Generalized splines on arbitrary graphs

TL;DR

This paper introduces a broad framework for generalized splines on graphs with edges labeled by ideals, proving their existence, describing their structure, and providing tools for their analysis in algebraic and topological contexts.

Contribution

It establishes that generalized splines always exist on any edge-labeled graph and characterizes their structure, including explicit descriptions for trees and a new GKM matrix tool.

Findings

Generalized splines always contain a free submodule with rank equal to the number of vertices.

The structure of generalized splines is fully described for trees.

A new GKM matrix tool is introduced for analyzing splines.

Abstract

Let G be a graph whose edges are labeled by ideals of a commutative ring. We introduce a generalized spline, which is a vertex-labeling of G by elements of the ring so that the difference between the labels of any two adjacent vertices lies in the corresponding edge ideal. Generalized splines arise naturally in combinatorics (em algebraic splines of Billera and others) and in algebraic topology (certain equivariant cohomology rings, described by Goresky-Kottwitz-MacPherson and others). The central question of this manuscript asks when an arbitrary edge-labeled graph has nontrivial generalized splines. The answer is `always', and we prove the stronger result that generalized splines contain a free submodule whose rank is the number of vertices in G. We describe all generalized splines when G is a tree, and give several ways to describe the ring of generalized splines as an intersection of generalized splines for simpler subgraphs of G. We also present a new tool which we call the GKM matrix, an analogue of the incidence matrix of a graph, and end with open questions.