ORCID
0009-0006-9395-4103
Date of Award
4-18-2024
Degree Name
Doctor of Philosophy (PhD)
Degree Type
Dissertation
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 polynomials labeling adjacent vertices 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 flag, regular semisimple Hessenberg, and permutohedral 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, we generalize the graded S_n-representations from the cohomologies of the above varieties to splines on Cayley graphs of S_n, then (1) give explicit module and ring generators for whenever the S_n-generating set is minimal, (2) give a combinatorial characterization of when graded pieces of one S_n-representation is trivial, and (3) compute the first degree piece of both graded characters for all generating sets.
Language
English (en)
Chair and Committee
Martha Precup
Committee Members
John Shareshian
Recommended Citation
Lesnevich, Nathan, "Splines on Cayley Graphs of the Symmetric Group" (2024). Arts & Sciences Electronic Theses and Dissertations. 3038.
https://openscholarship.wustl.edu/art_sci_etds/3038