Minimal free resolutions of the 𝐺-parking function ideal and the toppling ideal

DOI10.1090/S0002-9947-2014-06248-XzbMath1310.13022arXiv1210.7569MaRDI QIDQ5496681

Madhusudan Manjunath, Frank-Olaf Schreyer, John Wilmes

Publication date: 2 February 2015

Published in: Transactions of the American Mathematical Society (Search for Journal in Brave)

Full work available at URL: https://arxiv.org/abs/1210.7569

zbMATH Keywords

minimal resolutionBetti numberschip firinglattice ideal\(G\)-parking function idealacyclic \(k\)-partitiontoppling ideal

Mathematics Subject Classification ID

Graphs and abstract algebra (groups, rings, fields, etc.) (05C25) Commutative rings defined by monomial ideals; Stanley-Reisner face rings; simplicial complexes (13F55) Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases) (13P10) Syzygies, resolutions, complexes and commutative rings (13D02) Combinatorial aspects of commutative algebra (05E40)

Related Items (10)

Divisors on graphs, binomial and monomial ideals, and cellular resolutions ⋮ Tutte short exact sequences of graphs ⋮ Minimal free resolutions of lattice ideals of digraphs ⋮ Skeleton ideals of certain graphs, standard monomials and spherical parking functions ⋮ Fourientations and the Tutte polynomial ⋮ Explicit deformation of lattice ideals via chip-firing games on directed graphs ⋮ An integer sequence and standard monomials ⋮ Another proof of Wilmes' conjecture ⋮ Laplacian ideals, arrangements, and resolutions ⋮ Sandpiles and dominos

Uses Software

• Macaulay2
• SageMath

Cites Work

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• Laplacian ideals, arrangements, and resolutions
• \(G\)-parking functions, acyclic orientations and spanning trees
• Source reversal and chip firing on graphs
• Polynomial ideals for sandpiles and their Gröbner bases
• Chip-firing games, potential theory on graphs, and spanning trees
• Riemann-Roch for sub-lattices of the root lattice \(A_n\)
• Monomials, binomials and Riemann-Roch
• Riemann-Roch and Abel-Jacobi theory on a finite graph
• Another proof of Wilmes' conjecture
• Minimal free resolutions that are not supported by a CW-complex
• Primer for the algebraic geometry of sandpiles
• Chip-Firing and Rotor-Routing on Directed Graphs
• Self-organized critical state of sandpile automaton models
• Trees, parking functions, syzygies, and deformations of monomial ideals

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