Relations between the 2x2 minors of a generic matrix

TL;DR

This paper proves a specific case of a conjecture regarding the minimal algebraic relations among 2x2 minors of a generic matrix, using advanced algebraic geometry and homological techniques.

Contribution

It establishes the t=2 case of the Bruns-Conca-Varbaro conjecture, linking relations among minors to generalized permanents via polynomial functors and duality.

Findings

Proves the t=2 case of the conjecture.

Connects relations among minors to generalized permanents.

Uses Koszul homology, subspace varieties, and Kempf-Weyman techniques.

Abstract

We prove the case t = 2 of a conjecture of Bruns-Conca-Varbaro, describing the minimal relations between the t x t minors of a generic matrix. Interpreting these relations as polynomial functors, and applying transpose duality as in the work of Sam-Snowden, this problem is equivalent to understanding the relations satisfied by t x t generalized permanents. Our proof follows by combining Koszul homology calculations on the minors side, with a study of subspace varieties on the permanents side, and with the Kempf-Weyman technique (on both sides).