A Bijection Between All Shi Regions and Core Partitions

Abstract

We extend the bijection of Fishel-Vazirani on dominant regions of the $m$-Shi arrangement. Our map puts the set of all minimal chambers of the $m$-Shi arrangement of Type $A_n$ in bijection with a certain set of (equivalence classes of) $n$-core partitions. As a step to our proof, we give a potentially interesting classification of the alcoves in the $S_n$-orbit of an $m$-minimal alcove which are themselves $m$-minimal.