Enriched order polytopes and enriched Hibi rings

Abstract

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.

Introduction

A lattice polytope in \({\mathbb {R}}^n\) is a convex polytope all of whose vertices are in \({\mathbb {Z}}^n\). In [18], Stanley introduced a class of lattice polytopes associated to finite partially ordered sets (poset for short). Let \((P, <_P )\) be a finite poset on . The order polytope of P is the convex polytope consisting of the set of points \((x_1,\ldots ,x_n) \in {\mathbb {R}}^n\) such that

  • \(0 \leqslant x_i \leqslant 1\) for \(1 \leqslant i \leqslant n\),

  • if .

Then is a lattice polytope of dimension n. In fact, each vertex of corresponds to a filter of P. Here, a subset F of P is called a filter of P if \(i \in F\) and \(j \in P\) together with guarantee \(j \in F\). For a subset , we define the (0, 1)-vector , where \({\mathbf {e}}_1,\ldots ,{\mathbf {e}}_n\) are the canonical unit coordinate vectors of \({\mathbb {R}}^n\). Then one has , where is the set of filters of P. Moreover, there is a close interplay between the combinatorial structure of P and the geometric structure of . Assume that P is naturally labeled, i.e., \(i < j\) if . Let \({\mathbb {Z}}_{\geqslant 0}\) be the set of nonnegative integers. A map \(f :P \rightarrow {\mathbb {Z}}_{\geqslant 0}\) is called a P-partition if for all \(x,y \in P\) with , f satisfies \(f(x) \leqslant f(y)\). We identify a P-partition f with a lattice point \((f(1),\ldots ,f(n)) \in {\mathbb {Z}}^n\). Since every P-partition \(f :P \rightarrow {\mathbb {Z}}_{\geqslant 0}\) with \(f(i) \leqslant 1\) is a filter of P, the set of P-partitions \(f :P \rightarrow {\mathbb {Z}}_{\geqslant 0}\) with \(f(i) \leqslant 1\) coincides with . Moreover, the set of P-partitions \(f :P \rightarrow {\mathbb {Z}}_{\geqslant 0}\) with \(f(i) \leqslant m\) coincides with for \(0 < m \in {\mathbb {Z}}\). Here, for a convex polytope , is the m-th dilated polytope.

In the present paper, we define a new class of lattice polytopes associated to finite posets from a viewpoint of the theory of enriched P-partitions. For a filter F of P, we set and , where is the set of minimal elements of F. For a subset and a vector \(\varepsilon =(\varepsilon _1,\ldots ,\varepsilon _r) \in \{-1,1\}^{r}\), we define the \((-1,0,1)\)-vector . The enriched order polytope of a finite (not necessarily naturally labeled) poset P on is the lattice polytope of dimension n which is the convex hull of

(1)

Then coincides with the set (1) above, Lemma 4.1. Now, we discuss a relation between and the theory of enriched P-partitions. Again, we assume that P is naturally labeled. A map is called an enriched P-partition [19] if, for all \(x, y \in P\) with , f satisfies

  • \(|f(x)| \leqslant |f(y)|\);

  • \(|f(x)| = |f(y)| \, \Rightarrow f(y) > 0\).

On the other hand, Petersen [17] introduced a slightly different notion “left enriched P-partitions” as follows. A map \(f:P \rightarrow {\mathbb {Z}}\) is called a left enriched P-partition if, for all \(x, y \in P\) with , f satisfies the following conditions:

  1. (i)

    \(|f(x)| \leqslant |f(y)|\);

  2. (ii)

    \(|f(x)| = |f(y)| \, \Rightarrow f(y) \geqslant 0\).

Then the set of left enriched P-partitions \(f:P \rightarrow {\mathbb {Z}}\) with \(|f(i)| \leqslant 1\) coincides with . Contrary to the case of order polytopes, the set of left enriched P-partitions \(f :P \rightarrow {\mathbb {Z}}\) with \(|f(i)| \leqslant m\) does not always coincide with the set of lattice points for \(m >1\), Example 4.2. However, we will show that the number \(\Omega _P^{(\ell )}(m)\) of left enriched P-partitions \(f:P \rightarrow {\mathbb {Z}}\) with \(|f(i)| \leqslant m\) is equal to . Namely,

Theorem 1.1

For a naturally labeled finite poset P on , let

be the Ehrhart polynomial of , and let \(\Omega _P^{(\ell )}(m)\) be the left enriched order polynomial of P. Then one has

where \({\overline{P}}\) is the dual poset of P.

In this paper, in order to show Theorem 1.1, we investigate the toric ring of the enriched order polytope . In [5], Hibi studied the toric ring of the order polytope . The toric ideal possesses a squarefree quadratic Gröbner basis, that is a Gröbner basis consisting of binomials whose initial monomials are squarefree and of degree 2. This implies that the toric ring is a normal Cohen–Macaulay domain and Koszul. In particular, possesses a flag regular unimodular triangulation. The toric ring is called the Hibi ring of P. See [4, Chapter 6]. We call the toric ring the enriched Hibi ring of P.

