Document Type
Article
Publication Date
4-2016
Publication Title
Pacific Journal of Mathematics
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 (algebraic splines of Billera and others) and in algebraic topology (certain equivariant cohomology rings, described by Goresky, Kottwitz, and MacPherson, among others). The central question of this paper asks when an arbitrary edge-labeled graph has nontrivial generalized splines. The answer is “always”, and we prove the stronger result that the module of generalized splines contains a free submodule whose rank is the number of vertices in G. We describe the module of 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.
Volume
281
Issue
2
First Page
333
Last Page
364
DOI
dx.doi.org/10.2140/pjm.2016.281.333
ISSN
0030-8730
Rights
© 2016 Mathematical Sciences Publishers
Recommended Citation
Gilbert, Simcha; Tymoczko, Julianna; and Viel, Shira, "Generalized Splines on Arbitrary Graphs" (2016). Mathematics and Statistics: Faculty Publications, Smith College, Northampton, MA.
https://scholarworks.smith.edu/mth_facpubs/7
Comments
Archived as published. First published in Pacific Journal of Mathematics in 2016, published by Mathematical Sciences Publishers.