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

Alfeld (1984) introduced a tetrahedral analog A S (_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 ^{C r}-splines on the cone over A S (_δn) 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.