Thesis Defense: Splines on Cayley Graphs of the Symmetric Group

Abstract: A spline is an assignment of polynomials to the vertices of a graph whose edges are labeled by ideals, where the difference of two adjacent polynomials must belong to the corresponding ideal. The set of splines forms a ring. We consider spline rings where the underlying graph is the Cayley graph of a symmetric group generated by a collection of transpositions. 

These rings generalize the GKM construction for equivariant cohomology rings of regular semisimple Hessenberg varieties. These cohomology rings carry two actions of the symmetric group $S_n$ whose graded characters are both of general interest in algebraic combinatorics.

In this dissertation presentation, we generalize the graded $S_n$-representations from the cohomologies of the above varieties to splines on Cayley graphs of $S_n$, give a combinatorial characterization of when graded pieces of one $S_n$-representation is trivial, and compute the first degree piece of both graded characters for all generating sets.

Advisors: Martha Precup and John Shareshian