On the Varchenko determinant formula for oriented braid arrangements

In this paper, we first consider the arrangement of hyperplanes and then the corresponding oriented arrangement of hyperplanes in n-dimensional real space. Following the work of Varchenko, who studied the determinant of the quantum bilinear form of a real configuration and the determinant formula for a matroid bilinear form, we discuss here first some of the main properties of the braid arrangement and then of the oriented braid arrangements in n-dimensional real space. The main result of this study is a theorem that provides an explicit formula for determining the determinant of the matrix associated with the oriented braid arrangement. The proof of this theorem is based on the results of two different approaches. One is to determine the space of all constants in the multiparametric quon algebra equipped with a multiparametric q-differential structure, and the other is to study the feasibility of multiparametric quon algebras in Hilbert space

The Fundamental Group of the Complement of the Complexification of a Real Arrangement of Hyperplanes

AbstractLet A be an arrangement of hyperplanes (i.e., a finite set of 1-codimension vector subspaces) in Rd. We say that the linear ordering of the hyperplanes A,H1≺H2≺···≺Hn, is ashellabilityorder of A, if there is an oriented affine linelcrossing the hyperplanes of A on the given linear order. The intersection latticeL(A) is the set of all intersections of the hyperplanes of A partially ordered by reversed inclusion. Set M(Ac)≔Cd\⋃{H⊗C:H∈A}. We will prove:Suppose that there are shellability orders H1≺H2≺···≺Hnand H′1≺′H′2≺′···≺′H′n, respectively, ofAandA′,such that the bijective map Hi→H′i, i∈[n]determines an isomorphism of the intersection lattices L(A)and L(A′).Then the fundamental groupsπ1(M(Ac))andπ1(M(A′c)) are isomorphic

Tropical types and associated cellular resolutions

An arrangement of finitely many tropical hyperplanes in the tropical torus leads to a notion of `type' data for points, with the underlying unlabeled arrangement giving rise to `coarse type'. It is shown that the decomposition of the tropical torus induced by types gives rise to minimal cocellular resolutions of certain associated monomial ideals. Via the Cayley trick from geometric combinatorics this also yields cellular resolutions supported on mixed subdivisions of dilated simplices, extending previously known constructions. Moreover, the methods developed lead to an algebraic algorithm for computing the facial structure of arbitrary tropical complexes from point data.Comment: minor correction