Gr\"obner bases of ideals cogenerated by Pfaffians

TL;DR

This paper characterizes a class of Pfaffian ideals with generators forming a Gr"obner basis under any anti-diagonal term order, describing their initial ideals, simplicial complexes, and multiplicity.

Contribution

It introduces a complete characterization of one-cogenerated Pfaffian ideals with universal Gr"obner bases and analyzes their algebraic and combinatorial properties.

Findings

Initial ideals are shellable and Cohen-Macaulay.

Provides a formula for the multiplicity of these ideals.

Describes the associated simplicial complexes explicitly.

Abstract

We characterise the class of one-cogenerated Pfaffian ideals whose natural generators form a Gr\"obner basis with respect to any anti-diagonal term-order. We describe their initial ideals as well as the associated simplicial complexes, which turn out to be shellable and thus Cohen-Macaulay. We also provide a formula for computing their multiplicity.