Facets of the r-Stable (n, k)-Hypersimplex

Let k, n, and r be positive integers with k < n and r≤⌊nk⌋\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${r \leq \lfloor \frac{n}{k} \rfloor}$$\end{document}. We determine the facets of the r-stable n, k-hypersimplex. As a result, it turns out that the r-stable n, k-hypersimplex has exactly 2n facets for every r<⌊nk⌋\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${r < \lfloor \frac{n}{k} \rfloor}$$\end{document}. We then utilize the equations of the facets to study when the r-stable hypersimplex is Gorenstein. For every k > 0 we identify an infinite collection of Gorenstein r-stable hypersimplices, consequently expanding the collection of r-stable hypersimplices known to have unimodal Ehrhart δ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\delta}$$\end{document}-vectors.


Introduction
The (n, k)-hypersimplices are an important collection of integer polytopes arising naturally in the settings of convex optimization, matroid theory, combinatorics, and algebraic geometry. Generalizing the standard (n − 1)-simplex, the (n, k)-hypersimplices serve as a useful collection of examples in these various contexts. While these polytopes are well studied, there remain interesting open questions about their properties in the field of Ehrhart theory, the study of integer point enumeration in dilations of rational polytopes (see, for example, [4]). The r-stable (n, k)-hypersimplices are a collection of lattice polytopes within the (n, k)-hypersimplex that were introduced in [2] for the purpose of studying unimodality of the Ehrhart δ -polynomials of the (n, k)-hypersimplices. However, they also exhibit interesting geometric similarities to the (n, k)-hypersimplices which they generalize. For example, it is shown in [2] that a regular unimodular triangulation of the (n, k)-hypersimplex, called the circuit triangulation, restricts to a triangulation of each r-stable (n, k)-hypersimplex. 1

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T.HibiandL.Solus 2 T. Hibi and L. Solus In the present paper, we compute the facets of the r-stable (n, k)-hypersimplices for 1 ≤ r < n k and then study when they are Gorenstein. In Section 2, we compute their facet-defining inequalities (Theorem 2.1). From these computations, we see that the geometric similarities between the (n, k)-hypersimplex and the r-stable (n, k)hypersimplices within are apparent in their minimal H-representations. Moreover, it turns out that each r-stable (n, k)-hypersimplex has exactly 2n facets for 1 ≤ r < n k (Corollary 2.2). In Section 3, we classify 1 ≤ r < n k for which these polytopes are Gorenstein (Theorem 3.6). We conclude that the Ehrhart δ -vector of each Gorenstein r-stable hypersimplex is unimodal (Corollary 3.7), thereby expanding the collection of r-stable hypersimplices known to have unimodal δ -polynomials.

The H-Representation of the r-Stable (n, k)-Hypersimplex
We first recall the definitions of the (n, k)-hypersimplices and the r-stable (n, k)hypersimplices. For integers 0 < k < n let [n] := {1, 2, . . . , n} and let [n] k denote the collection of all k-subsets of [n]. The characteristic vector of a subset I of [n] is the (0, 1)-vector ε I = (ε 1 , . . . , ε n ) for which ε i = 1 for i ∈ I and ε i = 0 for i / ∈ I. The (n, k)-hypersimplex is the convex hull in R n of the collection of characteristic vectors k }, and it is denoted ∆ n, k . Label the vertices of a regular n-gon embedded in R 2 in a clockwise fashion from 1 to n. Given a third integer 1 ≤ r ≤ n k , a subset I ⊂ [n] (and its characteristic vector) is called r-stable if, for each pair i, j ∈ I, the path of shortest length from i to j about the n-gon uses at least r edges. The r-stable n, khypersimplex, denoted by ∆ stab(r) n, k , is the convex polytope in R n which is the convex hull of the characteristic vectors of all r-stable k-subsets of [n]. For fixed n and k the r-stable (n, k)-hypersimplices form the nested chain of polytopes Notice that ∆ n, k is precisely the 1-stable (n, k)-hypersimplex.
The definitions of ∆ n, k and ∆ stab(r) n, k provided are V -representations of these polytopes. In this section, we provide the minimal H-representation of ∆ stab(r) n, k , i.e., its collection of facet-defining inequalities. It is well known that the facet-defining in- In the definition of H , r the indices i of the coordinates x 1 , . . . , x n are taken to be elements of Z/nZ. We also let H and H , r denote the (n − 2)-flats given by strict equality in the above definitions. In the following we will say an (n − 2)-flat is facet- Facetsofther-Stable(n,k)-Hypersimplex The following is an immediate corollary to these results.
Corollary 2.2. All but possibly the smallest polytope in the nested chain n, k has 2n facets.
This is an interesting geometric property since the number of vertices of these polytopes strictly decreases down the chain. To prove Theorem 2.1 we will utilize the geometry of the circuit triangulation of ∆ n, k as defined in [8], the construction of which we will now recall.

