The convex hull of the roots of a classical root lattice is called a root polytope. We determine explicit unimodular triangulations of the boundaries of the root polytopes associated to the root lattices A_n, C_n, and D_n, and compute their f-and h-vectors. This leads us to recover formulae for the growth series of these root lattices, which were first conjectured by Conway-Mallows-Sloane and Baake-Grimm and proved by Conway-Sloane and Bacher-de la Harpe-Venkov.
In this paper we show that the pipe dream complex associated to the permutation 1n(n-1)...2 can be geometrically realized as a triangulation of the vertex figure of a root polytope. Leading up to this result we show that the Grothendieck polynomial specializes to the h-polynomial of the corresponding pipe dream complex, which in certain cases equals the h-polynomial of canonical triangulations of root (and flow) polytopes, which in turn equals a specialization of the reduced form of a monomial in the subdivision algebra of root (and flow) polytopes. Thus, we connect Grothendieck polynomials to…
In this paper, we study the Ehrhart polynomial of the dual of the root polytope of type C of dimension $d$, denoted by $C_d^*$. We prove that the roots of the Ehrhart polynomial of $C_d^*$ have the same real part $-1/2$, and we also prove that the Ehrhart polynomials of $C_d^*$ for $d=1,2,\cdots$ has the interlacing property.
Let $Φ$ be an irreducible crystallographic root system and $\mathcal P$ its root polytope, i.e., its convex hull. We provide a uniform construction, for all root types, of a triangulation of the facets of $\mathcal P$. We also prove that, on each orbit of facets under the action of the Weyl gruop, the triangulation is unimodular with respect to a root sublattice that depends on the orbit.
We completely characterize the faces of the root polytope $\tilde Q_G = \text{conv}\{\mathbf 0, \mathbf e_i - \mathbf e_j\: (i,j) \in E(G)\}$ combinatorially. Our results specialize to state of the art results in a straightforward way.
Subword complexes are simplicial complexes introduced by Knutson and Miller to illustrate the combinatorics of Schubert polynomials and determinantal ideals. They proved that any subword complex is homeomorphic to a ball or a sphere and asked about their geometric realizations. We show that a family of subword complexes can be realized geometrically via regular triangulations of root polytopes. This implies that a family of $β$-Grothendieck polynomials are special cases of reduced forms in the subdivision algebra of root polytopes. We can also write the volume and Ehrhart series of root polyto…
We show that lattice polytopes cut out by root systems of classical type are normal and Koszul, generalizing a well-known result of Bruns, Gubeladze, and Trung in type A. We prove similar results for Cayley sums of collections of polytopes whose Minkowski sums are cut out by root systems. The proofs are based on a combinatorial characterization of diagonally split toric varieties.
The symmetric edge polytopes of odd cycles (del Pezzo polytopes) are known as smooth Fano polytopes. In this paper, we show that if the length of the cycle is 127, then the Ehrhart polynomial has a root whose real part is greater than the dimension. As a result, we have a smooth Fano polytope that is a counterexample to the two conjectures on the roots of Ehrhart polynomials.
Given arbitrary integers $k$ and $d$ with $0 \leq 2k \leq d$, we construct a Gorenstein Fano polytope $\Pc \subset \RR^d$ of dimension $d$ such that (i) its Ehrhart polynomial $i(\Pc, n)$ possesses $d$ distinct roots; (ii) $i(\Pc, n)$ possesses exactly $2k$ imaginary roots; (iii) $i(\Pc, n)$ possesses exactly $d - 2k$ real roots; (iv) the real part of each of the imaginary roots is equal to $- 1 / 2$; (v) all of the real roots belong to the open interval $(-1, 0)$.
V. Golyshev conjectured that for any smooth polytope P of dimension at most five, the roots $z\in\C$ of the Ehrhart polynomial for P have real part equal to -1/2. An elementary proof is given, and in each dimension the roots are described explicitly. We also present examples which demonstrate that this result cannot be extended to dimension six.
