The Frobenius morphism on flag varieties, II

In this paper, which is the sequel to arXiv:1410.3742, we study the Frobenius pushforward of the structure sheaf on the adjoint varieties in type ${\mathbf{A}}_3$ and ${\mathbf{A}}_4$. We show that this pushforward sheaf decomposes into a direct sum of indecomposable bundles and explicitly determine this set that does not depend of the characteristic. In accordance with the results of arXiv:0707.0913, this set forms a strong full exceptional collection in the derived category of coherent sheaves. These computations lead to a natural conjectural answer in the general case that we state at the end.