Theorem 1.2

Let P be a finite poset on . Then is a reflexive polytope with a flag regular unimodular triangulation. Moreover, the toric ring is normal, Gorenstein, and Koszul.

First, in Sect. 2, we introduce known results on two poset polytopes introduced by Stanley [18], that is, the order polytope and the chain polytope of a poset P. A squarefree quadratic Gröbner basis of the toric ideal of each of and its applications will be extended to “enriched case” in the following sections. In Sect. 3, we study the notion of enriched chain polytopes [16] because we need to compare the toric ideals of enriched order polytopes and that of enriched chain polytopes in order to prove Theorem 1.1. In Sect. 4, we discuss fundamental properties of enriched order polytopes. In Sect. 5, we study the toric ideals of enriched order polytopes and their applications. By proving that the toric ideal of possesses a squarefree quadratic Gröbner basis consisting of binomials whose initial monomials do not contain the variable corresponding to the origin, we show Theorem 1.2 (Corollary 5.3). Moreover, by comparing the initial ideals of toric ideals of two enriched poset polytopes, Theorem 5.4, we will complete the proof of Theorem 1.1. Note that Theorem 1.1 implies the existence of a bijection between and . In Sect. 6, towards such a bijection, we consider an elementary geometric property, the facet representations of enriched order and chain polytopes, Proposition 6.1 and Theorem 6.2. The number of facets is discussed in Corollary 6.3 and Proposition 6.5. Finally, we show that is rarely unimodularly equivalent to , Proposition 6.6.

Two poset polytopes

In this section, we review properties of order polytopes and chain polytopes. Let \({(P,<_P )}\) be a finite poset on . Recall that the order polytope is the convex hull of

In [18], Stanley introduced another lattice polytope associated to P as well as the order polytope . An antichain of P is a subset of P consisting of pairwise incomparable elements of P. Let denote the set of antichains of P. Note that the empty set \(\varnothing \) is an antichain of P. The chain polytope of P is the convex hull of

Then is a lattice polytope of dimension n. The order polytope and the chain polytope have similar properties.

First, we study the Ehrhart polynomials of and . Let be a general lattice polytope of dimension n. Given a positive integer m, we define

The study on originated in Ehrhart [3] who proved that is a polynomial in m of degree n with the constant term 1. Moreover, the leading coefficient of coincides with the usual Euclidean volume of . We say that is the Ehrhart polynomial of . An Ehrhart polynomial often coincides with a counting function of a combinatorial object. A map \(f :P \rightarrow {\mathbb {Z}}_{\geqslant 0}\) is called an order preserving map if for all \(x,y \in P\) with , f satisfies \(f(x) \leqslant f(y)\). Let \(\Omega _P(m)\) denote the number of order preserving maps \(f :P \rightarrow {\mathbb {Z}}_{\geqslant 0}\) with \(f(i) \leqslant m\). Then \(\Omega _P(m)\) is a polynomial in m of degree n and called the order polynomial of P. Stanley showed a relation between the Ehrhart polynomials of and and the order polynomial \(\Omega _P(m)\). In fact,

Proposition 2.1

([18, Theorem 4.1]) Let P be a finite poset on . Then one has

On the other hand, and are not always unimodularly equivalent. Here, two lattice polytopes of dimension n are unimodularly equivalent if there exist a unimodular matrix \(U \in {\mathbb {Z}}^{n \times n}\) and a lattice point \({\mathbf {w}}\in {\mathbb {Z}}^n\) such that , where \(f_U\) is the linear transformation in \({\mathbb {R}}^n\) defined by U, i.e., \(f_U({\mathbf {x}})={\mathbf {x}}U\) for all \({\mathbf {x}}\in {\mathbb {R}}^n\). In [7], Hibi and Li characterized when and are unimodularly equivalent. In fact,

Proposition 2.2

([7, Corollary 2.3]) Let P be a finite poset on . Then the following conditions are equivalent:

  1. (i)

    The order polytope and the chain polytope are unimodularly equivalent.

  2. (ii)

    The number of the facets of is equal to that of .

  3. (iii)

    The following poset is not a subposet of P.

    figurea

Next, we review the toric ideals of order polytopes and chain polytopes. First, we recall basic materials and notation on toric ideals. Let be the Laurent polynomial ring in \(n+1\) variables over a field K. If \({\mathbf {a}}= (a_{1}, \ldots , a_{n}) \in {\mathbb {Z}}^{n}\), then \(\mathbf{t}^{{\mathbf {a}}}s\) is the Laurent monomial . Let be a lattice polytope and . Then, the toric ring of is the subalgebra of generated by over K. We regard as a homogeneous algebra by setting each \(\deg {\mathbf {t}}^{{\mathbf {a}}_i}s=1\). Let denote the polynomial ring in d variables over K with each . The toric ideal of is the kernel of the surjective homomorphism defined by \(\pi (x_i)={\mathbf {t}}^{{\mathbf {a}}_i}s\) for \(1 \leqslant i \leqslant d\). It is known that is generated by homogeneous binomials. See, e.g., [4, 20].

