The combinatorial mutation $\mathrm{mut}_w(P,F)$ for a lattice polytope $P$ was introduced in the context of mirror symmetry for Fano manifolds in [1]. It was also proved in \cite{ACGK} that for a lattice polytope $P \subseteq N_\mathbb{R}$ containing the origin in its interior, the polar dual $P^* \subseteq M_\mathbb{R}$ and $\mathrm{mut}_w(P,F)^* \subseteq M_\mathbb{R}$ have the same Ehrhart quasi-polynomial. To extend this framework, we introduce combinatorial mutation for rational pointed polyhedra in $N_\mathbb{R}$ containing the origin in their interiors. Such polyhedra are Minkowski sums of rational polytopes and rational polyhedral pointed cones. On the dual side $M_\mathbb{R}$, the construction applies to full-dimensional rational polytopes containing the origin, not necessarily in their interiors. As an application of this extension of the combinatorial mutation, we prove that the chain polytope of a poset $Π$ can be obtained by a sequence of combinatorial mutations in $M_\mathbb{R}$ from the order polytope of $Π$. Namely, the order polytope and the chain polytope of the same poset $Π$ are mutation-equivalent.
We provide a Minkowski sum decomposition of marked chain-order polytopes into building blocks associated to elementary markings and thus give an explicit minimal set of generators of an associated semi-group algebra. We proceed by characterizing the reflexive polytopes among marked chain-order polytopes as those with the underlying marked poset being ranked.
For any marked poset we define a continuous family of polytopes, parametrized by a hypercube, generalizing the notions of marked order and marked chain polytopes. By providing transfer maps, we show that the vertices of the hypercube parametrize an Ehrhart equivalent family of lattice polytopes. The combinatorial type of the polytopes is constant when the parameters vary in the relative interior of each face of the hypercube. Moreover, with the help of a subdivision arising from a tropical hyperplane arrangement associated to the marked poset, we give an explicit description of the vertices of…
This work regards the order polytopes arising from the class of generalized snake posets and their posets of meet-irreducible elements. Among generalized snake posets of the same rank, we characterize those whose order polytopes have minimal and maximal volume. We give a combinatorial characterization of the circuits in these order polytopes and then conclude that every triangulation is unimodular. For a generalized snake word, we count the number of flips for the canonical triangulation of these order polytopes. We determine that the flip graph of the order polytope of the poset whose lattice…
There are two related poset structures, the higher Stasheff-Tamari orders, on the set of all triangulations of the cyclic $d$ polytope with $n$ vertices. In this paper it is shown that both of them have the homotopy type of a sphere of dimension $n-d-3$. Moreover, we resolve positively a new special case of the \emph{Generalized Baues Problem}: The Baues poset of all polytopal decompositions of a cyclic polytope of dimension $d \leq 3$ has the homotopy type of a sphere of dimension $n-d-2$.
We prove the unimodality of the Ehrhart $δ$-polynomial of the chain polytope of the zig-zag poset, which was conjectured by Kirillov. First, based on a result due to Stanley, we show that this polynomial coincides with the $W$-polynomial for the zig-zag poset with some natural labeling. Then, its unimodality immediately follows from a result of Gasharov, which states that the $W$-polynomials of naturally labeled graded posets of rank $1$ or $2$ are unimodal.
We introduce Lipschitz functions on a finite partially ordered set $P$ and study the associated Lipschitz polytope $L(P)$. The geometry of $L(P)$ can be described in terms of descent-compatible permutations and permutation statistics that generalize descents and big ascents. For ranked posets, Lipschitz polytopes are centrally-symmetric and Gorenstein, which implies symmetry and unimodality of the statistics. Finally, we define $(P,k)$-hypersimplices as generalizations of classical hypersimplices and give combinatorial interpretations of their volumes and $h^*$-vectors.
