A representation-valued relative Riemann-Hurwitz theorem and the Hurwitz-Hodge bundle

TL;DR

This paper develops a formula that describes the G-module structure of the Hurwitz-Hodge bundle in families of admissible G-covers, extending classical Riemann-Hurwitz concepts to a representation-theoretic context.

Contribution

It introduces a representation-ring-valued relative Riemann-Hurwitz formula for admissible G-covers, linking the G-module structure of the Hurwitz-Hodge bundle to the base curve's Hodge bundle.

Findings

Derived a formula for the G-module structure of the Hurwitz-Hodge bundle

Extended Riemann-Hurwitz theorem to a representation-ring context

Applicable to sheaves on families of admissible G-covers

Abstract

We provide a formula describing the G-module structure of the Hurwitz-Hodge bundle for admissible G-covers in terms of the Hodge bundle of the base curve, and more generally, for describing the G-module structure of the push-forward to the base of any sheaf on a family of admissible G-covers. This formula can be interpreted as a representation-ring-valued relative Riemann-Hurwitz formula for families of admissible G-covers.