Now, we study the toric ideals of and . Remark that and are unimodularly equivalent and . A subset I of P is called a poset ideal of P if \(i \in I\) and \(j \in P\) together with guarantee \(j \in I\). Let denote the set of poset ideals of P, ordered by inclusion. If and are incomparable in , then we write \(I \not \sim J\). Then the order polytope is the convex hull of

Let denote the polynomial ring over K in variables \(x_I\), where . In particular, the origin corresponds to the variable \(x_{\varnothing }\). Then the toric ideal is the kernel of the ring homomorphism defined by . Let be a reverse lexicographic order on such that \(x_I <_{P} x_J\) if \(I \subsetneq J\). In [5], Hibi essentially proved that possesses a squarefree quadratic Gröbner basis. In fact,

Proposition 2.3

([5]) Work with the same notation as above. Then

is a Gröbner basis of with respect to a reverse lexicographic order . Moreover, is a normal Cohen–Macaulay domain and Koszul.

Recently, the toric ring is called the Hibi ring of P and studied by many authors from several viewpoints. One can find some of them in [4, Note of Chapter 6].

For a poset ideal I of P, we denote by the set of maximal elements of I. Then is an antichain of P and every antichain of P is the set of maximal elements of a poset ideal. On the other hand, for an antichain A of P, the poset ideal of P generated by A is the smallest poset ideal of P which contains A. Every poset ideal of P can be obtained by this way. Hence and have a one-to-one correspondence. Let denote the polynomial ring over K in variables \(x_{\max (I)}\), where . Then the toric ideal is the kernel of the ring homomorphism defined by \(\pi _{{\mathscr {C}}}(x_{\max (I)})=s \prod _{i \in \max (I)} t_i\). Let be a reverse lexicographic order on such that if \(I \subsetneq J\). Given poset ideals , let denote the poset ideal of P generated by . Note that . The following lemma is fundamental and important.

Lemma 2.4

Let P be a finite poset and . For , the following conditions are equivalent:

  1. (i)

    \(p \in J\);

  2. (ii)

    ;

  3. (iii)

    ;

  4. (iv)

    .

Proof

First, (ii) \(\Rightarrow \) (i) is trivial. Suppose \(p \in J\). Since p does not belong to , p is not a maximal element in J. Hence we have . Thus (i) \(\Rightarrow \) (iv) holds. Suppose . Then there exists an element \(q \in I \cup J\) such that . If q belongs to I, then \(p \notin \), a contradiction. Thus \(q \in J\), and hence \(p \in I \cap J\). If holds, then there exists an element such that . This contradicts to the hypothesis . Thus (iv) \(\Rightarrow \) (ii) holds. Finally, we have (ii) \(\Leftrightarrow \) (iii) by .\(\square \)

In [6], Hibi and Li essentially proved that possesses a squarefree quadratic Gröbner basis. In fact,

Proposition 2.5

([6]) Work with the same notation as above. Then

is a Gröbner basis of with respect to a reverse lexicographic order . Moreover, is a normal Cohen–Macaulay domain and Koszul.

From Propositions 2.3 and 2.5 we can prove the following.

Proposition 2.6

Work with the same notation as above. Then one has

Furthermore, we obtain .

Proof

From Propositions 2.3 and 2.5, we have

Hence it follows that the map \(x_{I} \mapsto x_{\max (I)}\) induces an isomorphism from to . Therefore, the first claim follows.

Since both and are squarefree, both and possess a unimodular triangulation, and hence the Ehrhart polynomial coincides with the Hilbert polynomial of its toric ring for each of and , see [4, Section 4.2] or [20, Chapters 8 and 13]. Moreover, for an ideal I of \(K[{\mathbf {x}}]\) and a monomial order < on \(K[{\mathbf {x}}]\), the Hilbert polynomial of \(K[{\mathbf {x}}]/I\) is equal to that of \(K[{\mathbf {x}}]/\mathrm{in}_{<}(I)\). Therefore, the second claim follows.\(\square \)

Enriched chain polytopes

In this section, we recall the definition and properties of enriched chain polytopes given in [16]. Let \((P, <_P )\) be a finite poset on . The enriched chain polytope of P is the convex hull of

Then is a lattice polytope of dimension n. It is easy to see that is centrally symmetric (i.e., for any facet of , is also a facet of ), and the origin of \({\mathbb {R}}^n\) is the unique interior lattice point of . Remark that .

A lattice polytope of dimension n is called reflexive if the origin of \({\mathbb {R}}^n\) is a unique lattice point belonging to the interior of and its dual polytope

is also a lattice polytope, where \(\langle {\mathbf {x}},{\mathbf {y}}\rangle \) is the usual inner product of \({\mathbb {R}}^n\). It is known that reflexive polytopes correspond to Gorenstein toric Fano varieties, and they are related to mirror symmetry, see, e.g., [1, 2]. In each dimension there exist only finitely many reflexive polytopes up to unimodular equivalence [13] and all of them are known up to dimension 4 [12]. Recently, several classes of reflexive polytopes were constructed by an algebraic technique on Gröbner bases, cf., [10, 11, 15]. The algebraic technique is based on the following lemma that follows from the argument in [9, Proof of Lemma 1.1].

