EQUIVARIANT GROBNER BASES AND THE GAUSSIAN TWO-FACTOR MODEL

Exploiting symmetry in Grobner 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 clue to the fact that the group elements, being invertible, cannot preserve a term order. By contrast, inspired by work of Aschenbrenner and Hi liar, we introduce the concept of eguivariant Grobner 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 Grobner 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 is an element of N, i > j] -> R[s(i), t(i) | i is an element of N], y(ij) -> s(i)s(j) + t(i)t(j) is generated by two types of polynomials: off-diagonal 3 x 3-minors and pentads. This confirms a conjecture by Drton, Sturmfels, and Sullivant on the Gaussian two-factor model from algebraic statistics.