Splines on the Alfeld split of a simplex and type A root systems
Numerical Analysis
2014-07-14 v1
Abstract
Alfeld introduced a subdivision AS(n) of an n-simplex, generalizing the Clough-Tocher split of a triangle. A formula for the dimension of the spline space C^r_k(AS(n)) was conjectured recently by Foucart-Sorokina. We prove that the graded module of C^r-splines on the cone over AS(n) is isomorphic to the module D^{r+1}(A_n) of multiderivations on the type A_n Coxeter arrangement. A theorem of Terao shows that the module of multiderivations of a Coxeter arrangement is free and gives an explicit basis. As a consequence the conjectured formula holds.
Cite
@article{arxiv.1402.6924,
title = {Splines on the Alfeld split of a simplex and type A root systems},
author = {Hal Schenck},
journal= {arXiv preprint arXiv:1402.6924},
year = {2014}
}