Equivariant Groebner bases and the Gaussian two-factor model

We show that the kernel I of the ring homomorphism R[y_ij | i,j in N, i > j] -> R[s_i,t_i | i in N] determined by y_ij -> s_is_j + t_it_j is generated by two types of polynomials: off-diagonal 3-times-3 and so-called 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-Groebner basis, where G is a monoid acting on an infinite set of variables, and we report on a computation that yielded a finite G-Groebner basis of I relative to the monoid G of strictly increasing functions N -> N.