Parking Spaces in Coxeter-Catalan combinatorics


The collection of parking functions under a natural Sn-action (which has Catalan-many orbits) has been a central object in Algebraic Combinatorics since the work of Haiman more than 30 years ago. One of the lines of research spawned around it was towards defining and studying analogous objects for real and complex reflection groups W; the main candidates are known as the W-non-nesting and W-non-crossing parking functions.

The W-non-nesting parking functions are relatively well understood; they form the so called algebraic W-parking-space which has a concrete interpretation as a quotient ring (of the ambient polynomial algebra over a system of homogeneous parameters that carry the reflection representation of W). The W-non-crossing parking functions on the other hand have defied unified explanations while simultaneously proving themselves central in the study of Coxeter and Artin groups (their geometric group theory and combinatorics) and in the representation theory of Dynkin quivers. One of the main open problems in the field since the early 2000’s has been to give a type-independent proof of the W-isomorphism between the algebraic and the non-crossing W-parking spaces. In this talk, I will present the first such proof, solving the more general Fuss version of the problem, that proceeds by comparing a collection of recursions that are shown to be satisfied by both objects. This relies on a variety of recent techniques we introduced, in particular a parabolic decomposition of the algebraic parking space and its relation with the spectrum of Laplacian matrices for reflection arrangements.