Group Actions on Riemann-Roch Space

TL;DR

This paper studies how a group acting on a Riemann surface influences the structure of the associated Riemann-Roch space, providing explicit formulas for decomposing this space into irreducible representations.

Contribution

It offers new results on decomposing the G-action on Riemann-Roch spaces, including explicit formulas for multiplicities of irreducible components, for effective G-invariant divisors.

Findings

Derived explicit formulas for irreducible representation multiplicities.

Analyzed decomposition of G-action on Riemann-Roch spaces.

Provided examples on well-known families of curves.

Abstract

Let $ \; G \; $ be a group acting on a compact Riemann surface $ \; {\mathcal X} \; $ and $ \; D \; $ be a $ \; G$-invariant divisor on $\; {\mathcal X}. \; $ The action of $ \; G \; $ on $ \; {\mathcal X} \; $ induces a linear representation $ \; L_G(D) \; $ of $ \; G \; $ on the Riemann-Roch space associated to $ \; D.$ In this paper we give some results on the decomposition of $ \; L_G(D) \; $ as sum of complex irreducible representations of $ \; G, \; $ for $ \; D \; $ an effective non-special $\; G$-invariant divisor. In particular, we give explicit formulae for the multiplicity of each complex irreducible factor in $ \; L_G(D) \; $. We work out some examples on well known families of curves.