Equivariant Gröbner bases and the Gaussian two-factor model

Abstract

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 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 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 ∈ ℕ, i>j] → ℝ[s i , t i | i ∈ 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.