The Chain Matrix of Bouquets of Geometric Lattices and its Determinant

TL;DR

This paper extends determinant formulas for bilinear forms from hyperplane arrangements to complexes of oriented matroids and bouquets of geometric lattices, revealing new algebraic properties and generalizations.

Contribution

It generalizes the determinant formula for a bilinear form to complexes of oriented matroids and bouquets of geometric lattices, expanding the algebraic framework.

Findings

Determinant formula successfully extended to COMs

Generalization to bouquets of geometric lattices achieved

Supports broader applications in combinatorial algebra

Abstract

This work builds on Varchenko et al's introduction of bilinear forms for hyperplane arrangements, where the determinant of the associated matrices factorizes into simple components. While one of the determinant formula developed by Varchenko has been generalized to complexes of oriented matroids (COMs) already, this question was open for another, distinct form. Motivated by work from Varchenko and Brylawski, who generalized the alternative bilinear form and its determinant formula from hyperplane arrangements to matroids, we examine whether this formula can similarly be generalized to COMs. Our findings affirm this generalization, and we further extend the determinant formula to bouquets of geometric lattices as introduced by Laurent et al.