Combinatorics 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. These spline modules are generalizations of the GKM construction for equivariant cohomoloy rings of the flag, Hessenberg, and permutohedral varieties. Each graded piece of these spline modules admits an S_n representation via the dot action. In this talk, we concretely describe generators for the spline modules and the symmetric group representations on certain graded pieces for Cayley graphs constructed using a minimal set of generating transpositions.