Equivariant Groebner bases and the Gaussian two-factor model

Exploiting symmetry in Groebner basis computations is difficult when the symmetry takes the form of a group acting by automorphisms on monomials in finitely many variables. This is largely due to the fact that the group elements, being invertible, cannot preserve a term order. By contrast, inspired by work of Aschenbrenner and Hillar, we introduce the concept of equivariant Groebner basis in a setting where a_monoid_ acts by_homomorphisms_ on monomials in potentially infinitely many variables. We require that the action be compatible with a term order, and under some further assumptions derive a Buchberger-type algorithm for computing equivariant Groebner bases. Using this algorithm and the monoid of strictly increasing functions N -> N we prove that the kernel of the ring homomorphism R[y_{ij} | i,j in N, i > j] -> R[s_i,t_i | i in N], y_{ij} -> s_i s_j + t_i t_j is generated by two types of polynomials: off-diagonal 3x3-minors and pentads. This confirms a conjecture by Drton, Sturmfels, and Sullivant on the Gaussian two-factor model from algebraic statistics.