Splines on Cayley graphs of the symmetric group
Year: 2025
Type: article
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 paper, we generalize the graded $S_n$ -representations from the cohomologies of the above varieties to splines on Cayley graphs of $S_n$ and 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.
Source: Forum of Mathematics Sigma
Author: Nathan R. T. Lesnevich
Language: English
Subfield: Discrete Mathematics and Combinatorics
Field: Mathematics
Domain: Physical Sciences
Open Access status: gold
Estimated APC paid: $2,650
Funder: National Science Foundation
OpenAlex ID: W4411599163