Determinantal Facet Ideals

The Michigan Mathematical Journal (Impact Factor: 0.41). 08/2011; 62(1). DOI: 10.1307/mmj/1363958240
Source: arXiv


We consider ideals generated by general sets of $m$-minors of an $m\times
n$-matrix of indeterminates. The generators are identified with the facets of
an $(m-1)$-dimensional pure simplicial complex. The ideal generated by the
minors corresponding to the facets of such a complex is called a determinantal
facet ideal. Given a pure simplicial complex $\Delta$, we discuss the question
when the generating minors of its determinantal facet ideal $J_\Delta$ form a
Gr\"obner basis and when $J_\Delta$ is a prime ideal.

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    • "For example, let G 1 , G 2 be closed graphs on the vertex set [5] with the maximal cliques F 11 = [1] [3], F 12 = [2] [4], F 13 = [3] [5] and F 21 = [1] [3], F 22 = [2] [5], respectively. One can easily see that the maximal cliques F 13 = [3] [5] and F 21 = [1] [3] give the clique [3] [6] in the associated graph G which is not maximal. Actually, the maximal cliques of G are [1] [3], [2] [6], [3] [7], and [4] [8]. (2) The cliques [a, b] and [c, d] in the above proposition are not necessarily uniquely determined by the maximal clique of G. "
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    ABSTRACT: We study ideals generated by $2$--minors of generic Hankel matrices.
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    ABSTRACT: We consider determinantal facet ideals i.e., ideals generated by those minors of a generic matrix corresponding to facets of simplicial complexes. Using these ideals we give a combinatorial description for minimal primes of some mixed determinantal ideals. At the end, we discuss the minimal free resolution of these ideals.