Splines on the Alfeld split of a simplex and type A root systems

Abstract

Alfeld (1984) introduced a tetrahedral analog AS(Δ3) of the Clough–Tocher split of a triangle. A formula for the dimension of the spline space Ckr(AS(Δn)) is conjectured in Foucart and Sorokina (2013). We prove that the graded module of Cr-splines on the cone over AS(Δn) is isomorphic to the module Dr+1(An) of multiderivations on the type An 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.