A Bijection Between All Shi Regions and Core Partitions

TL;DR

This paper establishes a bijection between all Shi regions and certain core partitions, extending previous work and providing a new classification of alcoves in the $m$-Shi arrangement of Type $A_{n}$.

Contribution

It extends the bijection of Fishel-Vazirani to all Shi regions and introduces a classification of alcoves in the $S_{n}$-orbit of $m$-minimal alcoves.

Findings

Bijection between all Shi regions and equivalence classes of $n$-core partitions.

Classification of $m$-minimal alcoves within the $S_{n}$-orbit.

Extension of the bijection to all minimal chambers of the $m$-Shi arrangement.

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.