Triangulations of a product of two simplices and, more generally, of root polytopes are closely related to Gelfand-Kapranov-Zelevinsky's theory of discriminants, to tropical geometry, tropical oriented matroids, and to generalized permutohedra. We introduce a new approach to these objects, identifying a triangulation of a root polytope with a certain bijection between lattice points of two generalized permutohedra. In order to study such bijections, we define trianguloids as edge-colored graphs satisfying simple local axioms. We prove that trianguloids are in bijection with triangulations of r…
Let $\mathcal P_Φ$ be the root polytope of a finite irreducible crystallographic root system $Φ$, i.e., the convex hull of all roots in $Φ$. The polar of $\mathcal P_Φ$, denoted $\mathcal P_Φ^*$, coincides with the union of the orbit of the fundamental alcove under the action of the Weyl group. In this paper, we establishes which polytopes $\mathcal P_Φ^*$ are zonotopes and which are not. The proof is constructive.
We study the root polytope $\mathcal P_Φ$ of a finite irreducible crystallographic root system $Φ$ using its relation with the abelian ideals of a Borel subalgebra of a simple Lie algebra with root system $Φ$. We determine the hyperplane arrangement corresponding to the faces of codimension 2 of $\mathcal P_Φ$ and analyze its relation with the facets of $\mathcal P_Φ$. For $Φ$ of type $A_n$ or $C_n$, we show that the orbits of some special subsets of abelian ideals under the action of the Weyl group parametrize a triangulation of $\mathcal P_Φ$. We show that this triangulation restricts to a t…
Generalized alcoved polytopes are polytopes whose facet normals are roots in a given root system. We call a set of points in an alcoved polytope a generating set if there does not exist a strictly smaller alcoved polytope containing it. The type $A$ alcoved polytopes are precisely the tropical polytopes that are also convex in the usual sense. In this case the tropical generators form a generating set. We show that for any root system other than $F_4$, every alcoved polytope invariant under the natural Weyl group action has a generating set of cardinality equal to the Coxeter number of the roo…
Recent work has focused on the roots z of the Ehrhart polynomial of a lattice polytope P. The case when Re(z) = -1/2 is of particular interest: these polytopes satisfy Golyshev's "canonical line hypothesis". We characterise such polytopes when dim(P) <= 7. We also consider the "half-strip condition", where all roots z satisfy -dim(P)/2 <= Re(z) <= dim(P)/2-1, and show that this holds for any reflexive polytope with dim(P) <= 5. We give an example of a 10-dimensional reflexive polytope which violates the half-strip condition, thus improving on an example by Ohsugi--Shibata in dimension 34.
Several polytopes arise from finite graphs. For edge and symmetric edge polytopes, in particular, exhaustive computation of the Ehrhart polynomials not merely supports the conjecture of Beck {\it et al.}\ that all roots $α$ of Ehrhart polynomials of polytopes of dimension $D$ satisfy $-D \le \Re(α) \le D - 1$, but also reveals some interesting phenomena for each type of polytope. Here we present two new conjectures: (1) the roots of the Ehrhart polynomial of an edge polytope for a complete multipartite graph of order $d$ lie in the circle $|z+\tfrac{d}{4}| \le \tfrac{d}{4}$ or are negative int…
Let $Φ$ be a finite crystallographic irreducible root system and $\mathcal P_Φ$ be the convex hull of the roots in $Φ$. We give a uniform explicit description of the polytope $\mathcal P_Φ$, analyze the algebraic-combinatorial structure of its faces, and provide connections with the Borel subalgebra of the associated Lie algebra. We also give several enumerative results.
The Ehrhart polynomial of a convex lattice polytope counts integer points in integral dilates of the polytope. We present new linear inequalities satisfied by the coefficients of Ehrhart polynomials and relate them to known inequalities. We also investigate the roots of Ehrhart polynomials. We prove that for fixed d, there exists a bounded region of C containing all roots of Ehrhart polynomials of d-polytopes, and that all real roots of these polynomials lie in [-d, [d/2]). In contrast, we prove that when the dimension d is not fixed the positive real roots can be arbitrarily large. We finish …
In this paper, we study the root distributions of Ehrhart polynomials of free sums of certain reflexive polytopes. We investigate cases where the roots of the Ehrhart polynomials of the free sums of $A_d^\vee$'s or $A_d$'s lie on the canonical line $\mathrm{Re}(z)=-\frac{1}{2}$ on the complex plane $\mathbb{C}$, where $A_d$ denotes the root polytope of type A of dimension $d$ and $A_d^\vee$ denotes its polar dual. For example, it is proved that $A_m^\vee \oplus A_n^\vee$ with $\min\{m,n\} \leq 1$ or $m+n \leq 7$, $A_2^\vee \oplus (A_1^\vee)^{\oplus n}$ and $A_3^\vee \oplus (A_1^\vee)^{\oplus n…
We use the method of steepest descents to study the root distribution of the Ehrhart polynomial of the $d$-dimensional cross-polytope, namely $\mathcal{L}_{d}$, as $d\rightarrow \infty$. We prove that the distribution function of the roots, approximately, as $d$ grows, by variation of argument of the generating function $\sum_{m\geq 0}\mathcal{L}_{d}(m)t^{m+x-1}=(1+t)^{d}(1-t)^{-d-1}t^{x-1}$, as $t$ varies appropriately on the segment of the imaginary line contained inside the unit disk.