The Circuit Triangulation
Fix 0 < k < n, and let G n, k be the labeled, directed graph with the following vertices and edges. The vertices of G n, k are all the vectors ε I ∈ R n where I is a k-subset of [n]. We think of the indices of a vertex of G n, k modulo n. Now suppose that ε and ε are two vertices of G n, k such that for some i ∈ [n] (ε i , ε i+1 ) = (1, 0) and ε is obtained from ε by switching the order of ε i and ε i+1 . Then the directed and labeled edge ε i → ε is an edge of G n, k . Hence, an edge of G n, k corresponds to a move of a single 1 in a vertex ε one spot to the right, and such a move can be done if and only if the next spot is occupied by a 0.
We are interested in the circuits of minimal length in the graph G n, k . Such a circuit is called a minimal circuit. Suppose that ε is a vertex in a minimal circuit of G n, k . Then the minimal circuit can be thought of as a sequence of edges in G n, k that moves each 1 in ε into the position of the 1 directly to its right (modulo n). It follows that a minimal circuit in G n, k has length n. An example of a minimal circuit in G 9, 3 is provided in Figure 1. Notice that for a fixed initial vertex of the minimal circuit the labels of the edges form a permutation ω = ω 1 ω 2 · · · ω n ∈ S n , the symmetric group on n elements. Following the convention of [8], we associate a minimal circuit in G n, k with the permutation consisting of the labels of the edges of the circuit for which ω n = n. Let (ω) denote the minimal circuit in G n, k corresponding to the permutation ω ∈ S n with ω n = n. Let σ (ω) denote the convex hull in R n of the set of vertices of (ω). Notice that σ (ω) will always be an (n − 1)-simplex. [8]) The collection of simplices σ (ω) given by the minimal circuits in G n, k are the maximal simplices of a triangulation of the hypersimplex ∆ n, k . We call this triangulation the circuit triangulation.

Theorem 2.3. (Lam and Postnikov
Denote the circuit triangulation of ∆ n, k by ∇ n, k , and let max ∇ n, k denote the set of maximal simplices of ∇ n, k . To simplify notation we will write ω to denote the simplex σ (ω) ∈ max ∇ n, k . In [2] it is shown that the collection of simplices in ∇ n, k that lie forms a triangulation of this polytope. We let ∇ r n, k denote this triangulation of ∆ stab(r) n, k and let max ∇ r n, k denote the set of maximal simplices of ∇ r n, k . In the following, we compute the facet-defining inequalities for ∆ stab(r) n, k using the nesting of triangulations: The method by which we will do this is outlined in the following remark.

Remark 2.4.
To compute the facet-defining inequalities of ∆ stab(r) n, k we first consider the geometry of their associated facet-defining (n − 2)-flats. Suppose that ∆ . Therefore, to compute the facet-defining (n − 2)-flats of ∆ stab(r) n, k it suffices to compute the former and latter collections of (n − 2)-flats independently. To identify the former collection we will use an induction argument on r.
To identify the latter collection we work with pairs of adjacent (n − 1)-simplices in the set max ∇ r n, k . Note that two simplices u, ω ∈ max ∇ r n, k are adjacent (i.e., share a common facet) if and only if they differ by a single vertex. Therefore, their common vertices span an (n − 2)-flat which we will denote by H[u, ω]. Thus, we will identify adjacent pairs of simplices u ∈ max ∇ r−1 n, k and ω ∈ max ∇ r−1 n, k \ max ∇ r n, k for which H[u, ω] is facet-defining.