Lemma 3.1

Let be a lattice polytope of dimension n such that the origin of \({\mathbb {R}}^n\) is contained in its interior. Suppose that any lattice point in \({\mathbb {Z}}^n\) is a linear integer combination of the lattice points in . If there exists a monomial order such that the initial ideal of is generated by squarefree monomials which do not contain the variable corresponding to the origin, then is reflexive and has a regular unimodular triangulation. Moreover, is a normal Gorenstein domain.

In order to use Lemma 3.1 for enriched chain polytopes , we study the toric ideal of . Let denote the polynomial ring over K in variables \(x_{A}^{\varepsilon }\), where and \(\varepsilon =(\varepsilon _1,\ldots ,\varepsilon _n) \in \{-1,0,1\}^{n}\) with

$$\begin{aligned} |\varepsilon _i|= {\left\{ \begin{array}{ll} \,1, &{} i \in A,\\ \,0, &{} i \notin A. \end{array}\right. } \end{aligned}$$

Then the toric ideal is the kernel of a ring homomorphism defined by . In addition,

is the toric ideal . For \(\varepsilon =(\varepsilon _1,\ldots ,\varepsilon _n) \in \{-1,0,1\}^n\), we write . We identify the variable \(x^{\varepsilon ^{+}}_A\) on with the variable \(x_A\) on . It is known [20, Proposition 1.11] that there exists a nonnegative weight vector such that . Then we define the weight vector on such that the weight of each variable \(x_A^\varepsilon \) with respect to is the weight of the variable \(x_A^{\varepsilon ^+}\) with respect to . In addition, let \({\mathbf {w}}_{\#}\) be the weight vector on such that the weight of each variable \(x_A^\varepsilon \) with respect to \({\mathbf {w}}_{\#}\) is |A|. Fix any monomial order \(\prec \) on as a tie-breaker. Let be a monomial order on such that if and only if one of the following holds:

  • The weight of u is less than that of v with respect to \({\mathbf {w}}_\#\).

  • The weight of u is the same as that of v with respect to \({\mathbf {w}}_\#\), and the weight of u is less than that of v with respect to .

  • The weight of u is the same as that of v with respect to \({\mathbf {w}}_\#\) and , and \(u \prec v\).

The following proposition was given in [16, Theorem 1.3]:

Proposition 3.2

([16]) Work with the same notation as above. Let be the set of all binomials

where , \(\varepsilon _p\ne \mu _p\), and , together with all binomials

where with \(I \not \sim J\), and

  1. (a)

    for any , we have \(\varepsilon _p =\varepsilon '_p =\mu _p=\mu '_p \);

  2. (b)

    for any , we have

  3. (c)

    for any , we have

Then is a Gröbner basis of with respect to a monomial order . The initial monomial of each binomial is the first monomial. In particular, the initial ideal is generated by squarefree quadratic monomials which do not contain the variable \(x_\varnothing ^\mathbf{0}\).

By Lemma 3.1 and Proposition 3.2, we have the following immediately.

Corollary 3.3

([16]) Let P be a finite poset on . Then is a reflexive polytope with a flag regular unimodular triangulation. Moreover, is a normal Gorenstein domain and Koszul.

Next, we study Ehrhart polynomials of enriched chain polytopes. Assume that P is naturally labeled. Let \(\Omega ^{(\ell )}_P(m)\) denote the number of left enriched P-partitions \(f :P \rightarrow {\mathbb {Z}}\) with \(|f(i)| \leqslant m\). Then \(\Omega ^{(\ell )}_P(m)\) is a polynomial in m of degree n and called the left enriched order polynomial of P.

Proposition 3.4

([16, Theorem 0.2]) Let P be a naturally labeled finite poset on . Then one has

Fundamental properties of enriched order polytopes

In this section, we discuss some fundamental properties of enriched order polytopes. First, we consider the set of lattice points in enriched order polytopes.

Lemma 4.1

Let P be a finite poset on . Then one has

In addition, the origin is the unique interior lattice point in .

Proof

Let . It is enough to show that . Let . Since is the convex hull of X, there exist \({\mathbf {a}}_1,\dots , {\mathbf {a}}_s \in X\) such that \({\mathbf {x}}= \sum _{i=1}^s \lambda _i {\mathbf {a}}_i \), where \(\lambda _i > 0 \), \( \sum _{i=1}^s \lambda _i =1\). Then each \({\mathbf {a}}_i\) is a \((-1,0,1)\)-vector, and hence so is \({\mathbf {x}}\). It is easy to see that \(x_k = 1\) (resp. \(x_k = - 1\)) if and only if k-th component of \({\mathbf {a}}_i\) is equal to 1 (resp. \(-1\)) for all \(i = 1,2,\ldots ,s\). Suppose that . If \(x_k = 0\), then \(|x_k| \leqslant |x_\ell |\) and the equality holds if and only if \(x_\ell =0\). Suppose that \(|x_k| =1\). Then k-th component of \({\mathbf {a}}_i\) is equal to \(x_k \) for all \(i = 1,2,\ldots ,s\). Since each \({\mathbf {a}}_i\) is a left enriched P-partition, \(\ell \)-th component of \({\mathbf {a}}_i\) is equal to 1 for all \(i = 1,2,\ldots ,s\). Hence \(x_\ell = 1\). In particular, \(|x_k| = |x_\ell |\) and \(x_\ell \geqslant 0\). Thus \({\mathbf {x}}\) is a left enriched P-partition, that is, \({\mathbf {x}}\) belongs to X.