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…
The polytopes $\mathcal{U}_{I,\bar{J}}$ were introduced by Ceballos, Padrol, and Sarmiento to provide a geometric approach to the study of $(I,\bar{J})$-Tamari lattices. They observed a connection between certain $\mathcal{U}_{I,\bar{J}}$ and acyclic root polytopes, and wondered if Mészáros' subdivision algebra can be used to subdivide all $\mathcal{U}_{I,\bar{J}}$. We answer this in the affirmative from two perspectives, one using flow polytopes and the other using root polytopes. We show that $\mathcal{U}_{I,\bar{J}}$ is integrally equivalent to a flow polytope that can be subdivided using t…
The type A_n full root polytope is the convex hull in R^{n+1} of the origin and the points e_i-e_j for 1<= i<j <= n+1. Given a tree T on the vertex set [n+1], the associated root polytope P(T) is the intersection of the full root polytope with the cone generated by the vectors e_i-e_j, where (i, j) is an edge of T, i<j. The reduced forms of a certain monomial m[T] in commuting variables x_{ij} under the reduction x_{ij}x_{jk} --> x_{ik}x_{ij}+x_{jk}x_{ik}+βx_{ik}, can be interpreted as triangulations of P(T). Using these triangulations, the volume and Ehrhart polynomial of P(T) are obtained. I…
Consider a lattice in a real finite dimensional vector space. Here, we are interested in the lattice polytopes, that is the convex hulls of finite subsets of the lattice. Consider the group $G$ of the affine real transformations which map the lattice onto itself. Replacing the group of euclidean motions by the group $G$ one can define the notion of regular lattice polytopes. More precisely, a lattice polytope is said to be regular if the subgroup of $G$ which preserves the polytope acts transitively on the set of its complete flags. Recently, Karpenkov obtained a classification of the regular …
This paper presents an algorithm to compute the value of the inverse Laplace transforms of rational functions with poles on arrangements of hyperplanes. As an application, we present an efficient computation of the partition function for classical root systems.
The Voronoi conjecture on parallelohedra claims that for every convex polytope $P$ that tiles Euclidean $d$-dimensional space with translations there exists a $d$-dimensional lattice such that $P$ and the Voronoi polytope of this lattice are affinely equivalent. The Voronoi conjecture is still open for the general case but it is known that some combinatorial restriction for the face structure of $P$ ensure that the Voronoi conjecture holds for $P$. In this paper we prove that if $P$ is the Voronoi polytope of one of the dual root lattices $\mathsf{D}_d^*$, $\mathsf{E}_6^*$, $\mathsf{E}_7^*$ or…
The square root rank of a nonnegative matrix $A$ is the minimum rank of a matrix $B$ such that $A=B \circ B$, where $\circ$ denotes entrywise product. We show that the square root rank of the slack matrix of the correlation polytope is exponential. Our main technique is a way to lower bound the rank of certain matrices under arbitrary sign changes of the entries using properties of the roots of polynomials in number fields. The square root rank is an upper bound on the positive semidefinite rank of a matrix, and corresponds the special case where all matrices in the factorization are rank-one.
