Simple-root bases for 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, we know very little about the bases themselves except their existence. In this article, we prove the unique existence of two distinguished bases which we call the simple-root basis plus (SRB+) and the simple-root basis minus (SRB−). They are characterized by nice divisibility properties relative to the simple roots.