The Grothendieck ring of the structure group of the geometric Frobenius morphism

TL;DR

This paper investigates the structure group scheme associated with the geometric Frobenius morphism on smooth varieties, describing its irreducible representations and computing its Grothendieck ring of finite-dimensional representations.

Contribution

It provides a detailed description of the irreducible representations and explicitly computes the Grothendieck ring for the structure group scheme of the geometric Frobenius morphism.

Findings

Explicit classification of irreducible representations

Calculation of the Grothendieck ring structure

Insights into the representation theory of the structure group scheme

Abstract

The geometric Frobenius morphism on smooth varieties is an fppf-fiber bundle. We study representations of the structure group scheme. In particular, we describe irreducible representations and compute its Grothendieck ring of finite dimensional representations.