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Determinantal Facet Ideals for Smaller Minors
TL;DR: In this paper, the authors define and study the notion of determinantal facet ideal (DFI) which is generated by a subset of maximal minors of a generic matrix indexed by the facets of a simplicial complex.
Abstract: A determinantal facet ideal (DFI) is generated by a subset of the maximal minors of a generic $n\times m$ matrix indexed by the facets of a simplicial complex. We consider the more general notion of an $r$-DFI, which is generated by a subset of $r$-minors of a generic matrix indexed by the facets of a simplicial complex for some $1\leq r\leq n$. We define and study so-called lcm-closed and interval DFIs, and show that the minors parametrized by the facets of $\Delta$ form a reduced Grobner basis with respect to any diagonal term order in both of these cases. We also see that being lcm-closed generalizes conditions previously introduced in the literature, and conjecture that in the case $r=n$, lcm-closedness is necessary for being a Grobner basis. We also give conditions on the maximal cliques of $\Delta$ ensuring that lcm-closed and interval DFIs are Cohen-Macaulay. Finally, we conclude with a variant of a conjecture of Ene, Herzog, and Hibi on the Betti numbers of certain types of $r$-DFIs, and provide a proof of this conjecture for Cohen-Macaulay interval DFIs.
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TL;DR: In this paper, Grobner bases for the Rees algebra and special fiber ring of a closed determinantal facet ideal were found, and they were shown to be of fiber type and their special fiber rings are Koszul.
Abstract: Using SAGBI basis techniques, we find Grobner bases for the presentation ideals of the Rees algebra and special fiber ring of a closed determinantal facet ideal. In particular, we show that closed determinantal facet ideals are of fiber type and their special fiber rings are Koszul. Moreover, their Rees algebras and special fiber rings are normal Cohen-Macaulay domains, and have rational singularities.
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TL;DR: In this article, it was shown that the linear strand of the initial ideal of a determinantal facet ideal (DFI) is supported by a polyhedral cell complex obtained as an induced subcomplex of the complex of boxes.
Abstract: A determinantal facet ideal (DFI) is an ideal $J_\Delta$ generated by maximal minors of a generic matrix parametrized by an associated simplicial complex $\Delta$. In this paper, we construct an explicit linear strand for the initial ideal with respect to any diagonal term order $<$ of an arbitrary DFI. In particular, we show that if $\Delta$ has no \emph{1-nonfaces}, then the Betti numbers of the linear strand of $J_\Delta$ and its initial ideal coincide. We apply this result to prove a conjecture of Ene, Herzog, and Hibi on Betti numbers of closed binomial edge ideals in the case that the associated graph has at most $2$ maximal cliques. More generally, we show that the linear strand of the initial ideal (with respect to $<$) of \emph{any} DFI is supported on a polyhedral cell complex obtained as an induced subcomplex of the \emph{complex of boxes}, introduced by Nagel and Reiner.
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TL;DR: In this article, it was shown that the linear strand of the initial ideal of a determinantal facet ideal (DFI) is supported by a polyhedral cell complex obtained as an induced subcomplex of the complex of boxes.
Abstract: A determinantal facet ideal (DFI) is an ideal $J_\Delta$ generated by maximal minors of a generic matrix parametrized by an associated simplicial complex $\Delta$. In this paper, we construct an explicit linear strand for the initial ideal with respect to any diagonal term order $<$ of an arbitrary DFI. In particular, we show that if $\Delta$ has no \emph{1-nonfaces}, then the Betti numbers of the linear strand of $J_\Delta$ and its initial ideal coincide. We apply this result to prove a conjecture of Ene, Herzog, and Hibi on Betti numbers of closed binomial edge ideals in the case that the associated graph has at most $2$ maximal cliques. More generally, we show that the linear strand of the initial ideal (with respect to $<$) of \emph{any} DFI is supported on a polyhedral cell complex obtained as an induced subcomplex of the \emph{complex of boxes}, introduced by Nagel and Reiner.
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TL;DR: In this paper, a hierarchy of combinatorial properties for simplicial complexes that generalize unit-interval, interval, and co-comparability graphs is introduced, and the determinantal facet ideals of all under-closed and semi-closed complexes have a square-free initial ideal with respect to any diagonal monomial order.
Abstract: We study d-dimensional generalizations of three mutually related topics in graph theory: Hamiltonian paths, (unit) interval graphs, and binomial edge ideals. We provide partial high-dimensional generalizations of Ore and Posa's sufficient conditions for a graph to be Hamiltonian. We introduce a hierarchy of combinatorial properties for simplicial complexes that generalize unit-interval, interval, and co-comparability graphs. We connect these properties to the already existing notions of determinantal facet ideals and Hamiltonian paths in simplicial complexes. Some important consequences of our work are:
(1) Every almost-closed strongly-connected d-dimensional simplicial complex is traceable. (This extends the well-known result "unit-interval connected graphs are traceable".)
(2) Every almost-closed d-complex that remains strongly connected after the deletion of d or less vertices, is Hamiltonian. (This extends the fact that "unit-interval 2-connected graphs are Hamiltonian".)
(3) The minors defining the determinantal facet ideal of any almost-closed complex form a lex-Groebner basis. (This revises a recent theorem by Ene et al., and extends a result by Herzog and others.)
(4) The determinantal facet ideals of all under-closed and semi-closed complexes have a square-free initial ideal with respect to any diagonal monomial order. In positive characteristic, they are even Frobenius split. (This provides the largest known class of determinantal facet ideals that are radical.)
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TL;DR: It follows that all binomial edge ideals are radical ideals, and the results apply for the class of conditional independence ideals where a fixed binary variable is independent of a collection of other variables, given the remaining ones.
244 citations
TL;DR: In this article, the Grobner basis of I G of S generated by 2-minors [i, j] of X which correspond to edges of G was constructed.
Abstract: Let G be a finite graph on [n] = {1, 2,…, n}, X a 2 × n matrix of indeterminates over a field K, and S = K[X] a polynomial ring over K. In this article, we study about ideals I G of S generated by 2-minors [i, j] of X which correspond to edges {i, j} of G. In particular, we construct a Grobner basis of I G as a set of paths of G and compute a primary decomposition.
156 citations
TL;DR: In this article, the depth of classes of binomial edge ideals and classifications of closed graphs with Cohen-Macaulay edge ideal were studied and the binomial-edge ideal is defined.
Abstract: We study the depth of classes of binomial edge ideals and classify all closed graphs whose binomial edge ideal is Cohen-Macaulay.
116 citations
115 citations
01 Jan 1998
TL;DR: It is shown how primary decompositions of an ideal can give useful descriptions of components of a graph arising in problems from combinatorics, statistics, and operations research.
Abstract: This paper shows how primary decompositions of an ideal can give useful descriptions of components of a graph arising in problems from combinatorics, statistics, and operations research. We begin this introduction with the general formulation. Then we give the simplest interesting example of our theory, followed by a statistical example similar to that which provided our original motivation. Later on we study the primary decompositions corresponding to some natural combinatorial problems.
103 citations