Parking functions and vertex operators

TL;DR

This paper explores algebraic structures linked to parking functions, using vertex operators to derive dimension formulas, and connects these to symmetric group representations, Catalan numbers, and geometric conjectures.

Contribution

It introduces new associative algebras and vector spaces related to parking functions, providing explicit dimension and character formulas and linking to geometric and algebraic structures.

Findings

Derived dimension and character formulas for new algebraic spaces

Established isomorphisms between parking function modules and symmetric group representations

Connected constructed spaces to Catalan and Fuss--Catalan numbers

Abstract

We introduce several associative algebras and series of vector spaces associated to these algebras. Using lattice vertex operators, we obtain dimension and character formulae for these spaces. In particular, we a series of representations of symmetric groups which turn out to be isomorphic to parking function modules. We also construct series of vector spaces whose dimensions are Catalan numbers and Fuss--Catalan numbers respectively. Conjecturally, these spaces are related to spaces of global sections of vector bundles on (zero fibres of) Hilbert schemes and representations of rational Cherednik algebras.