123 research outputs found

    Notes on lattice points of zonotopes and lattice-face polytopes

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    Minkowski's second theorem on successive minima gives an upper bound on the volume of a convex body in terms of its successive minima. We study the problem to generalize Minkowski's bound by replacing the volume by the lattice point enumerator of a convex body. In this context we are interested in bounds on the coefficients of Ehrhart polynomials of lattice polytopes via the successive minima. Our results for lattice zonotopes and lattice-face polytopes imply, in particular, that for 0-symmetric lattice-face polytopes and lattice parallelepipeds the volume can be replaced by the lattice point enumerator.Comment: 16 pages, incorporated referee remarks, corrected proof of Theorem 1.2, added new co-autho

    Unimodality Problems in Ehrhart Theory

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    Ehrhart theory is the study of sequences recording the number of integer points in non-negative integral dilates of rational polytopes. For a given lattice polytope, this sequence is encoded in a finite vector called the Ehrhart h∗h^*-vector. Ehrhart h∗h^*-vectors have connections to many areas of mathematics, including commutative algebra and enumerative combinatorics. In this survey we discuss what is known about unimodality for Ehrhart h∗h^*-vectors and highlight open questions and problems.Comment: Published in Recent Trends in Combinatorics, Beveridge, A., et al. (eds), Springer, 2016, pp 687-711, doi 10.1007/978-3-319-24298-9_27. This version updated October 2017 to correct an error in the original versio

    On positivity of Ehrhart polynomials

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    Ehrhart discovered that the function that counts the number of lattice points in dilations of an integral polytope is a polynomial. We call the coefficients of this polynomial Ehrhart coefficients, and say a polytope is Ehrhart positive if all Ehrhart coefficients are positive (which is not true for all integral polytopes). The main purpose of this article is to survey interesting families of polytopes that are known to be Ehrhart positive and discuss the reasons from which their Ehrhart positivity follows. We also include examples of polytopes that have negative Ehrhart coefficients and polytopes that are conjectured to be Ehrhart positive, as well as pose a few relevant questions.Comment: 40 pages, 7 figures. To appear in in Recent Trends in Algebraic Combinatorics, a volume of the Association for Women in Mathematics Series, Springer International Publishin

    Coxeter submodular functions and deformations of Coxeter permutahedra

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    We describe the cone of deformations of a Coxeter permutahedron, or equivalently, the nef cone of the toric variety associated to a Coxeter complex. This family of polytopes contains polyhedral models for the Coxeter-theoretic analogs of compositions, graphs, matroids, posets, and associahedra. Our description extends the known correspondence between generalized permutahedra, polymatroids, and submodular functions to any finite reflection group.Comment: Minor edits. To appear in Advances of Mathematic

    On the sum of the Voronoi polytope of a lattice with a zonotope

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    A parallelotope PP is a polytope that admits a facet-to-facet tiling of space by translation copies of PP along a lattice. The Voronoi cell PV(L)P_V(L) of a lattice LL is an example of a parallelotope. A parallelotope can be uniquely decomposed as the Minkowski sum of a zone closed parallelotope PP and a zonotope Z(U)Z(U), where UU is the set of vectors used to generate the zonotope. In this paper we consider the related question: When is the Minkowski sum of a general parallelotope and a zonotope P+Z(U)P+Z(U) a parallelotope? We give two necessary conditions and show that the vectors UU have to be free. Given a set UU of free vectors, we give several methods for checking if P+Z(U)P + Z(U) is a parallelotope. Using this we classify such zonotopes for some highly symmetric lattices. In the case of the root lattice E6\mathsf{E}_6, it is possible to give a more geometric description of the admissible sets of vectors UU. We found that the set of admissible vectors, called free vectors, is described by the well-known configuration of 2727 lines in a cubic. Based on a detailed study of the geometry of PV(e6)P_V(\mathsf{e}_6), we give a simple characterization of the configurations of vectors UU such that PV(E6)+Z(U)P_V(\mathsf{E}_6) + Z(U) is a parallelotope. The enumeration yields 1010 maximal families of vectors, which are presented by their description as regular matroids.Comment: 30 pages, 4 figures, 4 table

    Brick polytopes, lattice quotients, and Hopf algebras

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    This paper is motivated by the interplay between the Tamari lattice, J.-L. Loday's realization of the associahedron, and J.-L. Loday and M. Ronco's Hopf algebra on binary trees. We show that these constructions extend in the world of acyclic kk-triangulations, which were already considered as the vertices of V. Pilaud and F. Santos' brick polytopes. We describe combinatorially a natural surjection from the permutations to the acyclic kk-triangulations. We show that the fibers of this surjection are the classes of the congruence ≡k\equiv^k on Sn\mathfrak{S}_n defined as the transitive closure of the rewriting rule UacV1b1⋯VkbkW≡kUcaV1b1⋯VkbkWU ac V_1 b_1 \cdots V_k b_k W \equiv^k U ca V_1 b_1 \cdots V_k b_k W for letters a<b1,…,bk<ca < b_1, \dots, b_k < c and words U,V1,…,Vk,WU, V_1, \dots, V_k, W on [n][n]. We then show that the increasing flip order on kk-triangulations is the lattice quotient of the weak order by this congruence. Moreover, we use this surjection to define a Hopf subalgebra of C. Malvenuto and C. Reutenauer's Hopf algebra on permutations, indexed by acyclic kk-triangulations, and to describe the product and coproduct in this algebra and its dual in term of combinatorial operations on acyclic kk-triangulations. Finally, we extend our results in three directions, describing a Cambrian, a tuple, and a Schr\"oder version of these constructions.Comment: 59 pages, 32 figure

    Lattices, Polytopes and Tilings

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