A level initial ideal of the 2-minors determinantal ideal

TL;DR

This paper constructs a specific monomial order for the 2-minors determinantal ideal such that its initial ideal is level, Cohen-Macaulay, and exhibits favorable algebraic and combinatorial properties, including shellability.

Contribution

It introduces a monomial order making the initial ideal of the 2-minors determinantal ideal level and Cohen-Macaulay, and compares Betti tables and shellability properties.

Findings

Initial ideal is level and Cohen-Macaulay.

Betti tables of the ideal and initial ideal are compared.

Shellability of associated simplicial complex is proven for m<n.

Abstract

For $\Bbbk$ a field, let $X$ a $m \times n$ matrix of variables and $S=\Bbbk[X].$ We consider the determinantal ideal $I_2 \subseteq S$ generated by the $2$-minors of $X.$ In this paper we find a suitable monomial order over $S$ such that $I$, the initial ideal of $I_2$ with respect to that order, is level, namely, it is Cohen-Macaulay and the socle of an Artinian reduction of the $\mathbb{N}$-graded algebra $S/I$ is concentrated in only one degree. Moreover, we compare the Betti tables of $I_2$ with the tables of its initial ideals. In the last section, we prove the shellability of the simplicial complex naturally associated to $S/I$ in the case $m<n.$