Since is an n-dimensional subpolytope of a cube \([-1,1]^n\), it follows that each nonzero vector \({\mathbf {x}}\in X\) belongs to the boundary of . Suppose that the origin \(\mathbf{0} \in {\mathbb {R}}^n\) belongs to the boundary of . Then there exists a facet of which contains \(\mathbf{0}\). Let with \(\mathbf{0} \ne {\mathbf {a}}= (a_1,\dots , a_n) \in {\mathbb {R}}^n\) be the supporting hyperplane of and let \((\ne \varnothing )\) be a subposet of P. We may assume that satisfies \(a_i >0\). Let be a filter of P. Then and hence satisfies \( \langle {\mathbf {a}},{\mathbf {y}}\rangle = a_i > 0\) and satisfies \( \langle {\mathbf {a}},{\mathbf {y}}' \rangle = - a_i < 0\). This contradicts that \({{\mathscr {H}}}\) is a supporting hyperplane of .\(\square \)

Next, we consider lattice points in the dilated polytopes of an enriched order polytope. The following example shows that, contrary to the case of order polytopes, the set of left enriched P-partitions \(f :P \rightarrow {\mathbb {Z}}\) with \(|f(i)| \leqslant m\) does not always coincide with the set of lattice points if \(m >1\).

Example 4.2

Let P be a poset on \(\{1,2\}\) with . Then the set of left enriched P-partitions \(f :P \rightarrow {\mathbb {Z}}\) with \(|f(i)| \leqslant 2\) is

$$\begin{aligned} \{ (0,0), (0, \pm 1), (0, \pm 2), (\pm 1, 1), (\pm 1, \pm 2), (\pm 2, 2) \}, \end{aligned}$$

and

Thus two sets are different. On the other hand, the cardinality of each set is the same. Moreover, it follows that .

The toric ideals of enriched order polytopes

In this section, we discuss the toric ideals of enriched order polytopes. Let P be a finite poset on . For a poset ideal I of P, we set and . Then lattice points in can be written by poset ideals of P:

Contrary to the case of order polytopes, the enriched order polytopes and are not always unimodularly equivalent.

Example 5.1

Let P be the following poset on \(\{1,2,3\}\):

figureb

Then has five facets and has six facets. Thus and are not unimodularly equivalent. On the other hand, it follows that

Now, we consider the toric ideals . Let be the polynomial ring over K in variables \(x_I^{\varepsilon }\), where and \(\varepsilon =(\varepsilon _1,\ldots ,\varepsilon _n) \in \{-1,0,1\}^{n}\) with

Then the toric ideal is the kernel of a ring homomorphism defined by . In addition,

is the toric ideal . We define a reverse lexicographic order on such that if \(I \subsetneq J\).

Theorem 5.2

Work with the same notation as above. Let be the set of all binomials

(2)

where , \(\varepsilon _p \ne \mu _p\), and , together with all binomials

(3)

where with \(I \not \sim J\), and

  1. (a)

    for any , we have \(\varepsilon _p =\varepsilon '_p =\mu _p=\mu '_p\);

  2. (b)

    for any , we have

  3. (c)

    for any , we have

Then is a Gröbner basis of with respect to a monomial order . The initial monomial of each binomial is the first monomial. In particular, the initial ideal is generated by squarefree quadratic monomials which do not contain the variable \(x_\varnothing ^\mathbf{0}\).

Proof

It is easy to see that any binomial of type (2) belongs to . By Lemma 2.4, it follows that any binomial of type (3) belongs to . Hence is a subset of . Moreover, the initial monomial of each binomial is the first monomial. Assume that is not a Gröbner basis of with respect to . Let

By [4, Theorem 3.11], there exists a nonzero irreducible homogeneous binomial such that neither u nor v belongs to . For and \(\varepsilon , \mu \in \{-1,0,1\}^n\), if satisfies \(\varepsilon _i \ne \mu _i\), then . On the other hand, for with \(I \not \sim J\) and for \(\varepsilon , \mu \in \{-1,0,1\}^n\), if \(\varepsilon _p=\mu _p\) for any , then . Hence u and v are of the form

where and for \(k =1,2,\dots , r\) such that

  1. (a)

    \(I_1 \subset \dots \subset I_r\) and \(J_1 \subset \dots \subset J_r\).

  2. (b)

    For any p and q, and for any , we obtain .

  3. (c)

    For any p and q, and for any , we obtain .

