Enriched order polytopes and enriched Hibi rings

Stanley introduced two classes of lattice polytopes associated to posets, which are called the order polytope OP\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathscr {O}}}_P$$\end{document} and the chain polytope CP\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathscr {C}}}_P$$\end{document} of a poset P. It is known that, given a poset P, the Ehrhart polynomials of OP\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathscr {O}}}_P$$\end{document} and CP\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathscr {C}}}_P$$\end{document} 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 R n is a convex polytope all of whose vertices are in 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 [n] . . = {1, . . . , n}. The order polytope O P of P is the convex polytope consisting of the set of points (x 1 , . . . , x n ) ∈ R n such that • 0 x i 1 for 1 i n, Then O P is a lattice polytope of dimension n. In fact, each vertex of O P corresponds to a filter of P. Here, a subset F of P is called a filter of P if i ∈ F and j ∈ P together with i < P j guarantee j ∈ F. For a subset X ⊂ [n], we define the (0, 1)vector e X . . = i∈X e i , where e 1 , . . . , e n are the canonical unit coordinate vectors of R n . Then one has O P ∩ Z n = {e F : F ∈ F(P)}, where F(P) is the set of filters of P. Moreover, there is a close interplay between the combinatorial structure of P and the geometric structure of O P . Assume that P is naturally labeled, i.e., i < j if i < P j. Let Z 0 be the set of nonnegative integers. A map f : P → Z 0 is called a P-partition if for all x, y ∈ P with x < P y, f satisfies f (x) f (y). We identify a P-partition f with a lattice point ( f (1), . . . , f (n)) ∈ Z n . Since every P-partition f : P → Z 0 with f (i) 1 is a filter of P, the set of P-partitions f : P → Z 0 with f (i) 1 coincides with O P ∩ Z n . Moreover, the set of P-partitions f : P → Z 0 with f (i) m coincides with mO P ∩ Z n for 0 < m ∈ Z. Here, for a convex polytope P ⊂ R n , mP . . = {mx : x ∈ P} 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 F min . . = min(F) and F comin . . = F \ F min , where min(F) is the set of minimal elements of F. For a subset X = {i 1 , . . . , i r } ⊂ [n] and a vector ε = (ε 1 , . . . , ε r ) ∈ {−1, 1} r , we define the (−1, 0, 1)-vector e ε X . . = r j=1 ε j e i j . The enriched order polytope O (e) P ⊂ R n of a finite (not necessarily naturally labeled) poset P on [n] is the lattice polytope of dimension n which is the convex hull of e ε F min + e F comin : F ∈ F(P), ε ∈ {−1, 1} |F min | .
Then O (e) P ∩ Z n coincides with the set (1) above, Lemma 4.1. Now, we discuss a relation between O where 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 O (e) P . In [5], Hibi studied the toric ring of the order polytope OP . The toric ideal I OP 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 K [OP ] is a normal Cohen-Macaulay domain and Koszul. In particular, OP possesses a flag regular unimodular triangulation. First, in Sect. 2, we introduce known results on two poset polytopes introduced by Stanley [18], that is, the order polytope O P and the chain polytope C P of a poset P. A squarefree quadratic Gröbner basis of the toric ideal of each of O P , C P and its applications will be extended to "enriched case" in the following sections. In Sect. 3, we study the notion of enriched chain polytopes C (e) P [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 O (e) P 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 mO (e) P ∩ Z n and mC (e) P ∩ Z n . 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 O (e) P is rarely unimodularly equivalent to C (e) P , 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 [n]. Recall that the order polytope O P ⊂ R n is the convex hull of In [18], Stanley introduced another lattice polytope associated to P as well as the order polytope O P . An antichain of P is a subset of P consisting of pairwise incomparable elements of P. Let A(P) denote the set of antichains of P. Note that the empty set ∅ is an antichain of P. The chain polytope C P ⊂ R n of P is the convex hull of Then C P is a lattice polytope of dimension n. The order polytope O P and the chain polytope C P have similar properties.
First, we study the Ehrhart polynomials of O P and C P . Let P ⊂ R n be a general lattice polytope of dimension n. Given a positive integer m, we define The study on L P (m) originated in Ehrhart [3] who proved that L P (m) is a polynomial in m of degree n with the constant term 1. Moreover, the leading coefficient of L P (m) coincides with the usual Euclidean volume of P. We say that L P (m) is the Ehrhart polynomial of P. An Ehrhart polynomial often coincides with a counting function of a combinatorial object. A map f : P → Z 0 is called an order preserving map if for all x, y ∈ P with x < P y, f satisfies f (x) f (y). Let P (m) denote the number of order preserving maps f : P → Z 0 with f (i) m. Then 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 O P and C P and the order polynomial P (m). In fact, On the other hand, O P and C P are not always unimodularly equivalent. Here, two lattice polytopes P, Q ⊂ R n of dimension n are unimodularly equivalent if there exist a unimodular matrix U ∈ Z n×n and a lattice point w ∈ Z n such that Q = f U (P) + w, where f U is the linear transformation in R n defined by U , i.e., f U (x) = xU for all x ∈ R n . In [7], Hibi and Li characterized when O P and C P are unimodularly equivalent. In fact, Next, we review the toric ideals of order polytopes and chain polytopes. First, we recall basic materials and notation on toric ideals. Let n , s] be the Laurent polynomial ring in n + 1 variables over a field K . If a = (a 1 , . . . , a n ) ∈ Z n , then t a s is the Laurent monomial t a 1 1 · · · t a n n s ∈ K [t ±1 , s]. Let P ⊂ R n be a lattice polytope and P ∩ Z n = {a 1 , . . . , a d }. Then, the toric ring of P is the subalgebra K [P] of K [t ±1 , s] generated by {t a 1 s, . . . , t a d s} over K . We regard K [P] as a homogeneous algebra by setting each deg t a i s = 1. Let K [x] = K [x 1 , . . . , x d ] denote the polynomial ring in d variables over K with each deg(x i ) = 1. The toric ideal I P of P is the kernel of the surjective homomorphism π : K [x] → K [P] defined by π(x i ) = t a i s for 1 i d. It is known that I P is generated by homogeneous binomials. See, e.g., [4,20]. Now, we study the toric ideals of OP and CP . Remark that O P and OP are unimodularly equivalent and C P = CP . A subset I of P is called a poset ideal of P if i ∈ I and j ∈ P together with i > P j guarantee j ∈ I . Let J(P) denote the set of poset ideals of P, ordered by inclusion. If I ∈ J(P) and J ∈ J(P) are incomparable in J(P), then we write I J . Then the order polytope OP is the convex hull of For a poset ideal I of P, we denote by max(I ) the set of maximal elements of I . Then max(I ) 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 J(P) and A(P) have a one-to-one correspondence. Let R[C] denote the polynomial ring over K in variables x max(I ) , where I ∈ J(P). Then the toric ideal I CP is the kernel of the ring homomorphism π C : The following lemma is fundamental and important.