Computing Facet-Defining Inequalities via a Nesting of Triangulations
Suppose 1 < k < n − 1. In order to prove Theorem 2.1 in the fashion outlined by Remark 2.4 we require a sequence of lemmas. Notice that ∆ stab(r) n, k is contained in , r for all ∈ [n]. So in the following we simply show that H and H , r form the complete set of facet-defining (n − 2)-flats.
Proof. First notice that the result clearly holds for r = 1. So we need only show that n − 1 affinely independent vertices of ∆ stab(r) n, k lie in H . Hence, to prove the claim it suffices to identify a simplex ω ∈ max ∇ r n, k such that H supports a facet of ω. Since r ≤ n k − 1 it also suffices to work with r = n k − 1. Fix ∈ [n]. For r = n k − 1 we construct a minimal circuit in the graph G n, k that corresponds to a simplex in max ∇ r n, k for which H is facet-supporting. To this end, consider the characteristic vector of the k-subset Denote this characteristic vector by ε , and think of its indices modulo n. Labeling the 1 in coordinate ( − 1) − (s − 1)r of ε as 1 s , we see that 1 s and 1 s+1 are separated by r − 1 zeros for s ∈ [k − 1]. That is, the coordinate ε i = 0 for every Moreover, since kr = k n k − 1 ≤ n then there are at least r − 1 zeros between 1 1 and 1 k . Hence, this vertex is r-stable. From ε we can now construct an r-stable circuit ω by moving the 1's in ε one coordinate to the right (modulo n), one at a time, in the following pattern: An example of ω for n = 9, k = 3, and = 5 is provided in Figure 1. This produces a minimal circuit in G n, k since each 1 s has moved precisely enough times to replace 1 s+1 . Moreover, since k > 1 then k n k − 1 ≤ n − 2. So there are at least r + 1 0's between 1 1 and 1 k in ε . From here, it is a straightforward exercise to check that every vertex in ω is r-stable. Therefore, ω ∈ max ∇ r n, k . Finally, since r > 1, the simplex ω has only one vertex satisfying x = 1, and this is the vertex following ε in the circuit ω . Hence, all other vertices of ω satisfy x = 0. So H supports a facet of ω . Thus, we conclude that H is facet-defining for ∆ stab(r) n, k for r < n k . The following theorem follows immediately from the construction of the (n − 1)simplex ω in the proof of Lemma 2.5, and it justifies the assumption on the dimension of ∆ x i = 1 except for exactly one vertex, say ε . Since all vertices in (ω) are (0, 1)-vectors, this means that all vertices other than ε have exactly one coordinate in the subvector (ε , ε +1 , . . . , ε +r−2 ) being 1 and all other coordinates are 0. Similarly, this subvector is the 0-vector for ε . Since (ω) is a minimal circuit this means that the move preceding the vertex ε in (ω) results in the only 1 in (ε , ε +1 , . . . , ε +r−2 ) exiting the subvector to the right. Similarly, the move following the vertex ε in (ω) results in a single 1 entering the subvector on the left. Suppose that ε = (. . . , ε −1 , ε , ε +1 , . . . , ε +r−2 , ε +r−1 , . . .) = (. . . , 1, 0, 0, . . . , 0, 1, . . .). Hence, neither the vertex preceding or following the vertex ε is r-stable. For example, in the vertex following ε there is a 1 in entries and + r − 1. This contradicts the fact that ω ∈ max ∇ r n, k . To see why Lemma 2.7 will be useful, suppose that Theorem 2.1 holds for ∆ stab(r−1) n, k for some 1 ≤ r < n k . Then Lemmas 2.5 and 2.7 tell us that the collection of facet-defining (n − 2)-flats for ∆ stab(r−1) n, k that are also facet-defining for ∆ stab(r) n, k is {H : ∈ [n]}. This is the nature of the induction argument mentioned in Remark 2.4. To identify the facet-defining (n − 2)-flats of ∆ stab(r) n, k that intersect relint∆ stab(r−1) n, k we will use the following definition.  is (n − 1)-dimensional we may assume, without loss of generality, that α lies in the relative interior of an (n − 2)-dimensional simplex in the triangulation of ∂ ∆ stab(r) n, k ∩relint ∆ stab(r−1) n, k induced by ∇ r n, k and ∇ r−1 n, k \∇ r n, k . Therefore, there exists some u ∈ max ∇ r n, k such that H F is facet-defining for u and α ∈ u ∩ H F , and there exists some ω ∈ max ∇ r−1 n, k \ max ∇ r n, k such that α ∈ ω ∩ H F . Since ∇ r−1 n, k is a triangulation of ∆ stab(r−1) n, k it follows that u ∩ H F = ω ∩ H F . Hence, u and ω are adjacent simplices that share the facet-defining (n − 2)-flat H F , and they form an r-supporting pair (u, ω) with H[u, ω] = H F . It will be helpful to understand the key vertex of an r-supporting pair (u, ω). To do so, we will use the following definition. Definition 2.10. Let ε ∈ R n be a vertex of ∆ n, k . A pair of 1's in ε is an ordered pair of two coordinates of ε, (i, j), such that ε i = ε j = 1, and ε t = 0 for all i < t < j (modulo n). A pair of 1's is called an r-stable pair if there are at least r − 1 0's separating the two 1's. Proof. We first show that ε has precisely one (r −1)-stable but not r-stable pair, ( , + r − 1). To see this, consider the minimal circuit (ω) in the graph G n, k associated with the simplex ω. Think of the key vertex ε as the initial vertex of this circuit, and recall that each edge of the circuit corresponds to a move of exactly one 1 to the right by exactly one entry. Hence, in the circuit (ω) the vertex following ε differs from ε by a single right move of a single 1. Since ε is the only vertex in (ω) that is (r − 1)-stable but not r-stable, then the move of this single 1 to the right by one entry must eliminate all pairs that are (r − 1)-stable but not r-stable. Moreover, this move cannot introduce any new (r − 1)-stable but not r-stable pairs. Since a single 1 can be in at most two pairs, and this 1 must move exactly one entry to the right, then this 1 must be in entry j in the pairs (i, j) and ( j, t) where (i, j) is (r − 1)-stable but not r-stable, and ( j, t) is (r + 1)-stable. Moreover, since the move of the 1 in entry j can only change the stability of the pairs (i, j) and ( j, t), then it must be that all other pairs are r-stable.

