Generalized Spline Modules on Arbitrary Graphs

Abstract: Generalized splines on a graph G with edge weighted by ideals a commutative ring R are R-vertex labelings such that if two vertices share an edge in G, the vertex labels are congruent modulo the edge ideal. When R is a principal ideal domain, we introduce collapsing operations that reduces any simple graph to a single vertex and carries along the edge ideal information. This corresponds to a sequence of surjective maps between the associated spline modules, and leads to an explicit construction of an R-module basis in terms of the edge ideals. We also solve an interpolation problem, i.e. given a partial vertex labeling, when can it can be extended to a generalized spline?

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commutative algebracombinatoricscategory theoryrepresentation theory

Audience: researchers in the topic

UC Davis algebra & discrete math seminar