Lemma 2.4
Let P be a finite poset and I , J ∈ J(P). For p ∈ max(I ) \ max(J ), the following conditions are equivalent: Since p does not belong to max(J ), p is not a maximal element in J . Hence we have p / ∈ max(I ∪ J ). Thus (i) ⇒ (iv) holds. Suppose p / ∈ max(I ∪ J ). Then there exists an element q ∈ I ∪ J such that p < P q. If q belongs to I , then p / ∈ max(I ), a contradiction. Thus q ∈ J , and hence p ∈ I ∩ J . If p / ∈ max(I ∩ J ) holds, then there exists an element q ∈ I ∩ J such that p < P q . This contradicts to the hypothesis p ∈ max(I ). Thus (iv) ⇒ (ii) holds.
In [6], Hibi and Li essentially proved that I CP 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 I CP with respect to a reverse lexicographic order < C . Moreover,

R[C]/I CP 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 .
Proof From Propositions 2.3 and 2.5, we have Hence it follows that the map . Therefore, the first claim follows. Since both in < O (I OP ) and in < C (I CP ) are squarefree, both OP and CP possess a unimodular triangulation, and hence the Ehrhart polynomial coincides with the Hilbert polynomial of its toric ring for each of OP and CP , see [4,Section 4.2] or [20,Chapters 8 and 13]. Moreover, for an ideal I of K [x] and a monomial order < on K [x], the Hilbert polynomial of K [x]/I is equal to that of K [x]/in < (I ). Therefore, the second claim follows.

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 [n]. The enriched chain polytope C (e) P is a lattice polytope of dimension n. It is easy to see that C (e) P is centrally symmetric (i.e., for any facet F of C (e) P , −F is also a facet of C (e) P ), and the origin of R n is the unique interior lattice point of C (e) A lattice polytope P ⊂ R n of dimension n is called reflexive if the origin of R n is a unique lattice point belonging to the interior of P and its dual polytope is also a lattice polytope, where x, y is the usual inner product of 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 [ In order to use Lemma 3.1 for enriched chain polytopes C (e) P , we study the toric ideal of C (e) Then the toric ideal 1 · · · t ε n n s. In addition, In addition, let w # be the weight vector on R[C (e) ] such that the weight of each variable x ε A with respect to w # is |A|. Fix any monomial order ≺ on K [C (e) ] as a tie-breaker. Let < C (e) be a monomial order on R[C (e) ] such that u < C (e) v if and only if one of the following holds: • The weight of u is less than that of v with respect to w # .
• The weight of u is the same as that of v with respect to w # , and the weight of u is less than that of v with respect to w C (e) . • The weight of u is the same as that of v with respect to w # and w C (e) , and u ≺ v.
The following proposition was given in [16,Theorem 1.3]: 16]) Work with the same notation as above. Let G C (e) be the set of all binomials where I , J ∈ J(P), ε p = μ p , and p ∈ max(I ) ∩ max(J ), together with all binomials where I , J ∈ J(P) with I J , and (a) for any p ∈ max(I ) ∩ max(J ), we have ε p = ε p = μ p = μ p ; Then G C (e) is a Gröbner basis of I C (e) P with respect to a monomial order < C (e) . 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 0 ∅ .
By Lemma 3.1 and Proposition 3.2, we have the following immediately.