The full root polytope of type $A$ is the convex hull of all pairwise differences of the standard basis vectors which we represent by forward and backward arrows. We completely classify all flag triangulations of this polytope that are uniform in the sense that the edges may be described as a function of the relative order of the indices of the four basis vectors involved. These fifteen triangulations fall naturally into three classes: three in the lex class, three in the revlex class and nine in the Simion class. We also consider a refined face count where we distinguish between forward and b…
Given a crystallographic reduced root system and an element v of the lattice generated by the roots we study the minimum number |v|, called the length of v, of roots needed to express v as sum of roots. This number is related to the linear functionals presenting the convex hull of the roots; the map v --> |v| turns out to be piecewise quasi-linear with quasi-linearity domains the cones over the facets of this convex hull. In order to show this relation we investigate the integral closure of the monoid generated by the roots in a facet. We study also the positive lenght, i.e. the minimum number…
The coordination sequence of a lattice $Ł$ encodes the word-length function with respect to $M$, a set that generates $Ł$ as a monoid. We investigate the coordination sequence of the cyclotomic lattice $Ł= \Z[ζ_m]$, where $ζ_m$ is a primitive $mþ$ root of unity and where $M$ is the set of all $mþ$ roots of unity. We prove several conjectures by Parker regarding the structure of the rational generating function of the coordination sequence; this structure depends on the prime factorization of $m$. Our methods are based on unimodular triangulations of the $mþ$ cyclotomic polytope, the convex hul…
Let G be a connected bipartite graph with color classes E and V and root polytope Q. Regarding the hypergraph (V,E) induced by G, we prove that its interior polynomial is equivalent to the Ehrhart polynomial of Q, which in turn is equivalent to the h-vector of any triangulation of Q. It follows that the interior polynomials of (V,E) and its transpose (E,V) agree. When G is a complete bipartite graph, our result recovers a well known hypergeometric identity due to Saalschütz. It also implies that certain extremal coefficients in the Homfly polynomial of a special alternating link can be read of…
We determine lattice polytopes of smallest volume with a given number of interior lattice points. We show that the Ehrhart polynomials of those with one interior lattice point have largest roots with norm of order n^2, where n is the dimension. This improves on the previously best known bound n and complements a recent result of Braun where it is shown that the norm of a root of an Ehrhart polynomial is at most of order n^2. For the class of 0-symmetric lattice polytopes we present a conjecture on the smallest volume for a given number of interior lattice points and prove the conjecture for cr…
Let $ρ$ be a metric on the set $X=\{1,2,\dots,n+1\}$. Consider the $n$-dimensional polytope of functions $f:X\rightarrow \mathbb{R}$, which satisfy the conditions $f(n+1)=0$, $|f(x)-f(y)|\leq ρ(x,y)$. The question on classifying metrics depending on the combinatorics of this polytope have been recently posed by A. M. Vershik \cite{V}. We prove that for any "generic" metric the number of $(n-m)$-dimensional faces, $0\leq m\leq n$, equals $\binom{n+m}{m,m,n-m}=(n+m)!/m!m!(n-m)!$. This fact is intimately related to regular triangulations of the root polytope (the convex hull of the roots of $A_n$…
As a discrete analog to Minkowski's theorem on convex bodies, Wills conjectured that the Ehrhart coefficients of a centrally symmetric lattice polytope with exactly one interior lattice point are maximized by those of the cube of side length two. We discuss several counterexamples to this conjecture and, on the positive side, we identify a family of lattice polytopes that fulfill the claimed inequalities. This family is related to the recently introduced class of $l$-reflexive polytopes.
The type C_n full root polytope is the convex hull in R^n of the origin and the points e_i-e_j, e_i+e_j, 2e_k for 1 <= i < j <= n, k \in [n]. Given a graph G, with edges labeled positive or negative, associate to each edge e of G a vector v(e) which is e_i-e_j if e=(i, j), i < j, is labeled negative and e_i+e_j if it is labeled positive. For such a signed graph G, the associated root polytope P(G) is the intersection of the full root polytope with the cone generated by the vectors v(e), for edges e in G. The reduced forms of a certain monomial m[G] in commuting variables x_{ij}, y_{ij}, z_k un…
We study a class of polynomials that has all of its roots on the critical line and shares many properties with Ehrhart polynomials. Braun showed that the roots of Ehrhart polynomials are bounded quadratically and Higashitani provided examples for polytopes whose Ehrhart polynomial roots come close to this bound. In the case of polytopes which have their roots on the critical line, we present an improved bound. As a side effect this confirms a special case of a conjecture of Braun.
