Equivariant Gröbner bases and the Gaussian two-factor model

We show that the kernel I of the ring homomorphism R[yij | i, j ∈ N, i > j]→ R[si, ti | i ∈ N] determined by yij 7→ sisj +titj is generated by two types of polynomials: off-diagonal 3 × 3-minors and pentads. This confirms a conjecture by Drton, Sturmfels, and Sullivant on the Gaussian two-factor model. Our proof is computational: inspired by work of Aschenbrenner and Hillar we introduce the concept of G-Gröbner basis, where G is a monoid acting on an infinite set of variables, and we report on a computation that yielded a finite G-Gröbner basis of I relative to the monoid G of strictly increasing functions N→ N.