Face Lattice Research Articles

Study of electronic and magnetic properties of MnS layers

Self-consistent ab initio calculations, based on the density functional theory (DFT) and using the full potential linear augmented plane wave (FLAPW) method, are performed to investigate both electronic and magnetic properties of the MnS layers. Polarized spin and spin—orbit coupling are included in the calculations within the framework of the antiferromagnetic state between two adjacent Mn layers. Magnetic moments considered to lie along axes are computed. Obtained data from ab initio calculations are used as input data for the high temperature series expansion (HTSE) calculations to compute other magnetic parameters. The zero-field high temperature static susceptibility series of the spin-4.39 nearest-neighbour Heisenberg model on centred face cubic (FCC) and lattices is thoroughly analysed by a power series coherent anomaly method (CAM). The exchange interactions between the magnetic atoms, the Néel temperature, and the critical exponent associated with the magnetic susceptibility are obtained for the MnS layer.

A combinatorial proof of the cyclic sieving phenomenon for faces of Coxeterhedra

For a Coxeter system (W,S), the subgroup W J generated by a subset J⊆S is called a parabolic subgroup of W. The Coxeterhedron PW associated to (W,S) is the finite poset of all cosets {wW J } w?W,J⊆S of all parabolic subgroups of W, ordered by inclusion. This poset can be realized by the face lattice of a simple polytope, constructed as the convex hull of the orbit of a generic point in ? n under an action of the reflection group W. In this paper, for the groups W=A n?1, B n and D n in a case-by-case manner, we present an elementary proof of the cyclic sieving phenomenon for faces of various dimensions of PW under the action of a cyclic group generated by a Coxeter element. This result provides a geometric, enumerative and combinatorial approach to re-prove a theorem in Reiner et al. (J. Comb. Theory, Ser. A 108:17---50, 2004). The original proof is proved by an algebraic method that involves representation theory and Springer's theorem on regular elements.

Cones and convex bodies with modular face lattices

If a convex body C has modular and irreducible face lattice (and is not strictly convex), there is a face-preserving homeomorphism from C to a section of a cone of hermitian matrices over R, C, or H, or C has dimension 8, 14 or 26.

Eigenvectors for a random walk on a hyperplane arrangement

We find explicit eigenvectors for the transition matrix of the Bidigare–Hanlon–Rockmore random walk, from Bidigare et al. (1999) [1]. This is accomplished by using Brown and Diaconisʼ (1998) analysis in [3] of the stationary distribution, together with some combinatorics of functions on the face lattice of a hyperplane arrangement, due to Gelʼfand and Varchenko (1987) [10].

ORBITOPES

An orbitope is the convex hull of an orbit of a compact group acting linearly on a vector space. These highly symmetric convex bodies lie at the crossroads of several fields, including convex geometry, algebraic geometry, and optimization. We present a self-contained theory of orbitopes, with particular emphasis on instances arising from the groups SO(n) and O(n); these include Schur–Horn orbitopes, tautological orbitopes, Carathéodory orbitopes, Veronese orbitopes, and Grassmann orbitopes. We study their face lattices, algebraic boundaries, and representations as spectrahedra or projected spectrahedra.

Quantum convex support

Convex support, the mean values of a set of random variables, is central in information theory and statistics. Equally central in quantum information theory are mean values of a set of observables in a finite-dimensional C ∗-algebra A , which we call (quantum) convex support. The convex support can be viewed as a projection of the state space of A and it is a projection of a spectrahedron. Spectrahedra are increasingly investigated at least since the 1990s boom in semi-definite programming. We recall the geometry of the positive semi-definite cone and of the state space. We write a convex duality for general self-dual convex cones. This restricts to projections of state spaces and connects them to results on spectrahedra. Our main result is an analysis of the face lattice of convex support by mapping this lattice to a lattice of orthogonal projections, using natural isomorphisms. The result encodes the face lattice of the convex support into a set of projections in A and enables the integration of convex geometry with matrix calculus or algebraic techniques.

Maximal admissible faces and asymptotic bounds for the normal surface solution space

The enumeration of normal surfaces is a key bottleneck in computational three-dimensional topology. The underlying procedure is the enumeration of admissible vertices of a high-dimensional polytope, where admissibility is a powerful but non-linear and non-convex constraint. The main results of this paper are significant improvements upon the best known asymptotic bounds on the number of admissible vertices, using polytopes in both the standard normal surface coordinate system and the streamlined quadrilateral coordinate system. To achieve these results we examine the layout of admissible points within these polytopes. We show that these points correspond to well-behaved substructures of the face lattice, and we study properties of the corresponding “admissible faces”. Key lemmata include upper bounds on the number of maximal admissible faces of each dimension, and a bijection between the maximal admissible faces in the two coordinate systems mentioned above.