Can one build an arbitrary polytope from any polytope inside by iteratively stacking pyramids onto facets, without losing the convexity throughout the process? We prove that this is indeed possible for (i) 3-polytopes, (ii) 4-polytopes under a certain infinitesimal quasi-pyramidal relaxation, and (iii) all dimensions asymptotically. The motivation partly comes from our study of K-theory of monoid rings and of certain posets of discrete-convex objects.
The conjecture on roots of Ehrhart polynomials, stated by Matsui et al. \cite[Conjecture 4.10]{MHNOH}, says that all roots $α$ of the Ehrhart polynomial of a Gorenstein Fano polytope of dimension $d$ satisfy $-\frac{d}{2} \leq \Re(α) \leq \frac{d}{2} -1$. In this paper, we observe the behaviors of roots of SSNN polynomials which are a wider class of the polynomials containing all the Ehrhart polynomials of Gorenstein Fano polytopes. As a result, we verify that this conjecture is true when the roots are real numbers or when $d \leq 5$.
We study real sequences $\{a_{n}\}_{n\in \mathbb{N}}$ that eventually agree with a polynomial. We show that if the numerator polynomial of its rational generating series is of degree $s$ and has only nonnegative coefficients, then the numerator polynomial of the subsequence $\{ a_{rn+i}\}_{n\in \mathbb{N}}$, $0\leq i<r$, has only nonpositive, real roots for all $r\geq s-i$. We apply our results to combinatorially positive valuations on polytopes and to Hilbert functions of Veronese submodules of graded Cohen-Macaulay algebras. In particular, we prove that the Ehrhart $h^\ast$-polynomial of the…
In this paper we study the role of planarity in generalized scattering amplitudes, through several closely interacting structures in combinatorics, algebraic and tropical geometry. The generalized biadjoint scalar amplitude, introduced recently by Cachazo-Early-Guevara-Mizera (CEGM), is a rational function of homogeneous degree $-(k-1)(n-k-1)$ in $\binom{n}{k}-n$ independent variables; its poles can be constructed directly from the rays of the positive tropical Grassmannian. We introduce for each pair of integers $(k,n)$ with $2\le k\le n-2$ a system of generalized positive roots which arises …
We study the well-known problem of combinatorial classification of fullerenes. By a (mathematical) fullerene we mean a convex simple three dimensional polytope with all facets pentagons and hexagons. We analyse approaches of construction of arbitrary fullerene from the dodecahedron (a fullerene $C_{20}$). A growth operation is a combinatorial operation that substitutes the patch with more facets and the same boundary for the patch on the surface of a simple polytope to produce a new simple polytope. It is known that an infinite set of different growth operations transforming fullerenes into fu…
We study the connection between triangulations of a type $A$ root polytope and the resonance arrangement, a hyperplane arrangement that shows up in a surprising number of contexts. Despite an elementary definition for the resonance arrangement, the number of resonance chambers has only been computed up to the $n=8$ dimensional case. We focus on data structures for labeling chambers, such as sign vectors and sets of alternating trees, with an aim at better understanding the structure of the resonance arrangement, and, in particular, enumerating its chambers. Along the way, we make connections w…
We investigate the Ehrhart polynomial for the class of 0-symmetric convex lattice polytopes in Euclidean $n$-space $\mathbb{R}^n$. It turns out that the roots of the Ehrhart polynomial and Minkowski's successive minima are closely related by their geometric and arithmetic mean. We also show that the roots of lattice $n$-polytopes with or without interior lattice points differ essentially. Furthermore, we study the structure of the roots in the planar case. Here it turns out that their distribution reflects basic properties of lattice polygons.
In this paper we prove that points in the space $X(k,n)$ of configurations of $n$ points in $\mathbb{CP}^{k-1}$ which are fixed under a certain cyclic action are the solutions to the generalized scattering equations on planar kinematics (PK). In the first part, we give a constructive upper bound: we show that these solutions inject into certain aperiodic k-element subsets of $\{1,\ldots, n\}$, and consequently that their number is bounded above by the number of Lyndon words with k one's and n-k zeros. The proof uses a somewhat surprising connection between the superpotential of the mirror of $…
