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Primer for the algebraic geometry of sandpiles. (English) Zbl 1320.05060
Amini, Omid (ed.) et al., Tropical and non-Archimedean geometry. Bellairs workshop in number theory, tropical and non-Archimedean geometry, Bellairs Research Institute, Holetown, Barbados, USA, May 6–13, 2011. Providence, RI: American Mathematical Society (AMS); Montreal: Centre de Recherches Mathématiques (ISBN 978-1-4704-1021-6/pbk). Contemporary Mathematics 605. Centre de Recherches Mathématiques Proceedings, 211-256 (2013).
Summary: The Abelian Sandpile Model (ASM) is a game played on a graph realizing the dynamics implicit in the discrete Laplacian matrix of the graph. The purpose of this primer is to apply the theory of lattice ideals from algebraic geometry to the Laplacian matrix, drawing out connections with the ASM. An extended summary of the ASM and of the required algebraic geometry is provided. New results include a characterization of graphs whose Laplacian lattice ideals are complete intersection ideals; a new construction of arithmetically Gorenstein ideals; a generalization to directed multigraphs of a duality theorem between elements of the sandpile group of a graph and the graph’s superstable configurations (parking functions); and a characterization of the top Betti number of the minimal free resolution of the Laplacian lattice ideal as the number of elements of the sandpile group of least degree. A characterization of all the Betti numbers is conjectured.
For the entire collection see [Zbl 1281.14002].

MSC:
05C25 Graphs and abstract algebra (groups, rings, fields, etc.)
05C57 Games on graphs (graph-theoretic aspects)
05E40 Combinatorial aspects of commutative algebra
13D02 Syzygies, resolutions, complexes and commutative rings
82C20 Dynamic lattice systems (kinetic Ising, etc.) and systems on graphs in time-dependent statistical mechanics
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