Nonrealizable Minimal Vertex Triangulations of Surfaces: Showing Nonrealizability Using Oriented Matroids and Satisfiability Solvers

We show that no minimal vertex triangulation of a closed, connected, orientable 2-manifold of genus 6 admits a polyhedral embedding in ℝ3. We also provide examples of minimal vertex triangulations of closed, connected, orientable 2-manifolds of genus 5 that do not admit any polyhedral embeddings. Correcting a previous error in the literature, we construct the first infinite family of such nonrealizable triangulations of surfaces. These results were achieved by transforming the problem of finding suitable oriented matroids into a satisfiability problem. This method can be applied to other geometric realizability problems, e.g., for face lattices of polytopes.

Cubical Convex Ear Decompositions

We consider the problem of constructing a convex ear decomposition for a poset. The usual technique, introduced by Nyman and Swartz, starts with a $CL$-labeling and uses this to shell the 'ears' of the decomposition. We axiomatize the necessary conditions for this technique as a "$CL$-ced" or "$EL$-ced". We find an $EL$-ced of the $d$-divisible partition lattice, and a closely related convex ear decomposition of the coset lattice of a relatively complemented finite group. Along the way, we construct new $EL$-labelings of both lattices. The convex ear decompositions so constructed are formed by face lattices of hypercubes. We then proceed to show that if two posets $P_{1}$ and $P_{2}$ have convex ear decompositions ($CL$-ceds), then their products $P_{1}\times P_{2}$, $P_{1}\check{\times} P_{2}$, and $P_{1}\hat{\times} P_{2}$ also have convex ear decompositions ($CL$-ceds). An interesting special case is: if $P_{1}$ and $P_{2}$ have polytopal order complexes, then so do their products.

The Alternating Sign Matrix Polytope

We define the alternating sign matrix polytope as the convex hull of $n\times n$ alternating sign matrices and prove its equivalent description in terms of inequalities. This is analogous to the well known result of Birkhoff and von Neumann that the convex hull of the permutation matrices equals the set of all nonnegative doubly stochastic matrices. We count the facets and vertices of the alternating sign matrix polytope and describe its projection to the permutohedron as well as give a complete characterization of its face lattice in terms of modified square ice configurations. Furthermore we prove that the dimension of any face can be easily determined from this characterization.

Serre's Condition Rℓ for Affine Semigroup Rings

In this note we characterize the affine semigroup rings K[S] over an arbitrary field K that satisfy condition Rℓ of Serre. Our characterization is in terms of the face lattice of the positive cone pos(S) of S. We start by reviewing some basic facts about the faces of pos(S) and consequences for the monomial primes of K[S]. After proving our characterization we turn our attention to the Rees algebras of a special class of monomial ideals in a polynomial ring over a field. In this special case, some of the characterizing criteria are always satisfied. We give examples of non-normal affine semigroup rings that satisfy R2.

Affine and Toric Hyperplane Arrangements

We extend the Billera-Ehrenborg-Readdy map between the intersection lattice and face lattice of a central hyperplane arrangement to affine and toric hyperplane arrangements. For arrangements on the torus, we also generalize Zaslavsky's fundamental results on the number of regions.

Finding a Simple Polytope from Its Graph in Polynomial Time

We show that one can compute the combinatorial facets of a simple polytope from its graph in polynomial time. Our proof relies on a primal-dual characterization (by Joswig, Kaibel, and Korner in Israel J. Math. 129:109–118, 2002) and a linear program, with an exponential number of constraints, which can be used to construct the solution and can be solved in polynomial time. We show that this allows one to characterize the face lattice of the polytope, via a simple face recognition algorithm. In addition, we use this linear program to construct several interesting polynomial-time computable sets of graphs which may be of independent interest.

Matching polytopes, toric geometry, and the totally non-negative Grassmannian

In this paper we use toric geometry to investigate the topology of the totally non-negative part of the Grassmannian, denoted (Gr k,n )≥0. This is a cell complex whose cells Δ G can be parameterized in terms of the combinatorics of plane-bipartite graphs G. To each cell Δ G we associate a certain polytope P(G). The polytopes P(G) are analogous to the well-known Birkhoff polytopes, and we describe their face lattices in terms of matchings and unions of matchings of G. We also demonstrate a close connection between the polytopes P(G) and matroid polytopes. We use the data of P(G) to define an associated toric variety X G . We use our technology to prove that the cell decomposition of (Gr k,n )≥0 is a CW complex, and furthermore, that the Euler characteristic of the closure of each cell of (Gr k,n )≥0 is 1.

