Determinantal Facet Ideals

TL;DR

This paper studies determinantal facet ideals generated by minors of matrices, exploring conditions under which these ideals have Gr"obner bases and are prime, linking algebraic properties to combinatorial structures.

Contribution

It characterizes when the minors of a determinantal facet ideal form a Gr"obner basis and when the ideal is prime, based on the structure of the underlying simplicial complex.

Findings

Conditions for minors to form a Gr"obner basis

Criteria for the ideal to be prime

Connection between algebraic properties and simplicial complexes

Abstract

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.