The Shi arrangements and the Bernoulli polynomials

TL;DR

This paper explores the structure of the Shi arrangement, an affine hyperplane arrangement related to the braid arrangement, and introduces a basis for its derivation module using Bernoulli polynomials.

Contribution

It provides an explicit basis construction for the derivation module of the cone over the Shi arrangement utilizing Bernoulli polynomials, advancing understanding of hyperplane arrangements.

Findings

Explicit basis for the derivation module constructed

Bernoulli polynomials are key to the basis construction

Enhances understanding of affine hyperplane arrangements

Abstract

The braid arrangement is the Coxeter arrangement of the type $A_\ell$. The Shi arrangement is an affine arrangement of hyperplanes consisting of the hyperplanes of the braid arrangement and their parallel translations. In this paper, we give an explicit basis construction for the derivation module of the cone over the Shi arrangement. The essential ingredient of our recipe is the Bernoulli polynomials.