Abstract:We study generalized splines from the perspective of the representation theory of the category of graphs with contractions. Our main theorem proves a kind of finite generation, which in turn implies the existence of a ``universal generating set'' for the module of splines over any graph with fixed combinatorial genus. This theorem holds over any Noetherian commutative ring with a chosen finite list of ideals for edge-labels. We then give several applications of this theorem, including showing that a particular generating function associated to splines on trees is algebraic when the base ring satisfies certain finiteness conditions. We illustrate our technical theorems explicitly by giving a classification of splines on graphs with combinatorial genus one and two.
| Subjects: | Combinatorics (math.CO); Category Theory (math.CT); Rings and Algebras (math.RA); Representation Theory (math.RT) |
| Cite as: | arXiv:2605.24348 [math.CO] |
| (or arXiv:2605.24348v1 [math.CO] for this version) | |
| https://doi.org/10.48550/arXiv.2605.24348 arXiv-issued DOI via DataCite (pending registration) |
From: Eric Ramos [view email]
[v1]
Sat, 23 May 2026 02:20:26 UTC (178 KB)
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