Rank one summands of Frobenius pushforwards of line bundles on G/P

TL;DR

This paper characterizes the line bundle summands of Frobenius pushforwards of line bundles on partial flag varieties over fields of positive characteristic, providing explicit descriptions and multiplicity calculations for large iterates.

Contribution

It explicitly describes all line bundle summands of Frobenius pushforwards on G/P and computes their multiplicities, addressing a question by Gros-Kaneda.

Findings

Identifies all line bundle summands of Frobenius pushforwards on G/P.

Calculates multiplicities of the trivial line bundle as a summand for large r.

Answers Gros-Kaneda's question on multiplicity of a specific line bundle.

Abstract

Let $X=G/P$ be a partial flag variety, where $G$ is a semi-simple, simply connected algebraic group defined over an algebraically closed field $K$ of positive characteristic. Let $\mathsf{F}\colon X\to X$ be the absolute Frobenius morphism. Given a line bundle $\mathscr{L}$ on $X$ and an integer $r\geq1$, we describe all line bundles that are direct summands of the pushforward $\mathsf{F}_{*}^{r}\mathscr{L}$. For $\mathscr{L}$ corresponding to a dominant weight, we also compute, for $r$ sufficiently large, the multiplicity of $\mathscr{O}_{X}$ as a summand of $\mathsf{F}_{*}^{r}\mathscr{L}$. As an application we answer a question of Gros-Kaneda about the multiplcity of $\mathscr{L}(-\rho)$ as a direct summand of $\mathsf{F}_{*}\mathscr{O}_{G/B}$.