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

摘要

Alfeld introduced a subdivision AS(n) of an n-simplex, generalizing theClough-Tocher split of a triangle. A formula for the dimension of the splinespace C^r_k(AS(n)) was conjectured recently by Foucart-Sorokina. We prove thatthe graded module of C^r-splines on the cone over AS(n) is isomorphic to themodule D^{r+1}(A_n) of multiderivations on the type A_n Coxeter arrangement. Atheorem of Terao shows that the module of multiderivations of a Coxeterarrangement is free and gives an explicit basis. As a consequence theconjectured formula holds.