Equivariant Gröbner bases and the Gaussian two-factor model

Abstract:

Exploiting symmetry in Gröbner 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 Gröbner 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 Gröbner bases.

Using this algorithm and the monoid of strictly increasing functions $\mathbb {N} \to \mathbb {N}$ we prove that the kernel of the ring homomorphism \[ \mathbb {R}[y_{ij} \mid i,j \in \mathbb {N}, i > j] \to \mathbb {R}[s_i,t_i \mid i \in \mathbb {N}],\ y_{ij} \mapsto s_is_j + t_it_j \] is generated by two types of polynomials: off-diagonal $3 \times 3$-minors and pentads. This confirms a conjecture by Drton, Sturmfels, and Sullivant on the Gaussian two-factor model from algebraic statistics.