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.
In addition, the origin is the unique interior lattice point in O  Then each a i is a (−1, 0, 1)-vector, and hence so is x. It is easy to see that x k = 1 (resp.  = (a 1 , . . . , a n ) ∈ R n be the supporting hyperplane of F and let P = {i ∈ P : a i = 0} ( = ∅) be a subposet of P. We may assume that i ∈ max(P ) satisfies a i > 0. Let F = { j ∈ P : i P j } be a filter of P. Then F min = {i} and hence y = e (1) F min + e F comin satisfies a, y = a i > 0 and y = e (−1) F min + e F comin satisfies a, y = −a i < 0. This contradicts that H is a supporting hyperplane of O

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 Then the toric ideal 1 1 · · · t ε n n s. In addition, is the toric ideal I OP . We define a reverse lexicographic order Theorem 5.2 Work with the same notation as above. Let G O (e) be the set of all binomials where I , J ∈ J(P), ε p = μ p , and p ∈ max(I ) ∩ max(J ), together with all binomials where I , J ∈ J(P) with I J , and (a) for any p ∈ max(I ) ∩ max(J ), we have ε p = ε p = μ p = μ p ; (b) for any p ∈ max(I ) \ max(J ), we have ε p = μ p if p ∈ max(I ∪ J ), μ p if p ∈ max(I ∩ J ); (c) for any p ∈ max(J ) \ max(I ), we have ε p = μ p if p ∈ max(I ∪ J ), μ p if p ∈ max(I ∩ J ).

Then G O (e) is a Gröbner basis of I O (e)
P with respect to a monomial order < O (e) . 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 0 ∅ . Proof It is easy to see that any binomial of type (2) . On the other hand, for I , J ∈ J(P) with I J and for ε, . Hence u and v are of the form 1, 2, . . . , r such that (a) I 1 ⊂ · · · ⊂ I r and J 1 ⊂ · · · ⊂ J r . (b) For any p and q, and for any i ∈ max(I p ) ∩ max(I q ), we obtain ε (c) For any p and q, and for any j ∈ max(J p ) ∩ max(J q ), we obtain μ Since u and v satisfy conditions (b) and (c) and since f belongs to I O (e) P , it then follows that max(I r ) = max(J r ) and ε r = μ r . Hence one has x (ε r ) J r . This contradicts the assumption that f is irreducible.
By Lemma 3.1 and Theorem 5.2, we have the following immediately.
. where I , J ∈ J(P), ε p = μ p , and p ∈ max(I )∩max(J ) together with all monomials where I , J ∈ J(P) with I J and ε p = ε p for each p ∈ max(I ) ∩ max(J ). Moreover, from Proposition 3.2, in < C (e) (I C (e) P ) is generated by all monomials where I , J ∈ J(P), ε p = μ p , and p ∈ max(I )∩max(J ) together with all monomials where I , J ∈ J(P) with I J and ε p = ε p for each p ∈ max(I ) ∩ max(J ). Hence it follows that the map x Let P be a finite poset on [n]. Given elements i, j of P, we say that j covers i if i < j and there exists no k ∈ P such that i < k < j. If j covers i in P, then we write i j. A chain of P is a totally ordered subset of P. A chain of the form i 1 i 2 · · · i r is called a saturated chain. A saturated chain i 1 i 2 · · · i r is called maximal if i 1 ∈ min(P) and i r ∈ max(P). 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]. where i 1 i 2 · · · i r is a maximal chain of P, and ε j ∈ {1, −1}. 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 [n]. Then O (e)
P ⊂ R n is the solution set of the following linear inequalities:

