On Perverse sheaves of a Coxeter hyperplane arrangement of type $\mathcal{A}_n$

TL;DR

This paper explores the structure of perverse sheaves on Coxeter hyperplane arrangements of type A_n, revealing how complex sheaves relate to simpler ones on lower-dimensional arrangements.

Contribution

It establishes a relation between perverse sheaves on different dimensions of Coxeter arrangements, showing how certain simple sheaves are induced from lower-dimensional cases.

Findings

Simple perverse sheaves with zero stalks are induced from lower dimensions.

Relation between perverse sheaves on arrangements of different sizes.

Extension of quiver descriptions to Coxeter hyperplane arrangements.

Abstract

Kapranov and schechtman gave quiver description of perverse sheaves on real hyperplane arrangements. We used this description to relate the perverse sheaves on Coxeter hyperplane arrangements of type $\mathcal A_n$ for different values of $n$. As a consequence we prove that the simple perverse sheaves whose stalk on open cells are zero are induced from the perverse sheaves on lower dimension arrangements.