Since u and v satisfy conditions (b) and (c) and since f belongs to , it then follows that and \(\varepsilon _r=\mu _r\). Hence one has . This contradicts the assumption that f is irreducible.\(\square \)

By Lemma 3.1 and Theorem 5.2, we have the following immediately.

Corollary 5.3

Let P be a finite poset on . Then is a reflexive polytope with a flag regular unimodular triangulation. Moreover, is a normal Gorenstein domain and Koszul.

Theorem 5.4

Work with the same notation as above. Then one has

Furthermore, we obtain

Proof

From Theorem 5.2, is generated by all monomials

where , \(\varepsilon _p\ne \mu _p\), and together with all monomials

where with \(I \not \sim J\) and \(\varepsilon _p =\varepsilon '_p\) for each . Moreover, from Proposition 3.2, is generated by all monomials

where , \(\varepsilon _p\ne \mu _p\), and together with all monomials

where with \(I \not \sim J\) and \(\varepsilon _p =\varepsilon '_p\) for each . Hence it follows that the map , where \(\varepsilon _i' = \varepsilon _i\) for and \(\varepsilon _i' = 0\) for , induces an isomorphism for the first claim. By the argument in the last part of the proof of Proposition 2.6, we have and . Since , the second claim follows.\(\square \)

By Proposition 3.4 and Theorem 5.4, we have Theorem 1.1.

Facets of enriched order polytopes and enriched chain polytopes

Theorem 1.1 implies the existence of a bijection between and . Towards such a bijection, in this section, we consider an elementary geometric property, the facet representations of enriched order polytopes and enriched chain polytopes.

Let P be a finite poset on . Given elements ij of P, we say that j covers i if \(i < j\) and there exists no \(k \in P\) such that \(i< k < j\). If j covers i in P, then we write \(i \lessdot j\). A chain of P is a totally ordered subset of P. A chain of the form \(i_1 \lessdot i_2 \lessdot \cdots \lessdot i_r\) is called a saturated chain. A saturated chain \(i_1 \lessdot i_2 \lessdot \cdots \lessdot i_r\) is called maximal if and . First, we give the facet representations of enriched chain polytopes which easily follows from [16, Lemma 1.1] and the facet representations of chain polytopes [18].

Proposition 6.1

Let P be a finite poset on . Then is the solution set of the linear inequalities

where \(i_1 \lessdot i_2 \lessdot \cdots \lessdot i_r\) is a maximal chain of P, and . In addition, each of the above inequalities is facet defining.

On the other hand, the facet representations of enriched order polytopes are as follows.

Theorem 6.2

Let P be a finite poset on . Then is the solution set of the following linear inequalities:

  1. (a)

    , where \(i_1 \lessdot i_2 \lessdot \cdots \lessdot i_r\) is a saturated chain of P with ;

  2. (b)

    , where \(i_1 \lessdot i_2 \lessdot \cdots \lessdot i_r\) is a maximal chain of P.

In addition, each of the above inequalities is facet defining.

Proof

The proof is induction on n. If \(n=1\), then the assertion is trivial. Assume \(n \geqslant 2\).

Let \({{\mathscr {Q}}} \subset {\mathbb {R}}^n\) be the solution set of the above linear inequalities. Since \(2^{s-1} - \sum _{j=2}^{s} 2^{s-j} = 1\) holds for any positive integer s, it is easy to see that satisfies (a) and (b) for any filter F of P, and for any \(\varepsilon \in \{-1,1\}^{|F_{\min }|}\). Since is the convex hull of such vectors, we have . In order to prove , let \(\mathbf{x} = (x_1, \dots , x_n) \in {{\mathscr {Q}}}\). First, we will show that \(|x_i| \leqslant 1\) for each . Let \(i= i_1 \lessdot i_2 \lessdot \cdots \lessdot i_r\) be a saturated chain of P with . Then \(\mathbf{x}\) satisfies the following r inequalities:

figurec

If \(r=1\), then \(x_i \leqslant 1\) is trivial. Let \(r \geqslant 2\). Then the inequality given by a linear combination of the above inequalities is \(2^{r-1} x_{i_1} \leqslant 2^{r-1}\), and hence . Suppose that i belongs to a maximal chain \(i_1 \lessdot i_2 \lessdot \cdots \lessdot i_r\), say, \(i = i_k\). Then \(\mathbf{x}\) satisfies above and

figured

Then the inequality given by a linear combination

of the above inequalities is , and hence we have .

We now prove that \(\mathbf{x}\) belongs to by induction on n. Suppose that \(x_i = 0\) for some . Then \((x_1,\dots ,x_{i-1},x_{i+1},\dots ,x_n) \in {\mathbb {R}}^{n-1}\) satisfies inequalities (a) and (b) for the subposet of P. By the assumption of induction, \((x_1,\dots ,x_{i-1},x_{i+1},\dots ,x_n)\) belongs to . It then follows that \(\mathbf{x}\) belongs to . Thus we may assume that \(x_i \ne 0\) for any . Let . Note that \(0 < \lambda \leqslant 1\). Let

where , and \(\varepsilon \in \{-1,1\}^{|F_{\min } |}\) corresponds to the sign of \(x_i\) for each . We now show that the vector \(\mathbf{y}\) satisfies

  1. (c)

    , where \(i_1 \lessdot i_2 \lessdot \cdots \lessdot i_r\) is a saturated chain of P with ;

  2. (d)

    , where \(i_1 \lessdot i_2 \lessdot \cdots \lessdot i_r\) is a maximal chain of P.

