Parking Spaces for Complex Reflection Groups

TL;DR

This paper extends the theory of parking spaces to all irreducible well-generated complex reflection groups, establishing combinatorial and algebraic isomorphisms, and enumerating parking functions, with some exceptions.

Contribution

It introduces a new combinatorial and algebraic framework for parking spaces in complex reflection groups and proves their isomorphism, extending previous work to a broader class of groups.

Findings

Established a $(W imes C)$-equivariant isomorphism between combinatorial and algebraic $W$-parking spaces.

Enumerated $W$-noncrossing parking functions for these groups.

Extended results to the Fuss case, excluding groups $G_{34}$, $E_7$, and $E_8$.

Abstract

We answer an open problem of arXiv:1204.1760 and arXiv:1205.4293, extending their work to irreducible well--generated complex reflection groups $W$. We define a combinatorial $W$-noncrossing parking space and an algebraic $W$-parking space for such $W$, and exhibit a $(W \times C)$-equivariant isomorphism between the two. As a consequence of this isomorphism, we enumerate the $W$-noncrossing parking functions. Finally, we extend our results to the Fuss case. We prove the results for all such complex reflection groups except $G_{34}$, $E_7,$ and $E_8$.