Cluster parking functions II: $q,t$-dihedral sieving via diagonal coinvariants

Abstract

In a previous work, we defined the complex of cluster parking functions. On one side, they encode the type-refined enumeration of faces of the cluster complex, and on the other side, they have a reduced homology which is isomorphic to (ungraded) diagonal coinvariants. The goal of this work is to take into account the underlying dihedral symmetry. We thus have a product of a dihedral group and a symmetric group (there is a precise conjecture in the case of other finite Coxeter groups, but we focus on symmetric groups because of technicalities about diagonal coinvariants beyond this case). Under the action of the product group, the reduced homology of cluster parking functions is conjecturally isomorphic to diagonal coinvariants up to tensoring by a sign character of the dihedral group. This isomorphism can be reformulated as a dihedral sieving phenomenon. The main technical contribution is the definition of the dihedral automorphism group of cluster parking functions, and we discuss various features of the reduced homology character and its conjectural connection with diagonal coinvariants.