Symmetric Decompositions and the Veronese Construction

We study rational generating functions of sequences $\{a_n\}_{n\geq 0}$ that agree with a polynomial and investigate symmetric decompositions of the numerator polynomial for subsequences $\{a_{rn}\}_{n\geq 0}$. We prove that if the numerator polynomial for $\{a_n\}_{n\geq 0}$ is of degree $s$ and its coefficients satisfy a set of natural linear inequalities, then the symmetric decomposition of the numerator for $\{a_{rn}\}_{n\geq 0}$ is real-rooted whenever $r\geq \max \{s,d+1-s\}$. Moreover, if the numerator polynomial for $\{a_n\}_{n\geq 0}$ is symmetric, then we show that the symmetric decomposition for $\{a_{rn}\}_{n\geq 0}$ is interlacing. We apply our results to Ehrhart series of lattice polytopes. In particular, we obtain that the $h^\ast $-polynomial of every dilation of a $d$-dimensional lattice polytope of degree $s$ has a real-rooted symmetric decomposition whenever the dilation factor $r$ satisfies $r\geq \max \{s,d+1-s\}$. Moreover, if the polytope is Gorenstein, then this decomposition is interlacing.

Grossberg–Karshon Twisted Cubes and Hesitant Jumping Walk Avoidance

Let $G$ be a complex simply-laced semisimple algebraic group of rank $r$ and $B$ a Borel subgroup. Let $\mathbf i \in [r]^n$ be a word and let $\boldsymbol{\ell} = (\ell_1,\dots,\ell_n)$ be a sequence of non-negative integers. Grossberg and Karshon introduced a virtual lattice polytope associated to $\mathbf i$ and $\boldsymbol{\ell}$ called a twisted cube, whose lattice points encode the character of a $B$-representation. More precisely, lattice points in the twisted cube, counted with sign according to a certain density function, yields the character of the generalized Demazure module determined by $\mathbf i$ and $\boldsymbol{\ell}$. In recent work, the author and Harada described precisely when the Grossberg–Karshon twisted cube is untwisted, i.e., the twisted cube is a closed convex polytope, in the situation when the integer sequence $\boldsymbol{\ell}$ comes from a weight $\lambda$ of $G$. However, not every integer sequence $\boldsymbol{\ell}$ comes from a weight of $G$. In the present paper, we interpret the untwistedness of Grossberg–Karshon twisted cubes associated with any word $\mathbf i$ and any integer sequence $\boldsymbol{\ell}$ using the combinatorics of $\mathbf i$ and $\boldsymbol{\ell}$. Indeed, we prove that the Grossberg–Karshon twisted cube is untwisted precisely when $\mathbf i$ is hesitant-jumping-$\boldsymbol{\ell}$-walk-avoiding.

Ehrhart Polynomial Roots of Reflexive Polytopes

Recent work has focused on the roots $z\in\mathbb{C}$ 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 $\mathrm{dim}(P)\leq 7$. We also consider the "half-strip condition", where all roots $z$ satisfy $-\mathrm{dim}(P)/2\leq\Re{z}\leq \mathrm{dim}(P)/2-1$, and show that this holds for any reflexive polytope with $\mathrm{dim}(P)\leq 5$. 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 $34$.

Commutative and non-commutative bialgebras of quasi-posets and applications to Ehrhart polynomials

To any poset or quasi-poset is attached a lattice polytope, whose Ehrhart polynomial we study from a Hopf-algebraic point of view. We use for this two interacting bialgebras on quasi-posets. The Ehrhart polynomial defines a Hopf algebra morphism with values in \mathbb{Q}[X] . We deduce from the interacting bialgebras an algebraic proof of the duality principle, a generalization and a new proof of a result on B-series due to Whright and Zhao, using a monoid of characters on quasi-posets, and a generalization of Faulhaber’s formula. We also give non-commutative versions of these results, where polynomials are replaced by packed words. We obtain, in particular, a non-commutative duality principle.

On Counterexamples to a Conjecture of Wills and Ehrhart Polynomials whose Roots have Equal Real Parts

As a discrete analog to Minkowski's theorem on convex bodies, Wills conjectured that the Ehrhart coefficients of a $0$-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 BOUNDARY VOLUME OF A LATTICE POLYTOPE

For a d-dimensional convex lattice polytope P, a formula for the boundary volume vol(∂P) is derived in terms of the number of boundary lattice points on the first ⌊d/2⌋ dilations of P. As an application we give a necessary and sufficient condition for a polytope to be reflexive, and derive formulas for the f-vector of a smooth polytope in dimensions three, four, and five. We also give applications to reflexive order polytopes, and to the Birkhoff polytope.