Equivariant Syzygies of the Ideal of 2 x 2 Permanents of a 2 x n Matrix

TL;DR

This paper studies the algebraic structure of the ideal generated by 2x2 permanents of a 2xn matrix, revealing new methods to compute its Betti numbers using symmetry and group actions.

Contribution

It introduces a novel approach to determine equivariant syzygies and Betti numbers of the ideal of 2x2 permanents, expanding understanding of its algebraic properties.

Findings

Explicit description of equivariant syzygies

A new method for calculating Betti numbers

Connections to symmetry and group actions

Abstract

We describe the equivariant syzygies of the ideal of $2 \times 2$ permanents of a generic $2 \times n$ matrix under its natural symmetric and torus group actions. Our proof gives us a new method of finding the Betti numbers of this ideal, which were first described by Gesmundo, Huang, Schenck, and Weyman.