Relations between the 2 × 2 minors of a generic matrix

Abstract

We prove the case t=2 of a conjecture of Bruns–Conca–Varbaro, describing the minimal relations between the t×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×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).