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

Abstract Alfeld (1984) introduced a tetrahedral analog A S ( Δ 3 ) of the Clough–Tocher split of a triangle. A formula for the dimension of the spline space C k r ( A S ( Δ n ) ) is conjectured in Foucart and Sorokina (2013). We prove that the graded module of C r -splines on the cone over A S ( Δ 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.