we add a node to if it occurs in a part of . Note that and are distinct nodes. Similarly we add to and to if they appear in and , respectively. By Corollary 2.2, we have ensured that and for all .

For and , we add the directed edge if and for some . Similarly, we add the directed edge if and for some . We claim that every node except for maybe the special nodes has an incoming and an outgoing edge. For example, if , then . By Corollary 2.2(iii), there is an such that and by (ii), is not a maximal element in . Thus, there is some with and is an edge. It follows that every longest path either yields an alternating cycle or a proper alternating chain.

For an alternating cycle , we observe that

Since for every , is contained in some part of , we conclude that . Similarly, for all , is contained in some part of , and hence . Assuming that is rigid then shows that for some .

If does not contain a cycle, then let be a longest path in . In particular and . The same reasoning applies and shows that and hence for some . ∎

In general, not every alternating chain or cycle gives rise to a rigid linear function. Let be a poset that is not the antichain and define the double poset , where is the opposite order. In this case is, up to a shear, the ordinary prism over . Hence, the vertical facets of are prisms over the facets of . It follows from (2) that these facets correspond to cover relations in . Hence, every rigid is of the form for cycles where is a cover relation in .

We call a double poset compatible if and have a common linear extension. Note that a double poset is compatible if and only if it does not contain alternating cycles. Following [28], we call a double poset special if is a total order. At the other extreme, we call anti-special if is an anti-chain. A plane poset, as defined in [9] is a double poset such that distinct are -comparable if and only if they are not -comparable. For two posets and one classically defines the disjoint union and the ordinal sum as the posets on as follows. For set if and for some . For the ordinal sum, restricts to and on and respectively and for all and . The effect on order polytopes is and is a subdirect sum in the sense of McMullen [29]. Malvenuto and Reutenauer [28] define the composition of two double posets and as the double poset such that and .

The following is easily seen; for plane posets with the help of [9, Prop. 11].

Proposition 2.5.

Anti-special and plane posets are compatible. Moreover, the composition of two compatible double posets is a compatible double poset.

This defining property of compatible double posets assures us that in an alternating chain implies for any . In particular, a compatible double poset does not have alternating cycles. This also shows the following.

Lemma 2.6.

Let be a compatible double poset. If is part of an alternating chain with and , then there is no such that and .

For compatible double posets, we can give complete characterization of facets.

Theorem 2.7.

Let a compatible double poset. A linear function is rigid if and only if for some alternating chain . In particular, the facets of are in bijection with alternating chains.

Proof.

We already observed that and correspond to the improper alternating chains for . By Proposition 2.4 it remains to show that for any proper alternating chain the function is rigid. We only consider the case that is an alternating chain of the form

The other cases can be treated analogously. Let and and be the corresponding faces with reduced face partitions . Define and . Then for any set , we observe that . If is a filter of , then implies and hence and thus if and only if does not separate and for . Likewise, a filter is contained in if and only if does not separate and for . Lemma 2.6 implies that

To show that is rigid pick a linear function with and . Since the elements in and are exactly the minimal and maximal elements of the parts in , it follows from Corollary 2.2 that if , for . By Lemma 2.6, it follows that if , then is not contained in a part of the reduced face partition and vice versa. By Corollary 2.2(iii), it follows that for . Finally, for all by Proposition 2.1 and therefore for some finishes the proof. ∎

Example 1.

Let be a compatible double poset with .

  1. Let and be the -antichain. Then the only alternating chains are of the form for . The double order polytope is the -dimensional cube with vertices and .

  2. If and is the -chain , then any alternating chain can be identified with an element in . More precisely, is linearly isomorphic to the -dimensional crosspolytope .

  3. Let be the -chain and be the -antichain. Then any alternating chain is of the form for and any relation be can be completed to a unique alternating chain. Thus, is a -dimensional polytope with vertices and facets.

  4. The comb (see Figure 2) is the poset on elements such that if and for all . The -comb has filters and chains. Hence has vertices and facets.

  5. Generally, let be two posets and denote by and the number of filters and chains of for . Let be the trivial double poset induced by . Then has vertices and facets.

Example 2.

Consider the compatible ’XW’-double poset on five elements, whose Hasse diagrams are given in Figure 2. The polytope is six-dimensional with face vector

The facets correspond to the 28 alternating chains in , which are shown in Figure 3.

Figure 1. The ’XW’-double poset . The red and blue lines are the Hasse diagram of and , respectively. Striped lines are edges in both Hasse diagrams.
Figure 2. The comb .
Figure 3. The 28 alternating chains in .

For two particular types of posets, we wish to determine the combinatorics of in more detail.

Example 3 (Dimension- posets).

For , let be an ordered sequence of distinct numbers. We may define a partial order on by if and . Following Dushnik and Miller [8], these are, up to isomorphism, exactly the posets of order dimension . A chain in is a sequence with . Thus, chains in are in bijection to increasing subsequences of . Conversely, one checks that filters (via their minimal elements) are in bijection to decreasing subsequences. It follows from Theorem 2.7 that facets and vertices of are in 2-to-1 correspondence with increasing and decreasing sequences, respectively.

Example 4 (Plane posets).

Let be a compatible double poset. We may assume that are labelled such that for or implies . By [10, Prop. 15], is a plane poset, if and only if there is a sequence of distinct numbers such that for

This is to say, is canonically isomorphic to and is canonically isomorphic to . It follows that alternating chains in are in bijection to alternating sequences. That is, sequences such that . Hence, by Theorem 2.7, the facets of are in bijection to alternating sequences of whereas the vertices are in bijection to increasing and decreasing sequences of .

As a consequence of the proof of Theorem 2.7 we can determine a facet-defining inequality description of double order polytopes. For an alternating chain as in (4), let us write if the last relation in is .

Corollary 2.8.

Let be a compatible double poset. Then is the set of points such that

for all alternating chains of .

Proof.

Note that is in the interior of . Hence by Theorem 2.7 every facet-defining halfspace of is of the form for some alternating chain and with . If is an alternating chain with , then the maximal value of over is and over . The values are exchanged for . It then follows that and . ∎

With this, we can characterize the -level polytopes among compatible double order polytopes.

Corollary 2.9.

Let