Geodesic
Riemann-Roch for sub-lattices of the root lattice \(A_n\)
The electronic journal of combinatorics, Tome 17 (2010)

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Zbl arXiv
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 $A_n$ analogue to 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 obtain a geometric proof of the Riemann-Roch theorem for graphs and generalise the result to other sub-lattices of $A_n$. In particular, we provide a new 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.
DOI : 10.37236/396
Classification : 05C50, 05E99, 14H55, 17B22 
Omid Amini; Madhusudan Manjunath. Riemann-Roch for sub-lattices of the root lattice \(A_n\). The electronic journal of combinatorics, Tome 17 (2010). doi: 10.37236/396
@article{10_37236_396,
     author = {Omid Amini and Madhusudan Manjunath},
     title = {Riemann-Roch for sub-lattices of the root lattice {\(A_n\)}},
     journal = {The electronic journal of combinatorics},
     year = {2010},
     volume = {17},
     doi = {10.37236/396},
     zbl = {1277.05105},
     url = {http://geodesic.mathdoc.fr/articles/10.37236/396/}
}
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