Primer for the algebraic geometry of sandpiles

Perkinson, David; Perlman, Jacob; Wilmes, John

doi:10.1090/conm/605/12117

Cited by 41 publications

(79 citation statements)

References 25 publications

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“…One way to test whether or not a configuration is recurrent is to use a burning configuration. The following conditions were stated by Dhar [13] for undirected graphs and by Speer [32] for directed graphs, (see also [29,Thm. 2.27]).…”

Section: Avalanche-finite Matrices and Chip Firingmentioning

confidence: 99%

Chip firing on Dynkin diagrams and McKay quivers

2018

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Two classes of avalanche-finite matrices and their critical groups (integer cokernels) are studied from the viewpoint of chip-firing/sandpile dynamics, namely, the Cartan matrices of finite root systems and the McKay-Cartan matrices for finite subgroups G of general linear groups. In the root system case, the recurrent and superstable configurations are identified explicitly and are related to minuscule dominant weights. In the McKay-Cartan case for finite subgroups of the special linear group, the cokernel is related to the abelianization of the subgroup G. In the special case of the classical McKay correspondence, the critical group and the abelianization are shown to be isomorphic.1991 Mathematics Subject Classification. 17B22, 05E10, 14E16 .This says that row i ofC (= column i ofC t ) lies in ker(π) for each i, and hence im(C t ) ⊂ ker(π). Example 6.6. Example 5.12 considered a faithful representation γ :Example 6.7. Example 5.24, considered a faithful representation γ : G = Z/mZ ֒→ SL 2 (C) with K(γ) ∼ = Z/mZ ∼ = G(= G ab ).Example 6.8. On the other hand, Example 5.25 considered a different family of faithful representations γ : G = Z/mZ ֒→ GL n (C) that sent g to ω m I ∈ GL n (C), with K(γ) ∼ = (Z/nZ) m .Note that γ(G) ⊂ SL n (C) if and only if m divides n, which is exactly the same condition under which K(γ) ∼ = (Z/nZ) m can surject onto Z/mZ = G(= G ab ), as Theorem 6.2 would predict.

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Section: Avalanche-finite Matrices and Chip Firingmentioning

confidence: 99%

Chip firing on Dynkin diagrams and McKay quivers

2018

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“…The link between chip-firing and nonnegative rank is provided by the following variant of a theorem of Perkinson, Perlman and Wilmes [20]. The inspiration for Theorem 3.1 is [20,Theorem 4.11], which expresses an arbitrary (n − 1)-dimensional lattice in Z n−1 as a reduced Laplacian lattice ∆ s Z n−1 . Modifying its proof to express L ⊂ Z n 0 as a total Laplacian lattice is straightforward; we give the details below.…”

Section: Nonnegative Rankmentioning

confidence: 99%

CoEulerian graphs

Farrell

Levine

2015

Proc. Amer. Math. Soc.

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We suggest a measure of "Eulerianness" of a finite directed graph and define a class of "coEulerian" graphs. These are the graphs whose Laplacian lattice is as large as possible. As an application, we address a question in chip-firing posed by Björner, Lovász, and Shor in 1991, who asked for "a characterization of those digraphs and initial chip configurations that guarantee finite termination." Björner and Lovász gave an exponential time algorithm in 1992. We show that this can be improved to linear time if the graph is coEulerian, and that the problem is NP-complete for general directed multigraphs.

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“…Recurrent chip-distributions form an important subset of non-terminating chipdistributions. They are studied in the literature (see [11,14,12] and also [15] in a slightly different model). Here we mention some of their properties needed for the rest of the paper.…”

Section: Recurrent Chip-distributionsmentioning

confidence: 99%

Chip-firing based methods in the Riemann–Roch theory of directed graphs

Hujter

Tóthmérész

2019

European Journal of Combinatorics

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Baker and Norine proved a Riemann-Roch theorem for divisors on undirected graphs. The notions of graph divisor theory are in duality with the notions of the chip-firing game of Björner, Lovász and Shor. We use this connection to prove Riemann-Roch-type results on directed graphs. We give a simple proof for a Riemann-Roch inequality on Eulerian directed graphs, improving a result of Amini and Manjunath. We also study possibilities and impossibilities of Riemann-Roch-type equalities in strongly connected digraphs and give examples. We intend to make the connections of this theory to graph theoretic notions more explicit via using the chip-firing framework.

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Primer for the algebraic geometry of sandpiles

Abstract: Abstract. This is a draft of a primer on the algebraic geometry of the Abelian Sandpile Model. version: July, 11, 2009.

Cited by 41 publications

References 25 publications

Chip firing on Dynkin diagrams and McKay quivers

Chip firing on Dynkin diagrams and McKay quivers

CoEulerian graphs

Chip-firing based methods in the Riemann–Roch theory of directed graphs

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