2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/222477We 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.17 pages, 1 figure, proofs are made more detailed. To appear in Selecta MathematicaRepresentation TheoryCombinatorics17B69 (Primary) 05E05, 20C30 (Secondary)text