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Monomial and toric ideals associated to Ferrers graphs
Commutative Algebra · Submitted 13 Sep 2006
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim In fact, we prove that this property characterizes Ferrers graphs among bipartite graphs.
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The abstract, sentence by sentence
Each partition $\lambda = (\lambda_1, \lambda_2, ..., \lambda_n)$ determines a so-called Ferrers tableau or, equivalently, a Ferrers bipartite graph. Its edge ideal, dubbed Ferrers ideal, is a squarefree monomial ideal that is generated by quadrics. We show that such an ideal has a 2-linear minimal free resolution, i.e. it defines a small subscheme. In fact, we prove that this property characterizes Ferrers graphs among bipartite graphs. Furthermore, using a method of Bayer and Sturmfels, we provide an explicit description of the maps in its minimal free resolution: This is obtained by associating a suitable polyhedral cell complex to the ideal/graph. Along the way, we also determine the irredundant primary decomposition of any Ferrers ideal. We conclude our analysis by studying several features of toric rings of Ferrers graphs. In particular we recover/establish formulae for the Hilbert series, the Castelnuovo-Mumford regularity, and the multiplicity of these rings. While most of the previous works in this highly investigated area of research involve path counting arguments, we offer here a new and self-contained approach based on results from Gorenstein liaison theory.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | Each partition $\lambda = (\lambda_1, \lambda_2, ..., \lambda_n)$ determines a so-called Ferrers tableau or, equivalently, a Ferrers bipartite graph. | background 1.00 | 0.09 | 0.02 | asserted 1.00 |
| s1 | Its edge ideal, dubbed Ferrers ideal, is a squarefree monomial ideal that is generated by quadrics. | background 0.92 | 0.14 | 0.05 | asserted 1.00 |
| s2 | We show that such an ideal has a 2-linear minimal free resolution, i.e. it defines a small subscheme. | result 0.99 | 0.95 | 0.36 | asserted 1.00 |
| s3 | In fact, we prove that this property characterizes Ferrers graphs among bipartite graphs. | result 0.97 | 0.94 | 0.42 | asserted 1.00 |
| s4 | Furthermore, using a method of Bayer and Sturmfels, we provide an explicit description of the maps in its minimal free resolution: This is obtained by associating a suitable polyhedral cell complex to the ideal/graph. | method 0.57 | 0.95 | 0.08 | asserted 1.00 |
| s5 | Along the way, we also determine the irredundant primary decomposition of any Ferrers ideal. | result 0.88 | 0.97 | 0.07 | asserted 1.00 |
| s6 | We conclude our analysis by studying several features of toric rings of Ferrers graphs. | contribution 0.99 | 0.92 | 0.07 | asserted 0.87 |
| s7 | In particular we recover/establish formulae for the Hilbert series, the Castelnuovo-Mumford regularity, and the multiplicity of these rings. | result 0.98 | 0.91 | 0.13 | asserted 1.00 |
| s8 | While most of the previous works in this highly investigated area of research involve path counting arguments, we offer here a new and self-contained approach based on results from Gorenstein liaison theory. | method 0.59 | 0.87 | 0.06 | asserted 1.00 |
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Original abstract
Each partition $\lambda = (\lambda_1, \lambda_2, ..., \lambda_n)$ determines a so-called Ferrers tableau or, equivalently, a Ferrers bipartite graph. Its edge ideal, dubbed Ferrers ideal, is a squarefree monomial ideal that is generated by quadrics. We show that such an ideal has a 2-linear minimal free resolution, i.e. it defines a small subscheme. In fact, we prove that this property characterizes Ferrers graphs among bipartite graphs. Furthermore, using a method of Bayer and Sturmfels, we provide an explicit description of the maps in its minimal free resolution: This is obtained by associating a suitable polyhedral cell complex to the ideal/graph. Along the way, we also determine the irredundant primary decomposition of any Ferrers ideal. We conclude our analysis by studying several features of toric rings of Ferrers graphs. In particular we recover/establish formulae for the Hilbert series, the Castelnuovo-Mumford regularity, and the multiplicity of these rings. While most of the previous works in this highly investigated area of research involve path counting arguments, we offer here a new and self-contained approach based on results from Gorenstein liaison theory.
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1 arXiv paper cite this in 8 sentences
Papers citing it · 7 sentences without a confident label
Uses its method or tool 1 paper
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To determine the formula we follow the strategy employed in [ref] , where the authors use algebraic means to determine the Betti numbers.
Algebraic properties of edge ideals via combinatorial topology · Ferrers graphs · 1.00
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