A braid group action on parking functions

Abstract

We construct an action of the braid group on $n$ strands on the set of parking functions of $n$ cars such that elementary braids have orbits of length 2 or 3. The construction is motivated by a theorem of Lyashko and Looijenga stating that the number of the distinguished bases for $A_n$ singularity equals $(n+1)^{n-1}$ and thus equals the number of parking functions. We construct an explicit bijection between the set of parking functions and the set of distinguished bases, which allows us to translate the braid group action on distinguished bases in terms of parking functions.