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Explicit deformation of lattice ideals via chip-firing games on directed graphs. (English) Zbl 1328.05119
Summary: For a finite index sublattice \(L\) of the root lattice of type \(A\), we construct a deterministic algorithm to deform the lattice ideal \(I_{\mathrm{L}}\) to a nearby generic lattice ideal, answering a question posed by E. Miller and B. Sturmfels [Combinatorial commutative algebra. New York, NY: Springer (2005; Zbl 1066.13001)]. Our algorithm is based on recent results of D. Perkinson et al. [Contemp. Math. 605, 211–256 (2013; Zbl 1320.05060)] concerning commutative algebraic aspects of chip-firing on directed graphs. As an application of our deformation algorithm, we construct a cellular resolution of the lattice ideal \(I_{\mathrm{L}}\) by degenerating the Scarf complex of its deformation.

MSC:
05C57 Games on graphs (graph-theoretic aspects)
05C20 Directed graphs (digraphs), tournaments
06B99 Lattices
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[1] Asadi, A., Backman, S.: Chip-firing and Riemann-Roch theory for directed draphs (2010). arXiv:1012.0287 · Zbl 1274.05189
[2] Barany, I; Scarf, H, Matrices with identical sets of neighbors, Math. Oper. Res., 23, 863-873, (1998) · Zbl 0977.90023
[3] Bayer, D; Sturmfels, B, Cellular resolutions of monomial modules, J. Reine Angew. Math., 502, 123-140, (1998) · Zbl 0909.13011
[4] Cassels, J.W.S.: An Introduction to the Geometry of Numbers. Springer, Berlin (1959) · Zbl 0086.26203
[5] Cori, R; Rossin, D; Salvy, B, Polynomial ideals for sandpiles and their Gröbner bases, Theor. Comput. Sci., 276, 1-15, (2002) · Zbl 1002.68105
[6] Dochtermann, A; Sanyal, R, Laplacian ideals, and resolutions, J. Algebr. Comb., 40, 805-822, (2014) · Zbl 1303.05217
[7] Holroyd, AE; Levine, L; Msros, K; Peres, Y; Propp, J; Wilson, DB, Chip-firing and rotor-routing on directed graphs, in and out of equilibrium 2, Prog. Probab., 60, 331-364, (2008) · Zbl 1173.82339
[8] Hopkins, S, Another proof of wilmes’ conjecture, Discrete Math., 323, 43-48, (2013) · Zbl 1283.05215
[9] Kateri, M., Mohammadi, F., Sturmfels, B.: A Family of Quasisymmetry Models (2014). arXiv:1403.0547 · Zbl 1344.05155
[10] Manjunath, M; Sturmfels, B, Monomials, binomials and Riemann-Roch, J. Algebr. Comb., 37, 737-756, (2013) · Zbl 1272.13017
[11] Manjunath, M; Schreyer, F-O; Wilmes, J, Minimal free resolutions of the \(G\)-parking function ideal and the toppling ideal, Trans. Am. Math. Soc., 367, 2853-2874, (2015) · Zbl 1310.13022
[12] Miller, E., Sturmfels, B.: Combinatorial Commutative Algebra, Graduate Texts in Mathematics, vol. 227. Springer, Berlin (2005)
[13] Mohammadi, F; Shokrieh, F, Divisors on graphs, connected flags, and syzygies, Int. Math. Res. Not., 24, 6839-6905, (2012) · Zbl 1305.05132
[14] Peeva, I; Sturmfels, B, Generic lattice ideals, J. Am. Math. Soc., 11, 363-373, (1998) · Zbl 0905.13005
[15] Perkinson, D., Perlman, J., Wilmes, J.: Primer for the algebraic geometry of sandpiles, contemporary mathematics. In: Proceedings of the Bellairs Workshop on Tropical and Non-archimedean Geometry, p. 211 (2011) · Zbl 1320.05060
[16] Postnikov, A; Shapiro, B, Trees, parking functions, syzygies, and deformations of monomial ideals, Trans. Am. Math. Soc., 356, 3109-3142, (2004) · Zbl 1043.05038
[17] Speer, E, Asymmetric abelian sandpile models, J. Stat. Phys., 71, 61-74, (1993) · Zbl 0943.82551
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