Inequality (c). If either or holds, then

If and , then and hence

Inequality (d). If , then we have

If , then and hence

If \(\lambda =1\), then we have \(\mathbf{y}=\mathbf{0}\) by inequalities (c) and (d). Hence . If \(\lambda \ne 1\), then \(\frac{1}{1-\lambda }\, \mathbf{y}\) belongs to \({{\mathscr {Q}}}\) by inequalities (c) and (d). From the definition of \(\lambda \), there exists such that \(y_i=0\). By the assumption of induction, \(\frac{1}{1-\lambda }\, \mathbf{y}\) belongs to , and hence \(\mathbf{y}\) belongs to . Thus belongs to .

Finally, we will prove that each of inequalities (a) and (b) is facet defining. Let

where \(i_1 \lessdot i_2 \lessdot \cdots \lessdot i_r\) is a saturated chain of P with , and let

where \(i_1 \lessdot i_2 \lessdot \cdots \lessdot i_r\) is a maximal chain of P. It is enough to show that

Let \(i_1 \lessdot i_2 \lessdot \cdots \lessdot i_r\) be a saturated chain of P with . If , then let \(i = i_1\). If , then let i be an arbitrary element in . Note that, if , then \(i_2 \lessdot i_3 \lessdot \cdots \lessdot i_r\) is a maximal chain of . Let . Then

is unimodularly equivalent to a facet of by the assumption of induction. Hence . Since \((1,\dots ,1) \in {\mathbb {R}}^n\) belongs to , we have . On the other hand, for a maximal chain \(i_1 \lessdot i_2 \lessdot \cdots \lessdot i_r\) of P, let \(i = i_1\) if , and let i be an arbitrary element in otherwise. Note that, if , then \(i_2 \lessdot i_3 \lessdot \cdots \lessdot i_r\) is a maximal chain of . Then

is unimodularly equivalent to a facet of by the assumption of induction. Hence . Since \((1,\dots ,1) - 2 \mathbf{e}_i \in {\mathbb {R}}^n\) belongs to , we have as desired.\(\square \)

Given a polytope of dimension n, let \(f_{n-1} ({\mathscr {P}})\) be the number of the facets of . It is known [7, Corollary 1.2] that for any poset P.

Corollary 6.3

Let P be a finite poset on . Then we have the following:

  1. (a)

    Let (resp. ) be the number of saturated (resp. maximal) chains of P that contains a maximal element of P. Then .

  2. (b)

    Let \(\mathrm{mc}_\ell (P)\) be the number of maximal chains of P of length \(\ell \). Then .

Moreover, we have .

Proof

The formulas of the number of facets follows from Proposition 6.1 and Theorem 6.2. Each maximal chain of P of length \(\ell \) contains exactly \(\ell +1\) saturated chains of P that contains a maximal element of P. Since \(\ell + 2 \leqslant 2^{\ell +1}\) for any integer \(\ell \geqslant 0\), we have .\(\square \)

In [8, Lemma 3.8], tight upper bounds for and are given. Given an integer \(n\geqslant 2\), let

It is known [14, Theorem 1] that \(\mu _n\) is the maximum number of cliques possible in a graph with n vertices.

Proposition 6.4

([8, Lemma 3.8]) Let P be a finite poset on with \(n \geqslant 5\). Then we have , and In addition, both upper bounds are tight.

We give tight upper bounds for the number of facets of enriched order and chain polytopes.

Proposition 6.5

Let P be a finite poset on . Then we have and

In addition, both upper bounds are tight.

Proof

The proof for is induction on n. If \(n=1\), then has two facets. Let \(n \geqslant 2\) and let M be the set of all minimal elements of P. If \(|M| = m\), then we have

by the assumption of induction. Note that if P is a chain.

By explicit computation, for \(n=1, 2,3,4\), the maximum value of the number of facets of is 2, 4, 6, 10, respectively. (Note that if P is an antichain.) Thus the assertion for holds for \(n \leqslant 4\). Assume \(n \geqslant 5\). Let P be a poset on . Let \(P_1 = P\) and let \(M_1\) be the set of all maximal elements of \(P_1\). If \(P_1\) is not an antichain, then let and let \(M_2\) be the set of all maximal elements of \(P_2\). In general, if \(P_i\) is not an antichain, then and let \(M_{i+1}\) be the set of all maximal elements of \(P_{i+1}\). By this procedure, we get a sequence of posets \(P_1, \dots , P_r\) such that \(P_r\) is an antichain. Then we have

