Volume computation for polytopes and partition functions for classical root systems

We give an overview of some of the main results from the theories of hypergeometric and basic hypergeometric series and integrals associated with root systems. In particular, we list a number of summations, transformations and explicit evaluations for such multiple series and integrals. We concentrate on such results which do not directly extend to the elliptic level. This text is a provisional version of a chapter on hypergeometric and basic hypergeometric series and integra...

Let $\mathcal P_{\Phi}$ be the root polytope of a finite irreducible crystallographic root system $\Phi$, i.e., the convex hull of all roots in $\Phi$. The polar of $\mathcal P_{\Phi}$, denoted $\mathcal P_{\Phi}^*$, 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 $\mathcal P_{\Phi}^*$ are zonotopes and which are not. The proof is constructive.

We describe the computation of polytope volumes by descent in the face lattice, its implementation in Normaliz, and the connection to reverse-lexicographic triangulations. The efficiency of the algorithm is demonstrated by several high dimensional polytopes of different characteristics. Finally, we present an application to voting theory where polytope volumes appear as probabilities of certain paradoxa.

In a series of papers, George Andrews and various coauthors successfully revitalized seemingly forgotten, powerful machinery based on MacMahon's $\Omega$ operator to systematically compute generating functions $\sum_{\la \in P} z_1^{\la_1}...z_n^{\la_n}$ for some set $P$ of integer partitions $\la = (\la_1,..., \la_n)$. Our goal is to geometrically prove and extend many of the Andrews et al theorems, by realizing a given family of partitions as the set of integer lattice poin...

The Fourier transforms of polyhedral cones can be used, via Brion's theorem, to compute various geometric quantities of polytopes, such as volumes, moments, and lattice-point counts. We present a novel method of computing these conic Fourier transforms by polynomial interpolation. Given the fact that computing volumes of polytopes is #P-hard (Dyer--Frieze [DF88]), we cannot hope for fast algorithms in the general case. However, with extra assumptions on the combinatorics of t...

We survey the computation of polytope volumes by the algorithms of Normaliz to which the Lawrence algorithm has recently been added. It has enabled us to master volume computations for polytopes from social choice in dimension $119$. This challenge required a sophisticated implementation of the Lawrence algorithm.

We are interested in the fast computation of the exact value of integrals of polynomial functions over convex polyhedra. We present speed ups and extensions of the algorithms presented in previous work. We present the new software implementation and provide benchmark computations. The computation of integrals of polynomials over polyhedral regions has many applications; here we demonstrate our algorithmic tools solving a challenge from combinatorial voting theory.

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 ve...

In earlier work in collaboration with Pavel Galashin and Thomas McConville we introduced a version of chip-firing for root systems. Our investigation of root system chip-firing led us to define certain polynomials analogous to Ehrhart polynomials of lattice polytopes, which we termed the symmetric and truncated Ehrhart-like polynomials. We conjectured that these polynomials have nonnegative integer coefficients. Here we affirm "half" of this positivity conjecture by providing...

Motivated by representation theory and geometry, we introduce and develop an equivariant generalization of Ehrhart theory, the study of lattice points in dilations of lattice polytopes. We prove representation-theoretic analogues of numerous classical results, and give applications to the Ehrhart theory of rational polytopes and centrally symmetric polytopes. We also recover a character formula of Procesi, Dolgachev, Lunts and Stembridge for the action of a Weyl group on the ...