principal $6
\times 6$-minor lies in $F_{2,6}$. Theorem \[thm:Main\] implies an analogous statement for the Zariski closures of $F_{2,n}$ and $F_{2,6}$.

We sketch the proof of Theorem \[thm:Main\], which appears in Section \[sec:2f\]. We put a suitable elimination order on the monomials in $y_{ij},\ i,j \in {{\mathbb N}},\ i \geq j$, and report on a computation that yields a finite ${\mathrm{Inc}({{\mathbb N}})}$-Gröbner basis for the determinantal ideal generated by all $3 \times 3$-minors of $y$. Intersecting this ${\mathrm{Inc}({{\mathbb N}})}$-Gröbner basis with the ring in the off-diagonal matrix entries gives Theorem \[thm:Main\].

Acknowledgments {#acknowledgments .unnumbered}
===============

We thank Jan Willem Knopper and Rudi Pendavingh for motivating discussions on alternative computations that would prove Theorem \[thm:2factorGB\]. We also thank the referees for suggestions on improving the exposition.

An algorithm for equivariant Gröbner bases {#sec:GGB}
==========================================

We retain the setting of the introduction: $X$ is a potentially infinite set and ${\mathrm{Mon}}$ is the free commutative monoid generated by $X$, equipped with a term order (EGB1) preserved by the action of a monoid $G$ (EGB2) which also preserves least common multiples (EGB3). Condition EGB4 will be needed only later.

\[re:WhyMonoid\] Note that $G$ acts by injective maps on ${\mathrm{Mon}}$ by EGB2. It is essential 