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 2.
Let Q ⊂ R n be the solution set of the above linear inequalities. Since 2 s−1 − s j=2 2 s− j = 1 holds for any positive integer s, it is easy to see that e ε F min + e F comin satisfies (a) and (b) for any filter F of P, and for any ε ∈ {−1, 1} |F min | . Since O . Let i = i 1 i 2 · · · i r be a saturated chain of P with i r ∈ max(P). Then x satisfies the following r inequalities: . . .
If r = 1, then x i 1 is trivial. Let r 2. Then the inequality given by a linear combination (a 1 )+(a 2 )+2(a 3 )+· · ·+2 r −2 (a r ) of the above inequalities is 2 r −1 x i 1 2 r −1 , and hence x i = x i 1 1. Suppose that i belongs to a maximal chain i 1 i 2 · · · i r , say, i = i k . Then x satisfies (a 1 ), . . . , (a r ) above and − r j=1 2 r − j x i j 1.
(b 1 ) Then the inequality given by a linear combination (b 1 ) + (a 1 ) + 2(a 2 ) + · · · + 2 k−2 (a k−1 ) + 2 k−1 (a k+1 ) + · · · + 2 r −2 (a r ) of the above inequalities is −2 r −1 x i k 2 r −1 , and hence we have We now prove that x belongs to O (e) P by induction on n. Suppose that x i = 0 for some i ∈ min(P). Then (x 1 , . . . , x i−1 , x i+1 , . . . , x n ) ∈ R n−1 satisfies inequalities (a) and (b) for the subposet P \ {i} of P. By the assumption of induction, P\{i} . It then follows that x belongs to O (e) P . Thus we may assume that x i = 0 for any i ∈ min(P). Let λ = min {|x i | : i ∈ min(P)}. Note that 0 < λ 1. Let where F = [n], and ε ∈ {−1, 1} |F min | corresponds to the sign of x i for each i ∈ min(P) = F min . We now show that the vector y satisfies Inequality (c). If either x i 1 > 0 or i 1 / ∈ min(P) holds, then If x i 1 < 0 and i 1 ∈ min(P), then λ + x i 1 0 and hence If λ = 1, then we have y = 0 by inequalities (c) and (d). Hence x = e ε F min + e F comin ∈ O (e) P . If λ = 1, then 1 1−λ y belongs to Q by inequalities (c) and (d). From the definition of λ, there exists i ∈ min(P) such that y i = 0. By the assumption of induction, 1 1−λ y belongs to O Finally, we will prove that each of inequalities (a) and (b) is facet defining. Let where i 1 i 2 · · · i r is a saturated chain of P with i r ∈ max(P), and let where i 1 i 2 · · · i r is a maximal chain of P. It is enough to show that dim O (e) Let i 1 i 2 · · · i r be a saturated chain of P with i r ∈ max(P). If min(P) = {i 1 }, then let i = i 1 . If min(P) = {i 1 }, then let i be an arbitrary element in min(P) \ {i 1 }. Note that, if min(P) = {i 1 }, then i 2 i 3 · · · i r is a maximal chain of P \{i}. Let On the other hand, for a maximal chain i 1 i 2 · · · i r of P, let i = i 1 if min(P) = {i 1 }, and let i be an arbitrary element in min(P) \ {i 1 } otherwise. Note that, if min(P) = {i 1 }, then i 2 i 3 · · · i r is a maximal chain of P \{i}. Then is unimodularly equivalent to a facet of O  . Proof The formulas of the number of facets follows from Proposition 6.1 and Theorem 6.2. Each maximal chain of P of length contains exactly + 1 saturated chains of P that contains a maximal element of P. Since + 2 2 +1 for any integer 0, we have sc(P) + mc(P) In [8,Lemma 3.8], tight upper bounds for f n−1 (O P ) and f n−1 (C P ) are given. Given an integer n 2, let It is known [14, Theorem 1] that μ n is the maximum number of cliques possible in a graph with n vertices.
In addition, both upper bounds are tight.

Proof
The proof for C (e) P is induction on n. If n = 1, then C (e) P has two facets. Let n 2 and let M be the set of all minimal elements of P. If |M| = m, then we have  . 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 P 2 = P 1 \ M 1 and let M 2 be the set of all maximal elements of P 2 . In general, if P i is not an antichain, then P i+1 = P i \ M i 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 , . . . , P r such that P r is an antichain. Then we have f n−1 (O (e) P ) |M 1 | + |M 1 ||M 2 | + · · · + |M 1 ||M 2 | · · · |M r −1 | + 2|M 1 ||M 2 | · · · |M r |.
We show that max 2m 1 m 2 · · · m r + r −1 . This is a contradiction. Thus we have m 1 3. It is easy to see that m r = 1. Therefore 3 m 1 m 2 · · · m r 2.
Finally, we discuss when the numbers of facets of O P is equal to that of C (e) P . By the argument in the proof of Corollary 6.3, each maximal chain of P of length must satisfy + 2 = 2 +1 , and hence = 0. Thus P is an antichain.