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

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

2010-07-15

arxiv.org/abs/1007.2454v1

Mathematics

Combinatorics

43 pages, 2 figures, submitted in May 2009

Scientific paper

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.

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