Separating hyperplanes of edge polytopes

Let $G$ be a finite connected simple graph with $d$ vertices and let \Pc_G \subset \RR^d be the edge polytope of $G$. We call \Pc_G \emph{decomposable} if \Pc_G decomposes into integral polytopes \Pc_{G^+} and \Pc_{G^-} via a hyperplane. In this paper, we explore various aspects of decomposition of \Pc_G: we give an algorithm deciding the decomposability of \Pc_G, we prove that \Pc_G is normal if and only if both \Pc_{G^+} and \Pc_{G^-} are normal, and we also study how a condition on the toric ideal of \Pc_G (namely, the ideal being generated by quadratic binomials) behaves under decomposition.Comment: 15page

Volume of Hypercubes Clipped by Hyperplanes and Combinatorial Identities

There was an elegant expression for the volume of hypercube $[0,1]^n$ clipped by a hyperplane. We generalize the formula to the case of more than one hyperplane. Furthermore we derive several combinatorial identities from the volume expressions of clipped hypercubes

A vector partition function for the multiplicities of sl_k(C)

We use Gelfand-Tsetlin diagrams to write down the weight multiplicity function for the Lie algebra sl_k(C) (type A_{k-1}) as a single partition function. This allows us to apply known results about partition functions to derive interesting properties of the weight diagrams. We relate this description to that of the Duistermaat-Heckman measure from symplectic geometry, which gives a large-scale limit way to look at multiplicity diagrams. We also provide an explanation for why the weight polynomials in the boundary regions of the weight diagrams exhibit a number of linear factors. Using symplectic geometry, we prove that the partition of the permutahedron into domains of polynomiality of the Duistermaat-Heckman function is the same as that for the weight multiplicity function, and give an elementary proof of this for sl_4(C) (A_3).Comment: 34 pages, 11 figures and diagrams; submitted to Journal of Algebr

Coxeter submodular functions and deformations of Coxeter permutahedra

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