A description of the minimal elements of Shi regions in classical Weyl Groups

TL;DR

This paper explores the minimal elements of Shi regions in classical Weyl groups, establishing a combinatorial interpretation via parking functions and non-crossing arc diagrams, enhancing understanding of their structure.

Contribution

It introduces a bijection linking Shi regions to parking functions, enabling the computation of minimal elements through combinatorial models.

Findings

Bijection between parking functions and Shi regions

Combinatorial interpretation of minimal elements

Counting non-crossing arcs in arc diagrams

Abstract

In this extended abstract, we show how a bijection between parking functions and regions of the Shi arrangement from [Athanasiadis, Linusson '99] (in type $A_n$) and [Armstrong, Reiner, Rhoades '15] (in type $B_n, C_n, D_n$) allows for the computation of the minimal elements of the Shi regions. This gives a combinatorial interpretation of these minimal elements: they can be seen as counting non-crossing arcs in non-nesting arc diagrams.