We study restricted chain-order polytopes associated to Young diagrams using combinatorial mutations. These polytopes are obtained by intersecting chain-order polytopes with certain hyperplanes. The family of chain-order polytopes associated to a poset interpolate between the order and chain polytopes of the poset. Each such polytope retains properties of the order and chain polytope; for example its Ehrhart polynomial. For a fixed Young diagram, we show that all restricted chain-order polytopes are related by a sequence of combinatorial mutations. Since the property of giving rise to the peri…
Université du Québec à Montréal（蒙特利尔大学）；York University（约克大学）；Universitat de Barcelona（巴塞罗那大学）；Centre de Recerca Matemàtica（数学研究中心）；Osnabrück Universität（奥斯纳布吕克大学）；University of Michigan（密歇根大学）；University of Crete（克里特大学）
We consider the inversion enumerator I_n(q), which counts labeled trees or, equivalently, parking functions. This polynomial has a natural extension to generalized parking functions. Substituting q = -1 into this generalized polynomial produces the number of permutations with a certain descent set. In the classical case, this result implies the formula I_n(-1) = E_n, the number of alternating permutations. We give a combinatorial proof of these formulas based on the involution principle. We also give a geometric interpretation of these identities in terms of volumes of generalized chain polyto…
We analyze marked poset polytopes and generalize a result due to Hibi and Li, answering whether the marked chain polytope is unimodular equivalent to the marked order polytope. Both polytopes appear naturally in the representation theory of semi-simple Lie algebras, and hence we can give a necessary and sufficient condition on the marked poset such that the associated toric degenerations of the corresponding partial flag variety are isomorphic. We further show that the set of lattice points in such a marked poset polytope is the Minkowski sum of sets of lattice points for 0-1 polytopes. Moreov…
The Möbius polynomial is an invariant of ranked posets, closely related to the Möbius function. In this paper, we study the Möbius polynomial of face posets of convex polytopes. We present formulas for computing the Möbius polynomial of the face poset of a pyramid or a prism over an existing polytope, or of the gluing of two or more polytopes in terms of the Möbius polynomials of the original polytopes. We also present general formulas for calculating Möbius polynomials of face posets of simplicial polytopes and of Eulerian posets in terms of their $f$-vectors and some additional constraints.
In 1986 Stanley associated to a poset the order polytope. The close interplay between its combinatorial and geometric properties makes the order polytope an object of tremendous interest. Double posets were introduced in 2011 by Malvenuto and Reutenauer as a generalization of Stanleys labelled posets. A double poset is a finite set equipped with two partial orders. To a double poset Chappell, Friedl and Sanyal (2017) associated the double order polytope. They determined the combinatorial structure for the class of compatible double posets. In this paper we generalize their description to all d…
To every poset P, Stanley (1986) associated two polytopes, the order polytope and the chain polytope, whose geometric properties reflect the combinatorial qualities of P. This construction allows for deep insights into combinatorics by way of geometry and vice versa. Malvenuto and Reutenauer (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 …
We describe a family of shellings for the canonical triangulation of the order polytope of the zig-zag poset. This gives a new combinatorial interpretation for the coefficients in the numerator of the Ehrhart series of this order polytopein terms of the swap statistic on alternating permutations.
Stanley (1986) showed how a finite partially ordered set gives rise to two polytopes, called the order polytope and chain polytope, which have the same Ehrhart polynomial despite being quite different combinatorially. We generalize his result to a wider family of polytopes constructed from a poset P with integers assigned to some of its elements. Through this construction, we explain combinatorially the relationship between the Gelfand-Tsetlin polytopes (1950) and the Feigin-Fourier-Littelmann polytopes (2010), which arise in the representation theory of the special linear Lie algebra. We then…
Motivated by the graph associahedron KG, a polytope whose face poset is based on connected subgraphs of G, we consider the notion of associativity and tubes on posets. This leads to a new family of simple convex polytopes obtained by iterated truncations. These generalize graph associahedra and nestohedra, even encompassing notions of nestings on CW-complexes. However, these poset associahedra fall in a different category altogether than generalized permutohedra.
