Representations of finite groups on Riemann-Roch spaces

TL;DR

This paper investigates how finite groups act on Riemann-Roch spaces of divisors on algebraic curves, establishing bounds on the dimensions of irreducible components and exploring applications to coding theory.

Contribution

It provides a bound on the dimensions of irreducible constituents of the group action on Riemann-Roch spaces and demonstrates the sharpness of this bound with examples.

Findings

Irreducible constituents have dimension ≤ size of smallest G-orbit

The bound on dimensions is sharp, with explicit examples

Connections to permutation decoding of algebraic geometry codes

Abstract

We study the action of a finite group on the Riemann-Roch space of certain divisors on a curve. If $G$ is a finite subgroup of the automorphism group of a projective curve $X$ over an algebraically closed field and $D$ is a divisor on $X$ left stable by $G$ then we show the irreducible constituents of the natural representation of $G$ on the Riemann-Roch space $L(D)=L_X(D)$ are of dimension $\leq d$, where $d$ is the size of the smallest $G$-orbit acting on $X$. We give an example to show that this is, in general, sharp (i.e., that dimension $d$ irreducible constituents can occur). Connections with coding theory, in particular to permutation decoding of AG codes, are discussed in the last section. Many examples are included.