Related papers: An Intersection Matrix for Affine Hyperplane Arran…

We classify one-element extensions of a hyperplane arrangement by the induced adjoint arrangement. Based on the classification, several kinds of combinatorial invariants including Whitney polynomials, characteristic polynomials, Whitney…

We study the intersection numbers defined on twisted homology or cohomology groups that are associated with hypergeometric integrals corresponding to degenerate hyperplane arrangements in the projective $k$-space. We present formulas to…

In this article, we define and study a geometry and an order on the set of partitions of an even number of objects. One of the definitions involves the partition algebra, a structure of algebra on the set of such partitions depending on an…

Based on the notion of vectors and linear subspaces for a matroid, we develop a theory of flats and hyperplane arrangements for T-matroids, where T is a tract. This leads to several cryptomorphic descriptions of T-matroids: in terms of its…

We give an explicit formula for the Hilbert Series of an algebra defined by a linearly presented, standard graded, residual intersection of a grade three Gorenstein ideal.

This mainly tutorial paper is intended as a somewhat larger example for parts of the theory exposed in the Lectures on the dynamical Yang-Baxter equations by P. Etingof and O. Schiffmann, math.QA/9908064. We explicitly compute the matrix…

There is a q-deformation of the reflection representation of the affine symmetric group, which arises in the quantum geometric Satake equivalence, and in the study of the complex reflection groups $G(m,m,n)$. Demazure operators (often…

In this paper we construct a new factorized representation of the $R$-matrix related to the affine algebra $U_{q}(\widehat{sl_{n}})$ for symmetric tensor representations with arbitrary weights. Using the 3D approach we obtain explicit…

Bernstein polynomials, long a staple of approximation theory and computational geometry, have also increasingly become of interest in finite element methods. Many fundamental problems in interpolation and approximation give rise to…

We explore some connections between vectors of integers and integer partitions seen as bi-infinite words. This methodology enables us to give a combinatorial interpretation of the Macdonald identities for affine root systems of the seven…

This paper extends some results of Hatcher and Quinn beyond the metastable range. We give a bordism theoretic obstruction to deforming a map between manifolds simultaneously off of a collection of pairwise disjoint submanifolds under the…

The problem of expressing a specific polynomial as the determinant of a square matrix of affine-linear forms arises from algebraic geometry, optimisation, complexity theory, and scientific computing. Motivated by recent developments in this…

We analyze the Macdonald's $(q,t)$-deformed hypergeometric functions with one and two set variables and present their constraints. We prove the uniqueness to the solutions of these constraints. We propose a concise method to prove the…

We observe that there is an equivalence between the singularity category of an affine complete intersection and the homotopy category of matrix factorizations over a related scheme. This relies in part on a theorem of Orlov. Using this…

The Ehrhart polynomial and Ehrhart series count lattice points in integer dilations of a lattice polytope. We introduce and study a $q$-deformation of the Ehrhart series, based on the notions of harmonic spaces and Macaulay's inverse…

We present a semidefinite programming approach to bound the measures of cross-independent pairs in a bipartite graph. This can be viewed as a far-reaching extension of Hoffman's ratio bound on the independence number of a graph. As an…

We give a new residual intersection decomposition for the refined intersection products of Fulton-MacPherson. Our formula refines the celebrated residual intersection formula of Fulton, Kleiman, Laksov, and MacPherson. The new decomposition…

Hyperplanes of the form x_j = x_i + c are called affinographic. For an affinographic hyperplane arrangement in R^n, such as the Shi arrangement, we study the function f(M) that counts integral points in [1,M]^n that do not lie in any…

Motivated by Maulik-Okounkov stable maps associated to quiver varieties, we define and construct algebraic stable maps on tensor products of representations in the category O of the Borel subalgebra of an arbitrary untwisted quantum affine…

We use techniques from Gromov-Witten theory to construct new invariants of matroids taking value in the Chow groups of spaces of rational curves in the permutohedral toric variety. When the matroid is realizable by a complex hyperplane…