Matching fields were introduced by Sturmfels and Zelevinsky to study certain Newton polytopes and more recently have been shown to give rise to toric degenerations of various families of varieties. Whenever a matching field gives rise to a toric degeneration, the associated polytope of the toric variety coincides with the matching field polytope. We study combinatorial mutations, which are analogues of cluster mutations for polytopes, of matching field polytopes and show that the property of giving rise to a toric degeneration of the Grassmannians, is preserved by mutation. Moreover the polyto…
A mode of a probability vector is a local maximum with respect to some vicinity structure on the set of elementary events. The mode inequalities cut out a polytope from the simplex of probability vectors. Related to this is the concept of strong modes. A strong mode of a distribution is an elementary event that has more probability mass than all its direct neighbors together. The set of probability distributions with a given set of strong modes is again a polytope. We study the vertices, the facets, and the volume of such polytopes depending on the sets of (strong) modes and the vicinity struc…
Greene's rational function $Ψ_P({\bf x})$ is a sum of certain rational functions in ${\bf x}=(x_1, \ldots, x_n)$ over the linear extensions of the poset $P$ (which has $n$ elements), which he introduced in his study of the Murnaghan-Nakayama formula for the characters of the symmetric group. In recent work Boussicault, Féray, Lascoux and Reiner showed that $Ψ_P({\bf x})$ equals a valuation on a cone and calculated $Ψ_P({\bf x})$ for several posets this way. In this paper we give an expression for $Ψ_P({\bf x})$ for any poset $P$. We obtain such a formula using dissections of root polytopes. Mo…
Understanding the face structure of the balanced minimal evolution (BME) polytope, especially its top-dimensional facets, is crucially important to phylogenetic applications. We show that BME polytope has a sub-lattice of its poset of faces which is isomorphic to a quotient of the well-studied permutoassociahedron. This sub-lattice corresponds to compatible sets of splits displayed by phylogenetic trees, and extends the lattice of faces of the BME polytope found by Hodge, Haws, and Yoshida. Each of the maximal elements in our new poset of faces corresponds to a single split of the leaves. Near…
Let P be a poset, O(P) the order polytope of P and C(P) the chain polytope of P. In this paper, we study the canonical ideal of the Ehrhart ring K[C(P)] of C(P) over a field K and characterize the level (resp. anticanonical level) property of K[C(P)] by a combinatorial structure of P. In particular, we show that if K[C(P)] is level (resp. anticanonical level), then so is K[O(P)]. We exhibit examples which show the converse does not hold. Moreover, we show that the symbolic powers of the canonical ideal of K[C(P)] are identical with ordinary ones and degrees of the generators of the canonical a…
The diamond product is the poset operation that when applied to the face lattices of two polytopes results in the face lattice of the Cartesian product of the polytopes. Application of the diamond product to two Eulerian posets is a bilinear operation on the cd-indices of the two posets, yielding a product on cd-polynomials. A lattice path interpretation is provided for this product of two cd-monomials.
The Gelfand-Tsetlin and the Feigin-Fourier-Littelmann-Vinberg polytopes for the Grassmannians are defined, from the perspective of representation theory, to parametrize certain bases for highest weight irreducible modules. These polytopes are Newton-Okounkov bodies for the Grassmannian and, in particular, the GT-polytope is an example of a string polytope. The polytopes admit a combinatorial description as the Stanley's order and chain polytopes of a certain poset, as shown by Ardila, Bliem and Salaza. We prove that these polytopes occur among matching field polytopes. Moreover, we show that t…
We study a family of posets and the associated chain and order polytopes. We identify the order polytope as a maximal Kogan face in a Gelfand-Tsetlin polytope of a multiple of a fundamental weight. We show that the character of such a Kogan face equals to the character of a Demazure module which occurs in the irreducible representation of $\mathfrak{sl}_n$ having highest weight multiple of fundamental weight and for any such Demazure module there exists a corresponding poset and associated maximal Kogan face. We prove that the chain polytope parametrizes a monomial basis of the associated PBW-…
This is both an expository and research paper where we advocate a systematic study of continuous analogues of finite partially ordered sets, convex polytopes, oriented matroids, arrangements of subspaces, finite simplicial complexes, and other combinatorial structures. Among the illustrative examples are an Euler formula for a class of `continuous convex polytopes' (conjectured by Kalai and Wigderson), a duality result for a class of `continuous matroids', a calculation of the Euler characteristic of ideals in the Grassmannian poset (related to a problem of Gian-Carlo Rota), an exposition of t…
Given any connected poset $P$, we give a simple realization of Galashin's poset associahedron $\mathscr{A}(P)$ as a convex polytope in $\mathbb{R}^P.$ The realization is inspired by the description of $\mathscr{A}(P)$ as a compactification of the configuration space of order-preserving maps $P \to \mathbb{R}.$ In addition, we give an analogous realization for Galashin's affine poset cyclohedra.
Stanley introduced two classes of lattice polytopes associated to posets, which are called the order polytope ${\mathcal O}_P$ and the chain polytope ${\mathcal C}_P$ of a poset $P$. It is known that, given a poset $P$, the Ehrhart polynomials of ${\mathcal O}_P$ and ${\mathcal C}_P$ 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…
In this article we investigate a class of linear error correcting codes in relation with the order polytopes. In particular we consider the order polytopes of tree posets and bipartite posets. We calculate the parameters of the associated toric variety codes.