We show that

(4)

is equal to

$$\begin{aligned} {\left\{ \begin{array}{ll} \,\frac{47}{2} \cdot 3^{k-2} -\frac{3}{2} &{} \text{ if }\;\; n = 3k,\\ \,\frac{23}{2} \cdot 3^{k-1} -\frac{3}{2} &{} \text{ if } \;\; n = 3k+1,\\ \,\frac{11}{2} \cdot 3^k -\frac{3}{2}&{} \text{ if } \;\; n = 3k+2, \end{array}\right. } \end{aligned}$$

for \(n \geqslant 5\). Suppose that \(m_1, \dots , m_r\), where \(1 \leqslant r \leqslant n\), , and \(1 \leqslant m_i \in {\mathbb {Z}}\) give the maximum value of (4). If \(m_i < m_{i+1}\) for some i, then

$$\begin{aligned} 2 m_1 m_2 \cdots m_r + \sum _{j=1}^{r-1} \prod _{k=1}^j m_k < 2 m_1' m_2' \cdots m_r' + \sum _{j=1}^{r-1} \prod _{k=1}^j m_k', \end{aligned}$$

where \((m_i',m_{i+1}') = (m_{i+1},m_i)\) and \(m_k' = m_k\) if . This is a contradiction. Hence we have \(m_1 \geqslant m_2 \geqslant \cdots \geqslant m_r\). If \(m_1 \geqslant 4\), then

$$\begin{aligned} m_1 \leqslant \biggl \lfloor \frac{m_1+1}{2} \biggr \rfloor \biggl ( m_1- \biggl \lfloor \frac{m_1+1}{2} \biggr \rfloor \biggr ). \end{aligned}$$

Hence

$$\begin{aligned} 2 m_1 m_2 \cdots m_r + \sum _{j=1}^{r-1} \prod _{k=1}^j m_k < 2 m_0' m_1' \cdots m_r' + \sum _{j=0}^{r-1} \prod _{k=0}^j m_k', \end{aligned}$$

where \(m_0' = \lfloor ({m_1+1})/{2} \rfloor \), \(m_1' = m_1-m_0'\) and \(m_k' = m_k\) if \(k \notin \{0, 1\}\). This is a contradiction. Thus we have \(m_1 \leqslant 3\). It is easy to see that \(m_r \ne 1\). Therefore

$$\begin{aligned} 3 \geqslant m_1 \geqslant m_2 \geqslant \cdots \geqslant m_r \geqslant 2 . \end{aligned}$$

Since \(2 + 2 + 2 + 2 = 3 + 3 +2\) and , there are at most three \(m_i\)’s that are equal to 2. If \(n = 3k+1\), then \(m_1 = \cdots = m_{r-2} =3\) and \(m_{r-1} = m_r=2\). If \(n = 3k+2\), then \(m_1 = \cdots = m_{r-1} =3\) and \(m_r=2\). If \(n = 3k \geqslant 6\), then there are two possibilities:

$$\begin{aligned}&m_1 = \cdots = m_{r-3} =3, \quad \text{ and }\quad m_{r-2} = m_{r-1} = m_r=2, \nonumber \\&\qquad m_1 = \cdots = m_r=3. \end{aligned}$$
(5)

Since , it follows that \(m_1,\dots ,m_r\) satisfies (5).

Thus the maximum value is equal to

A poset that attains the maximum value is the ordinal sum of antichains \(A_1,\dots ,A_r\) such that \(|A_i| = m_i\).\(\square \)

Finally, we discuss when the numbers of facets of and coincide.

Proposition 6.6

Let P be a finite poset on . Then the following conditions are equivalent:

  1. (i)

    P is an antichain.

  2. (ii)

    and are unimodularly equivalent.

  3. (iii)

    is centrally symmetric.

  4. (iv)

    The number of the facets of is equal to that of .

Proof

First, (ii) \(\Rightarrow \) (iv) is trivial.

(ii) \(\Rightarrow \) (iii): Note that is always centrally symmetric, and that the origin is the unique interior lattice point in each of and . Hence if and are unimodularly equivalent, then is also centrally symmetric.

(iii) \(\Rightarrow \) (i): Assume that is centrally symmetric. Then since , one has . By the definition of , this implies that each element of P is a minimal element of P. Hence P is an antichain.

(i) \(\Rightarrow \) (ii): If P is an antichain, then we have .

(iv) \(\Rightarrow \) (i): Suppose that the number of the facets of is equal to that of . By the argument in the proof of Corollary 6.3, each maximal chain of P of length \(\ell \) must satisfy \(\ell + 2 = 2^{\ell +1}\), and hence \(\ell = 0\). Thus P is an antichain.\(\square \)

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Acknowledgements

The authors are grateful to an anonymous referee for his/her useful comments. In particular, the last section was added following referee’s advice.

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Ohsugi, H., Tsuchiya, A. Enriched order polytopes and enriched Hibi rings. European Journal of Mathematics 7, 48–68 (2021). https://doi.org/10.1007/s40879-020-00403-2

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Keywords

  • Reflexive polytope
  • Flag triangulation
  • Left enriched partition
  • Left enriched order polynomial
  • Gröbner basis
  • Toric ideal

Mathematics Subject Classification

  • 05A15
  • 13P10
  • 52B12
  • 52B20