Homological Properties of Determinantal Arrangements

TL;DR

This paper investigates the algebraic and topological properties of determinantal arrangements, showing they are free divisors, have specific homotopy group structures, and possess well-behaved combinatorial and topological invariants.

Contribution

It establishes that determinantal arrangements are free divisors and explores their topological and combinatorial properties, extending known results in arrangement theory.

Findings

Determinantal arrangements are free divisors.

Higher homotopy groups of their complements are isomorphic to those of S^3.

Poincaré polynomials of these complements factor nicely.

Abstract

We explore a natural extension of braid arrangements in the context of determinantal arrangements. We show that these determinantal arrangements are free divisors. Additionally, we prove that free determinantal arrangements defined by the minors of $2\times n$ matrices satisfy nice combinatorial properties. We also study the topology of the complements of these determinantal arrangements, and prove that their higher homotopy groups are isomorphic to those of $S^3$. Furthermore, we find that the complements of arrangements satisfying those same combinatorial properties above have Poincar\'e polynomials that factor nicely.