Then this situation looks like
Finally, since ω has the unique (r − 1)-stable but not r-stable vertex ε, and since ε has the unique (r − 1)-stable but not r-stable pair ( , + r − 1) then all other vertices in ω satisfy ∑ +r−1 Proof. First we note that the result is clearly true for r = 1. So in the following we assume r > 1. To prove the claim we show that H , r supports an (n − 1)-simplex ω ∈ max ∇ r n, k . To this end, consider the characteristic vector of the k-subset . Denote this characteristic vector by ε , and think of its indices modulo n. Labeling the 1 in coordinate ( − 1) + (s − 1)r of ε as 1 s , it is quick to see that 1 s and 1 s+1 are separated by r − 1 zeros for every s ∈ [k]. That is, ε i = 0 for every ( −1)+(s−1)r < i < ( −1)+sr (modulo n), and there are precisely r − 1 such coordinates. Moreover, since r < n k or n = kr + 1 then n ≥ kr + 1. So there are at least r zeros between 1 1 and 1 k . Hence, this vertex is r-stable. From ε we can now construct an r-stable circuit ω by moving the 1's in ε one coordinate to the right (modulo n), one at a time, in the following pattern: (1) Move 1 k . Then move 1 k−1 . Then move 1 k−2 . . . . Then move 1 1 .
Each move in this pattern produces a new r-stable vertex since there are always at least r − 1 zeros between each pair of 1's. So ω ∈ max ∇ r n, k and H , r supports ω since every vertex of ω lies in H , r except for the vertex preceding the first move of 1 1 in the circuit ω . , r for ∈ [n]. From Lemmas 2.12 and 2.13 we see that when 1 < r < n k the facet-defining (n − 2)-flats for ∆ stab(r) n, k that intersect relint∆ stab(r−1) n, k are precisely H , r for ∈ [n]. We are now ready to prove Theorem 2.1.

Proof of Theorem 2.1
First recall that Theorem 2.1 is known to be true for r = 1. Now let 1 < r < n k . By Theorem 2.6 we know that ∆ stab(r) n, k is (n − 1)-dimensional. First let r = 2. By , r , this proves the result for r = 2. Theorem 2.1 then follows by iterating this argument for 2 < r < n k .

