Parking functions on the cluster complex and Cyclic Sieving Phenomena
ABSTRACT
The associahedron K_n and its dual, the type-A cluster complex Y(S_n), with their many realizations and generalizations are central objects in algebraic combinatorics. Their faces are labeled by subdivisions of an (n+2)-gon, they have Catalan-many vertices/facets, and their f-vectors are given by the Kirkman numbers.
In past joint work with Josuat-Verges we gave a refinement of the f-vector of Y(n) that keeps track of a cycle type associated to each polygonal subdivision, and is expressed in terms of structure coefficients of Haiman's parking space of diagonal coinvariants. We will present in this talk two research works built on these structures.
In the first work, joint with Josuat-Verges, we give a topological, equivariant version of the combinatorial f-to-h reciprocity for Y(n). We label each polygonal subdivision with a set partition and get as a result a simplicial complex that carries a parking space representation in its top homology. This presents the result of the recent preprint arxiv:2402.03052.
In the second still in progress work, joint with Josuat-Verges and Sommers, we give q-versions of these refined f-vectors (obtained naturally via the parking space interpretation) and prove cyclic sieving phenomena (CSPs) for the natural cyclic rotation of the polygon. In particular, we show that the q-Kirkman numbers satisfy a new, weighted type of CSP. We present further a version that keeps track of dihedral symmetries via the qt-Schroder numbers, utilizing the full grading of the parking space.
Most of these results make sense for all real reflection groups but we will focus our presentation to the symmetric group case. We will end with open questions in each approach or possible common extensions.