Quadrics defined by skew-symmetric matrices

TL;DR

This paper introduces a new model for computing minimal free resolutions of ideals generated by products of skew-symmetric matrices and generic vectors, verified for small cases and conjectured to extend generally.

Contribution

The paper proposes a novel model for minimal free resolutions of specific ideals involving skew-symmetric matrices, verified for small sizes and conjectured for general cases.

Findings

Model verified for n=3 and n=4 cases

Conjectures proposed for general n

Potential to compute resolutions for all n

Abstract

In this paper we propose a model for computing a minimal free resolution for ideals of the form $I_{1}(X_{n}Y_{n})$, where $X_{n}$ is an $n\times n$ skew-symmetric matrix with indeterminate entries $x_{ij}$ and $Y_{n}$ is a generic column matrix with indeterminate entries $y_{j}$. We verify that the model works for $n=3$ and $n=4$ and pose some statements as conjectures. Answering the conjectures in affirmative would enable us to compute a minimal free resolution for general $n$.