On reduced polytopes and antipodality

AbstractLetBbe ano-symmetric convex body in ℝd, andMdbe the normed space with unit ballB. TheMd-thickness ΔB(K) of a convex bodyK⊆ ℝdis the smallest possibleMd-distance between two distinct parallel supporting hyperplanes ofK. Furthermore,Kis said to beMd-reduced if ΔB(K′) < ΔB(K) for every convex bodyK′ withK′ ⊆KandK′ ≠K. In our main theorems we describeMd-reduced polytopes as polytopes whose face lattices possess certain antipodality properties. As one of the consequences, we obtain that if the boundary ofBis regular, then ad-polytope withmfacets andnvertices is notMd-reduced providedm=d+ 2 orn=d+ 2 orn>m. The latter statement yields a new partial answer to Lassak's question on the existence of Euclidean reducedd-polytopes ford≥ 3.

Orthogonal Surfaces and Their CP-Orders

Orthogonal surfaces are nice mathematical objects which have interesting connections to various fields, e.g., integer programming, monomial ideals and order dimension. While orthogonal surfaces in one or two dimensions are rather trivial already the three dimensional case has a rich structure with connections to Schnyder woods, planar graphs and three-polytopes. Our objective is to detect more of the structure of orthogonal surfaces in four and higher dimensions. In particular we are driven by the question which non-generic orthogonal surfaces have a polytopal structure. We review the state of knowledge of the three-dimensional situation. On that basis we introduce terminology for higher dimensional orthogonal surfaces and continue with the study of characteristic points and the cp-orders of orthogonal surfaces, i.e., the dominance orders on the characteristic points. In the generic case these orders are (almost) face lattices of polytopes. Examples show that in general cp-orders can lack key properties of face lattices. We investigate extra requirements which may help to have cp-orders which are face lattices. Finally, we turn the focus and ask for the realizability of polytopes on orthogonal surfaces. There are criteria which prevent large classes of simplicial polytopes from being realizable. On the other hand we identify some families of polytopes which can be realized on orthogonal surfaces.

Affine and toric arrangements

We extend the Billera―Ehrenborg―Readdy map between the intersection lattice and face lattice of a central hyperplane arrangement to affine and toric hyperplane arrangements. For toric arrangements, we also generalize Zaslavsky's fundamental results on the number of regions. Nous étendons l'opérateur de Billera―Ehrenborg―Readdy entre le trellis d'intersection et la treillis de faces d'un arrangement hyperplans centraux aux arrangements affines et toriques. Pour les arrangements toriques, nous généralisons aussi les résultats fondamentaux de Zaslavsky sur le nombre de régions.

Cubes and orientability

We define and study a new class of matroids: cubic matroids. Cubic matroids include, as a particular case, all affine cubes over an arbitrary field. There is only one known orientable cubic matroid: the real affine cube. The main results establish as an invariant of orientable cubic matroids the structure of the subset of acyclic orientations with LV-face lattice isomorphic to the face lattice of the real cube or, equivalently, with the same signed circuits of length 4 as the real cube.

Schnyder Woods and Orthogonal Surfaces

In this paper we study connections between planar graphs, Schnyder woods, and orthogonal surfaces. Schnyder woods and the face counting approach have important applications in graph drawing and the dimension theory of orders. Orthogonal surfaces explain connections between these seemingly unrelated notions. We use these connections for an intuitive proof of the Brightwell-Trotter Theorem which says, that the face lattice of a 3-polytope minus one face has order dimension three. Our proof yields a linear time algorithm for the construction of the three linear orders that realize the face lattice. Coplanar orthogonal surfaces are in correspondence with a large class of convex straight line drawings of 3-connected planar graphs. We show that Schnyder’s face counting approach with weighted faces can be used to construct all coplanar orthogonal surfaces and hence the corresponding drawings. Appropriate weights are computable in linear time.

F-Vectors of Minkowski Additions of Convex Polytopes

The objective of this paper is to present two types of results on Minkowski sums of convex polytopes. The first is about a special class of polytopes we call perfectly centered and the combinatorial properties of the Minkowski sum with their own dual. In particular, we have a characterization of face lattice of the sum in terms of the face lattice of a given perfectly centered polytope. Exact face counting formulas are then obtained for perfectly centered simplices and hypercubes. The second type of results concerns tight upper bounds for the f-vectors of Minkowski sums of several polytopes.