Simple-root bases for Shi arrangements

TL;DR

This paper introduces two distinguished bases, SRB+ and SRB-, for the logarithmic derivation modules of extended Shi arrangements, providing new insights into their structure and properties related to simple roots.

Contribution

It defines and studies the simple-root basis plus and minus (SRB+ and SRB-) for Shi arrangements, advancing understanding beyond mere existence and degrees.

Findings

Defined and characterized SRB+ and SRB- bases

Established properties related to simple roots

Enhanced structural understanding of Shi arrangements

Abstract

In his affirmative answer to the Edelman-Reiner conjecture, Yoshinaga proved that the logarithmic derivation modules of the cones of the extended Shi arrangements are free modules. However, all we know about the bases is their existence and degrees. In this article, we introduce two distinguished bases for the modules. More specifically, we will define and study the simple-root basis plus (SRB+) and the simple-root basis minus (SRB-) when a primitive derivation is fixed. They have remarkable properties relevant to the simple roots and those properties characterize the bases.