Algebraic and combinatorial properties of ideals and algebras of uniform clutters of TDI systems

TL;DR

This paper explores algebraic and combinatorial properties of uniform clutters, linking their structural features to algebraic properties of associated ideals and algebras, with implications for Gr"obner bases, Ehrhart rings, and normality.

Contribution

It establishes new conditions under which uniform clutters exhibit properties like vertex criticality, diagonalization, and unimodularity, connecting combinatorial structures to algebraic and geometric properties.

Findings

C proves C is vertex critical under certain conditions.

A diagonalizes to an identity matrix if C satisfies the max-flow min-cut property.

Regular triangulations of the cone are unimodular when A is balanced.

Abstract

Let C be a uniform clutter, i.e., all the edges of C have the same size, and let A be the incidence matrix of C. We denote the column vectors of A by v1,...,vq. The vertex covering number of C, denoted by g, is the smallest number of vertices in any minimal vertex cover of C. Under certain conditions we prove that C is vertex critical. If C satisfies the max-flow min-cut property, we prove that A diagonalizes over the integers to an identity matrix and that v1,...,vq is a Hilbert basis. It is shown that if C has a perfect matching such that C has the packing property and g=2, then A diagonalizes over the integers to an identity matrix. If A is a balanced matrix we prove that any regular triangulation of the cone generated by v1,...,vq is unimodular. Some examples are presented to show that our results only hold for uniform clutters. These results are closely related to certain algebraic properties, such as the normality or torsion freeness, of blowup algebras of edge ideals and to finitely generated abelian groups. They are also related to the theory of Gr\"obner bases of toric ideals and to Ehrhart rings.