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$\DeclareMathOperator\Gr{Gr}$Let $\Gr(k,n)$ be the Grassmannian variety of $k$-planes in an $n$-dimensional vector space. The coordinate algebra $\mathbb{C}[\Gr(k,n)]$ is generated by Plücker coordinates subject to Plücker relations. The Plücker relations have $r$ summands with $r \ge 3$. In the case of $\Gr(3,6)$, it is verified in the book, Example 6.8.13, that when we multiply some monomial to a $4$-term Plücker relation, we can write it using monomials and $3$-term relations: \begin{align} & P_{124}(P_{134}P_{246}-P_{134}P_{256}-P_{136}P_{245}-P_{123}P_{456}) \\ & = P_{246}(P_{124}P_{135}-P_{123}P_{145}-P_{125}P_{134}) \\ & - P_{134}(P_{124}P_{256}-P_{125}P_{246}+P_{126}P_{245}) \\ & - P_{245}(P_{124}P_{136}-P_{123}P_{146}-P_{126}P_{134}) \\ & - P_{123}(P_{124}P_{456}-P_{145}P_{246}+P_{146}P_{245}). \end{align} Therefore this $4$-term polynomial is in the ideal generated by $3$-term polynomials given by $3$-term Plücker relations.

My question is: is this true in general? That is, can every $r$-term ($r \ge 4$) Plücker relation (we can multiply a monomial to the $r$-term Plücker relation) be written using monomials and $3$-term Plücker relations? Are there some references about this? Thank you very much.

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    $\begingroup$ Actually maybe I'm misunderstanding your question. Quoting from pg.3 of arxiv.org/abs/0808.2383: "The relations (1) [the 3 term Plücker relations] do not generate the Plücker ideal [...], but they always suffice to generate the image of [the ideal] in the Laurent polynomial ring." So maybe the "up to monomials" claim is true. $\endgroup$
    – Sam Hopkins
    Commented Sep 27, 2022 at 12:12
  • $\begingroup$ @SamHopkins, thank you very much! Yes, I assume that we can multiply a monomial to the $r$-term ($r \ge 4$) Plucker relation. $\endgroup$
    – Jianrong Li
    Commented Sep 27, 2022 at 14:30
  • $\begingroup$ It may interest you to know... to the best of my knowledge, the corresponding question for the quantum Grassmannian is still open. (The quantum case is harder due to the definition of the quantum Grassmannian being in terms of a particular subring of the ring of quantum matrices, which is not very helpful for questions of this type.) $\endgroup$
    – Jan Grabowski
    Commented Sep 27, 2022 at 14:54
  • $\begingroup$ @JanGrabowski, thank you very much! Is there some references which study the quantum version of this question? $\endgroup$
    – Jianrong Li
    Commented Sep 27, 2022 at 14:59
  • $\begingroup$ @JianrongLi I'm not sure exactly where to look for this specific question but if you start with the various papers by Launois, Lenagan and coauthors (in various combinations), they and their references ought to help. It's also possible that some of it is folklore, unfortunately. $\endgroup$
    – Jan Grabowski
    Commented Sep 28, 2022 at 15:09

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