  Lakshmibai, Mehta and Parameswaran (LMP) introduced the notion of maximal
multiplicity vanishing in Frobenius splitting. In this paper we define the
algebraic analogue of this concept and construct a Frobenius splitting
vanishing with maximal multiplicity on the diagonal of the full flag variety.
Our splitting induces a diagonal Frobenius splitting of maximal multiplicity
for a special class of smooth Schubert varieties first considered by Kempf.
Consequences are Frobenius splitting of tangent bundles, of blow-ups along the
diagonal in flag varieties along with the LMP and Wahl conjectures in positive
characteristic for the special linear group.
