Szego Seminar: "Splines on Cayley Graphs of the Symmetric Group"
Abstract: A spline is an assignment of polynomials to the vertices of a polynomial-edge-labeled graph, where the difference of two vertex polynomials along an edge must be divisible by the edge label. We consider spline modules where the underlying graph is the Cayley graph of a symmetric group generated by some transpositions. Each graded piece of this module admits a symmetric group representation via Tymoczko's dot action. These spline modules are generalizations of the GKM construction for equivariant cohomology rings of the flag, Hessenberg, and permutohedral varieties. Understanding the dot action representation on the equivariant cohomology of Hessenberg varieties is the primary concern of the graded Stanley-Stembridge conjecture. Our results give classes of Cayley graphs where we can concretely describe generators for these modules and the dot action representations on certain graded pieces.
Host: Jeremy Cummings