We study the poset topology of lattices arising from orientations of 1-skeleta of directionally simple polytopes, with Bruhat interval polytopes $Q_{e,w}$ as our main example. We show that the order complex $Δ((u,v)_w)$ of an interval therein is homotopy equivalent to a sphere if $Q_{u,v}$ is a face of $Q_{e,w}$ and is otherwise contractible. This significantly generalizes the known case of the permutahedron. We also show that saturated chains from $u$ to $v$ in such lattices are connected, and in fact highly connected, under moves corresponding to flipping across a 2-face. When $w$ is a Grassmannian permutation, this implies a strengthening of the restriction of Postnikov's move-equivalence theorem to the class of BCFW bridge decomposable plabic graphs.
We study the poset of Borel congruence classes of symmetric matrices ordered by containment of closures. We give a combinatorial description of this poset and calculate its rank function. We discuss the relation between this poset and the Bruhat poset of involutions of the symmetric group.
We investigate the poset of strata of a Schubert-like stratification of certain natural compactification of the space of hermitian $n\times n$ matrices. We prove that this poset is a modular ortholattice, we compute its Möbius function and we describe the topology of its order intervals.
The totally nonnegative Grassmannian $\mathrm{Gr}(k,n)_{\geq0}$ is the subset of the real Grassmannian $\mathrm{Gr}(k,n)$ consisting of points with all nonnegative Plücker coordinates. The circular Bruhat order is a poset isomorphic to the face poset of A. Postnikov's (2005) positroid cell decomposition of $\mathrm{Gr}(k,n)_{\geq0}$. We provide a closed formula for the sum of its weighted chains in the spirit of J. Stembridge (2002).
The higher Bruhat order is a poset of cubical tilings of a cyclic zonotope whose covering relations are cubical flips. For a 2-dimensional zonotope, the higher Bruhat order is isomorphic to a poset on commutation classes of reduced words for the longest element of a type A Coxeter system. For this case, we prove that the noncontractible intervals are in natural correspondence with the zonogonal tilings of a zonogon. Our proof uses some tools developed by Felsner and Weil to show that the two standard orderings on the rhombic tilings of a zonogon are identical.
The order complex of inclusion poset $PF_n$ of Borel orbit closures in skew-symmetric matrices is investigated. It is shown that $PF_n$ is an EL-shellable poset, and furthermore, its order complex triangulates a ball. The rank-generating function of $PF_n$ is computed and the resulting polynomial is contrasted with the Hasse-Weil zeta function of the variety of skew-symmetric matrices over finite fields.
Applying a classical theorem of Smith, we show that the poset property of being Gorenstein$^*$ over $\mathbb{Z}_2$ is inherited by the subposet of fixed points under an involutive poset automorphism. As an application, we prove that every interval in the Bruhat order on (twisted) involutions in an arbitrary Coxeter group has this property, and we find the rank function. This implies results conjectured by F. Incitti. We also show that the Bruhat order on the fixed points of an involutive automorphism induced by a Coxeter graph automorphism is isomorphic to the Bruhat order on the fixed subgrou…
Let u and v be permutations on n letters, with u <= v in Bruhat order. A Bruhat interval polytope Q_{u,v} is the convex hull of all permutation vectors z = (z(1), z(2),...,z(n)) with u <= z <= v. Note that when u=e and v=w_0 are the shortest and longest elements of the symmetric group, Q_{e,w_0} is the classical permutohedron. Bruhat interval polytopes were studied recently by Kodama and the second author, in the context of the Toda lattice and the moment map on the flag variety. In this paper we study combinatorial aspects of Bruhat interval polytopes. For example, we give an inequality descr…
We generalize chain enumeration in graded partially ordered sets by relaxing the graded, poset and Eulerian requirements. The resulting balanced digraphs, which include the classical Eulerian posets having an $R$-labeling, imply the existence of the (non-homogeneous) ${\bf cd}$-index, a key invariant for studying inequalities for the flag vector of polytopes. Mirroring Alexander duality for Eulerian posets, we show an analogue of Alexander duality for balanced digraphs. For Bruhat graphs of Coxeter groups, an important family of balanced graphs, our theory gives elementary proofs of the existe…
Let (W, S) be a Coxeter system. We investigate combinatorially certain partial orders, called extended Bruhat orders, on a (W x W)-set W(N,C), which depends on W, a subset N of S, and a component C of N. We determine the length of the maximal chains between two elements. These posets generalize W equipped with its Bruhat order. They include the (W x W)-orbits of the Renner monoids of reductive algebraic monoids and of some infinite dimensional generalizations which are equipped with the partial orders obtained by the closure relations of the Bruhat and Birkhoff cells. They also include the (W …
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…
When $G$ is a complex reductive algebraic group, MV polytopes are in bijection with the non-negative tropical points of the unipotent group of $G$. By fixing $w$ from the Weyl group, we can define MV polytopes whose highest vertex is labelled by $w$. We show that these polytopes are in bijection with the non-negative tropical points of the reduced double Bruhat cell labelled by $w^{-1}$. To do this, we define a collection of generalized minor functions $Δ_γ^\text{new}$ which tropicalize on the reduced Bruhat cell to the BZ data of an MV polytope of highest vertex $w$. We also describe the comb…
We make explict a description in terms of convex geometry of the higher Bruhat orders B(n,d) sketched by Kapranov and Voevodsky. We give an analogous description of the higher Stasheff-Tamari poset S_1(n,d). We show that the map f sketched by Kapranov and Voevodsky from B(n,d) to S([0,n+1],d+1) coincides with the map constructed by Rambau, and is a surjection for d<=2. We construct a map analogous to f from S_1(n,d) to B(n-1,d), and show that it is always a poset embedding. We also give an explicit criterion to determine if an element of B(n-1,d) is in the image of this map.
