The Frobenius morphism on flag varieties, II

TL;DR

This paper investigates the Frobenius pushforward of the structure sheaf on certain flag varieties, revealing a decomposition into indecomposable bundles and proposing a conjecture for the general case.

Contribution

It explicitly determines the decomposition of the Frobenius pushforward on specific adjoint varieties and formulates a conjecture for broader cases.

Findings

Decomposition into indecomposable bundles is characteristic-independent.

The set forms a strong full exceptional collection in the derived category.

Provides a conjectural framework for the general case.

Abstract

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 ${\bf A}_3$ and ${\bf A}_4$. 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.