Rees algebra of maximal order Pfaffians and its diagonal subalgebras

TL;DR

This paper explicitly describes the defining equations of the Rees algebra of maximal order Pfaffian ideals from generic skew-symmetric matrices and proves that their diagonal subalgebras are Koszul, with applications to sparse matrices.

Contribution

It provides explicit equations for the Rees algebra of Pfaffian ideals and establishes Koszulness of their diagonal subalgebras, including cases of sparse matrices and applications to powers.

Findings

Rees algebra of maximal Pfaffian ideals is explicitly characterized.

Diagonal subalgebras of these Rees algebras are Koszul.

Pfaffian ideals of certain sparse matrices are of Gr"obner linear type.

Abstract

Given a skew-symmetric matrix $X$, the Pfaffian of $X$ is defined as the square root of the determinant of $X$. In this article, we give the explicit defining equations of the Rees algebra of a Pfaffian ideal $I$ generated by the maximal order Pfaffians of a generic skew-symmetric matrix. We further prove that all diagonal subalgebras of the corresponding Rees algebra of $I$ are Koszul. We also look at Rees algebras of Pfaffian ideals of linear type associated with certain sparse skew-symmetric matrices. In particular, we consider the tridiagonal matrices and identify the corresponding Pfaffian ideals to be of Gr\"obner linear type and as the vertex cover ideals of unmixed bipartite graphs. As an application of our results, we conclude that all their ordinary and symbolic powers have linear quotients.