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 show that any lower Bruhat interval in a Coxeter group is a disjoint union of certain two-sided cosets as a consequence of Lifting Property and Subword Property. Furthermore, we describe these details in terms of Bruhat graphs, graded posets, and two-sided quotients altogether. For this purpose, we introduce some new ideas, quotient lower intervals and quotient Bruhat graphs.
We study the interplay between the discrete geometry of Bruhat poset intervals and subword complexes of finite Coxeter systems. We establish connections between the cones generated by cover labels for Bruhat intervals and of root configurations for subword complexes, culminating in the notion of brick polyhedra for general subword complexes.
The main purpose of this paper is to prove that the Bruhat-Chevalley ordering of the symmetric group when restricted to the fixed-point-free involutions forms an $EL$-shellable poset whose order complex triangulates a ball. Another purpose of this article is to prove that the Deodhar-Srinivasan poset is a proper, graded subposet of the Bruhat-Chevalley poset on fixed-point-free involutions.
In certain finite posets, the expected down-degree of their elements is the same whether computed with respect to either the uniform distribution or the distribution weighting an element by the number of maximal chains passing through it. We show that this coincidence of expectations holds for Cartesian products of chains, connected minuscule posets, weak Bruhat orders on finite Coxeter groups, certain lower intervals in Young's lattice, and certain lower intervals in the weak Bruhat order below dominant permutations. Our tools involve formulas for counting nearly reduced factorizations in 0-H…
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）
For two elements $v$ and $w$ of the symmetric group $\mathfrak{S}_n$ with $v\leq w$ in Bruhat order, the Bruhat interval polytope $Q_{v,w}$ is the convex hull of the points $(z(1),\ldots,z(n))\in \mathbb{R}^n$ with $v\leq z\leq w$. It is known that the Bruhat interval polytope $Q_{v,w}$ is the moment map image of the Richardson variety $X^{v^{-1}}_{w^{-1}}$. We say that $Q_{v,w}$ is \emph{toric} if the corresponding Richardson variety $X_{w^{-1}}^{v^{-1}}$ is a toric variety. We show that when $Q_{v,w}$ is toric, its combinatorial type is determined by the poset structure of the Bruhat interva…
Let $W$ be a finite reflection group. For a given $w \in W$, the following assertion may or may not be satisfied: (*) The principal Bruhat order ideal of $w$ contains as many elements as there are regions in the inversion hyperplane arrangement of $w$. We present a type independent combinatorial criterion which characterises the elements $w\in W$ that satisfy (*). A couple of immediate consequences are derived: (1) The criterion only involves the order ideal of $w$ as an abstract poset. In this sense, (*) is a poset-theoretic property. (2) For $W$ of type $A$, another characterisation of (*), …
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…
In [E. Tsukerman and L. Williams, {\em Bruhat Interval Polytopes}, Advances in Mathematics, 285 (2015), 766-810] it is shown that every Bruhat interval of the symmetric group satisfies the so-called generalized lifting property. In this paper we show that a Coxeter group satisfies this property if and only if it is finite and simply-laced.
Westwood High School（韦斯特伍德高中）；Cranbrook Kingswood Upper School（克兰布鲁克金伍兹高中部）；Oklahoma State University（俄克拉荷马州立大学）；Texas State University（德克萨斯州立大学）；West Windsor-Plainsboro High School North（西温莎-普莱恩斯伯勒北高中）；Seven Lakes High School（七湖高中）
We provide a weaker version of the generalized lifting property which holds in complete generality for all finite Coxeter groups, and we use it to show that every parabolic Bruhat interval of a finite Coxeter group is a Coxeter matroid. We also describe some combinatorial properties of the associated polytopes.
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 …
Motivated by the geometry of certain hyperplane arrangements, Manin and Schechtman defined for each positive integer n a hierarchy of finite partially ordered sets B(n, k), indexed by positive integers k, called the higher Bruhat orders. The poset B(n, 1) is naturally identified with the weak left Bruhat order on the symmetric group S_n, each B(n, k) has a unique maximal and a unique minimal element, and the poset B(n, k + 1) can be constructed from the set of maximal chains in B(n, k). Elias has demonstrated a striking connection between the posets B(n, k) for k = 2 and the diagrammatics of B…
In this article, we investigate the set of $γ$-sortable elements, associated with a Coxeter group $W$ and a Coxeter element $γ\in W$, under Bruhat order, and we denote this poset by $\mathcal{B}_γ$. We show that this poset belongs to the class of SB-lattices recently introduced by Hersh and Mészáros, by proving a more general statement, namely that all join-distributive lattices are SB-lattices. The observation that $\mathcal{B}_γ$ is join-distributive is due to Armstrong. Subsequently, we investigate for which finite Coxeter groups $W$ and which Coxeter elements $γ\in W$ the lattice $\mathcal…
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…
