Riemann-Roch for Sub-Lattices of the Root Lattice $A_n$

TL;DR

This paper extends Riemann-Roch theory to sub-lattices of the root lattice $A_n$, connecting it with Voronoi diagrams and providing geometric insights into graph Laplacians and lattice classification.

Contribution

It develops a general Riemann-Roch framework for $A_n$ sub-lattices, linking combinatorial graph theory with geometric lattice analysis.

Findings

Established a geometric Riemann-Roch theory for $A_n$ sub-lattices

Connected Riemann-Roch properties with Voronoi diagrams and simplicial distances

Provided insights into lattice classification and algorithmic challenges

Abstract

Recently, Baker and Norine {Advances in Mathematics, 215(2): 766-788, 2007} found new analogies between graphs and Riemann surfaces by developing a Riemann-Roch machinery on a finite graph $G$. In this paper, we develop a general Riemann-Roch Theory for sub-lattices of the root lattice $A_n$ by following the work of Baker and Norine, and establish connections between the Riemann-Roch theory and the Voronoi diagrams of lattices under certain simplicial distance functions. In this way, we rediscover the work of Baker and Norine from a geometric point of view and generalise their results to other sub-lattices of $A_n$. In particular, we provide a geometric approach for the study of the Laplacian of graphs. We also discuss some problems on classification of lattices with a Riemann-Roch formula as well as some related algorithmic issues.