Combinatorics Seminar: "Generating the nilpotent equivalence relation on Dyck paths"
Abstract: A Dyck path of length 2n determines a subspace of n x n upper triangular matrices. That subspace meets a largest conjugacy class of nilpotent matrices, indexed by a partition of n. In this way, the set of such subspaces, and so also Dyck paths, breaks up into equivalence classes indexed by partitions. There is a basic operation on subspaces that preserves the associated nilpotent conjugacy class. In the first part of the talk, we show, in joint work with Molly Fenn, that the operation is transitive on equivalence classes. The size of each equivalence class does not appear to be a known statistic on partitions but we'll discuss some conjectures. The second part of the talk, joint work with Martha Precup, concerns a modular law for the cohomology of certain varieties attached to each subspace. Properly translated, this modular law coincides with one that appears in the combinatorial literature for chromatic quasisymmetric functions. When combined with the result in the first part of the talk, we recover some results in recent work of Abreu and Nigro.
Host: Martha Precup