Representations of finite groups on Riemann-Roch spaces
<|reference_start|>Representations of finite groups on Riemann-Roch spaces: 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.<|reference_end|>
@article{joyner2002representations,
    title={Representations of finite groups on Riemann-Roch spaces},
    author={David Joyner and Will Traves},
    journal={arXiv preprint arXiv:math/0210408},
    year={2002},
    archivePrefix={arXiv},
    eprint={math/0210408},
    primaryClass={math.AG cs.IT math.GR math.IT}
}