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

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.