Consider an $n$-dimensional complex vector space $V$ with a chosen basis $e_1,\ldots,e_n$. This basis defines a Cartan decompostion of $GL(V)\cong GL_n$ and for an (integral dominant) highest weight $\lambda$ we may consider the irreducible representation $L_\lambda$. In $L_\lambda$ we have the Gelfand-Tsetlin basis for the chain $$GL(V)\supset GL(\mathrm{span}(e_1,\ldots,e_{n-1}))\supset\ldots\supset GL(\mathbb{C}e_1).$$

If we also assume that $\lambda$ is a partition (its coordinates are nonnegative integers), then we may consider the Schur module $\mathbb S^\lambda V$ as a factor of the tensor product $$\Lambda=(\wedge^1V)^{\otimes(\lambda_1-\lambda_2)}\otimes(\wedge^2V)^{\otimes(\lambda_2-\lambda_3)}\otimes\ldots\otimes(\wedge^nV)^{\otimes\lambda_n}.$$ Tableaux of shape $\lambda$ with pairwise distinct elements from $\{1,\ldots,n\}$ in each column enumerate the obvious basis in $\Lambda$. Images of the vectors corresponding to semistandard tableaux are known to comprise a basis in $\mathbb S^\lambda V$.

Now, the representations $L_\lambda$ and $\mathbb S^\lambda V$ of $GL(V)$ can be naturally identified up to a scalar multiple. Thus, we have two bases in the same space. For many years I kind of assumed these bases were the same simply via the standard weight preserving bijection between GT patterns and SSYTs. When I finally decided to take a closer look, however, they turned out to be distinct even for $GL_3$.

So my question is: is there some concrete connection between these bases? Some sort of duality, perhaps?