Decomposition of Frobenius pushforwards of line bundles on wonderful compactifications

TL;DR

This paper investigates the decomposition of Frobenius pushforwards of line bundles on wonderful compactifications of semisimple groups, providing conditions for direct summands, bounds on multiplicities, and methods for computing their classes.

Contribution

It offers a detailed analysis of the decomposition of Frobenius pushforwards on wonderful compactifications, including criteria for summands and new computational approaches.

Findings

Necessary and sufficient conditions for line bundles to be direct summands.

Lower bounds on multiplicities of certain line bundles.

Methods for computing classes in Grothendieck and Chow groups.

Abstract

De Concini-Procesi introduced varieties known as wonderful compactifications, which are smooth projective compactifications of semisimple adjoint groups $G$. We study the Frobenius pushforwards of invertible sheaves on the wonderful compactifications, and in particular its decomposition into locally free subsheaves. We give necessary and sufficient conditions for a specific line bundle to be a direct summand of the Frobenius pushforward of another line bundle, formulated in terms of the weight lattice of $\widetilde{G}$, the universal cover of $G$ (identified with the Picard group of the wonderful compactification). In the case of $G=\mathsf{PSL}_n$, we offer lower bounds on the multiplicities (as direct summands) for those line bundles satisfying the sufficient conditions. We also decompose Frobenius pushforwards of line bundles into a direct sum of vector subbundles, whose ranks are determined by invariants on the weight lattice of $G$. We study a particular block which decomposes as a direct sum of line bundles, and identify the line bundles which appear in this block. Finally, we present two approaches to compute the class of the Frobenius pushforward of line bundles on wonderful compactifications in the rational Grothendieck group and in the rational Chow group.