Gorenstein r-Stable Hypersimplices
In [2], the authors note that the r-stable hypersimplices appear to have unimodal Ehrhart δ -vectors, and they verify this observation for a collection of these polytopes in the k = 2 case. In [3], it is shown that a Gorenstein integer polytope with a regular unimodular triangulation has a unimodal δ -vector. In [2], it is shown that ∆ stab(r) n, k has a regular unimodular triangulation. One application for the equations of the facets of a rational convex polytope is to determine whether or not the polytope is Gorenstein [6]. We now utilize Theorem 2.1 to identify 1 ≤ r < n k for which ∆ stab(r) n, k is Gorenstein. We identify a collection of such polytopes for every k ≥ 2, thereby expanding the collection of r-stable hypersimplices known to have unimodal δ -vectors. In this section we let 1 < k < n − 1. This is because ∆ n, 1 and ∆ n, n−1 are simply copies of the standard (n − 1)-simplex, which are well known to be Gorenstein [1, p. 29].
First we recall the definition of a Gorenstein polytope. Let P ⊂ R N be a rational convex polytope of dimension d, and for an integer q ≥ 1 let qP := {qα : α ∈ P}. Let x 1 , x 2 , . . . , x N , and z be indeterminates over some field K. Given an integer q ≥ 1, let A(P) q denote the vector space over K spanned by the monomials x α 1 1 x α 2 2 · · · x α N N z q for (α 1 , α 2 , . . . , α N ) ∈ qP ∩ Z N . Since P is convex we have that A(P) p A(P) q ⊂ A(P) p+q for all p and q. It then follows that the graded algebra is finitely generated over K = A(P) 0 . We call A(P) the Ehrhart Ring of P, and we say that P is Gorenstein if A(P) is Gorenstein.
We now recall the combinatorial criterion given in [5] for an integral convex polytope P to be Gorenstein. Let ∂ P denote the boundary of P and let relint(P) = P − ∂ P. We say that P is of standard type if d = N and the origin in R d is contained in relint(P). When P ⊂ R d is of standard type we define its polar set The polar set P is again a convex polytope of standard type, and (P ) = P. We call P the dual polytope of P. Suppose (α 1 , α 2 , . . . , α d ) ∈ R d , and K is the hyperplane in R d defined by the equation ∑ d i=1 α i x i = 1. A well-known fact is that (α 1 , α 2 , . . . , α d ) is a vertex of P if and only if K ∩ P is a facet of P. It follows that the dual polytope of a rational polytope is always rational. However, it need not be that the dual of an integral polytope is always integral. If P is an integral polytope with integral dual 824 T.HibiandL.Solus 10 T. Hibi and L. Solus we say that P is reflexive. This idea plays a key role in the following combinatorial characterization of Gorenstein polytopes.
Notice that ϕ is also a lattice isomorphism. Hence, we have the isomorphism of Ehrhart Rings as graded algebras . In the following, it will be convenient to let T ( ) = { , + 1, + 2, . . . , + r − 1} for ∈ [n]. We also let T ( ) c denote the complement of T ( ) in [n]. Notice that for a fixed 1 ≤ r < n k and ∈ [n], the set T ( ) is precisely the set of summands in the defining equation of the (n − 2)-flat H , r . The defining inequalities of P stab(r) n, k corresponding to the (n − 2)-flats H , r come in two types, dependent on whether n / ∈ T ( ) or n ∈ T ( ). If n / ∈ T ( ) then If n ∈ T ( ) then = (x 1 , x 2 , . . . , x n−1 ) ∈ R n−1 : Thus, we may write P stab(r) n, k as the intersection of closed halfspaces in R n−1 P stab(r) n, k To denote the supporting hyperplanes corresponding to these halfspaces we simply drop the superscripts (+) and (−).
3.2. The Codegree of P stab(r) n, k Given the above description of P stab(r) n, k , we would now like to determine the smallest positive integer q for which qP stab(r) n, k contains a lattice point in its relative interior. To do so, recall that for a lattice polytope P of dimension d we can define the (Ehrhart) δ -polynomial of P. If we write this polynomial as δ P (z) = δ 0 + δ 1 z + δ 2 z 2 + · · · + δ d z d , then we call the coefficient vector δ (P) = (δ 0 , δ 1 , δ 2 , . . . , δ d ) the δ -vector of P. We let s denote the degree of δ P (z), and we call q = (d + 1) − s the codegree of P. It is a consequence of Ehrhart Reciprocity that q is the smallest positive integer such that qP contains a lattice point in its relative interior [1]. Hence, we would like to compute the codegree of P stab(r) n, k . To do so requires that we first prove two lemmas. In the following let q = n k . Our first goal is to show that there is at least one integer point in relint qP stab(r) n, k for 1 ≤ r < n k . We then show that q is the smallest positive integer for which this is true. Recall that q = n+α k for some α ∈ {0, 1, . . . , k − 1}. Also recall that for fixed n and k we have the nesting of polytopes for every 1 ≤ r < n k . Given these facts, we now prove two lemmas.