Let $\tilde{Gr}(k,n)$ be the affine cone of the Grassmannian $Gr(k,n)$. I think that the following set $S$ is an additive basis of $\mathbb{C}[\tilde{Gr}(k,n)]$: \begin{align} S = \{e_T: T \text{ is a rectangular semi-standard Young tableau with $k$ rows}\}, \end{align} where $e_T = P_{T_1} \cdots P_{T_n}$, where $T_i$'s are columns of $T$ and $P_{T_i}$ is the Plücker with indices from the entries of $T_i$. Are there some references about this? Thank you very much.
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asked May 4, 2019 at 15:15
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The result you mention is very classical, but it also fits within the more general and conceptual framework of Standard Monomial Theory: https://en.wikipedia.org/wiki/Standard_monomial_theory.
answered May 4, 2019 at 15:17
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$\begingroup$ thank you very much. I am trying to find an explicit place of the result. But I could not find it. Is there a more explicit reference? Thank you very much. $\endgroup$ Commented May 4, 2019 at 16:30
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1$\begingroup$ @JianrongLi: See for instance Chapter 1 of Seshadri's "Introduction to the Theory of Standard Monomials" (springer.com/us/book/9789811018138) which covers the Grassmannian. $\endgroup$ Commented May 4, 2019 at 16:44