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.



Reference request: additive basis of coordinate ring of Grassmannians
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>.


