A Note on Frobenius Splitting of Schubert Varities and Linear Syzygies

Abstract

Let G be a connected, simply connected group and B be a Borel subgroup of G. Let X be a Schubert variety in G/B. Let Lj,... ,Lr be invertible sheaves on G/B. For a e Nr, let LQ = L^1 ? ? ? ? ? L?r. The multi-graded coordinate rings R = ?aH?(G/B,La) and A = 0Q//?(X,LQ), were shown to be wonderful in [1] assuming each of L/ to be ample. In this note we extend the result obtained in [1] to effective invertible sheaves on G/B. An effective invertible sheaf on G/B is a pull-back of an ample invertible sheaf on G/P, for some parabolic subgroup P D B of G. In this note, we use this fact to obtain the necessary vanishing conditions for effective invertible sheaves on G/B. This will prove that R and A are multi-wonderful.