Let $V$ be a $n$-dimensional complex vector space.

Assume we have a $\mathbb C$-linear map $\varphi:\Gamma^{(a_1,\dots,a_n)}V\rightarrow \Gamma^{(b_1,\dots,b_n)}V$ between two irreducible representations of $SL_n$ that we know be given by the polar form $f\in {\rm Sym}^dV^*$ of a homogeneous polynomial $F$ (with $\sum_ia_i=d+\sum_ib_i$).

Each irreducible representation admits a basis parametrized by semistandard tableaux (with values in $[1,n]$) so it should be possible to see $\varphi$ in those basis but it is not easy to interpret the image in terms of tableaux.

For example, consider $\varphi:\Gamma^{(5,5,2,2)}V\rightarrow \Gamma^{(5,3,2,1)}V$ given by a $f\in {\rm Sym}^3V^*$ and the tableau $$\begin{aligned}1 \ 1\ 1\ 1\ 1\\ 2\ 2\ 2\ 2\ 2\\ 3\ 3\ \ \ \ \ \ \ \ \ \\ 4\ 4.\ \ \ \ \ \ \ \ \end{aligned}$$ The associated element of $(\wedge^4V)^{\otimes 2}\otimes \wedge^2V\otimes \wedge^2V\otimes \wedge^2V$ is $(e_1\wedge\cdots\wedge e_4)^{\otimes 2}\otimes (e_1\wedge e_2)^{\otimes 3}$ and its image should be $$\begin{aligned}\sum_{i=1}^4(-1)^if(e_i,e_1,e_1)det(V)\otimes \hat{e_i}\otimes (e_1\wedge e_2)\otimes e_2\otimes e_2 - \sum_{i=1}^4(-1)^if(e_i,e_2,e_1)det(V)\otimes \hat{e_i}\otimes (e_1\wedge e_2)\otimes e_1\otimes e_2\\ +\sum_{i=1}^4(-1)^if(e_i,e_2,e_2)det(V)\otimes \hat{e_i}\otimes (e_1\wedge e_2)\otimes e_1\otimes e_1-\sum_{i=1}^4(-1)^if(e_i,e_1,e_2)det(V)\otimes \hat{e_i}\otimes (e_1\wedge e_2)\otimes e_2\otimes e_1\end{aligned}$$
but the naive interpretation of the vectors in the last sum in terms of tableaux would give tableaux that are not weakly increasing in the first so not semistandard.

So, how to write those vectors in terms of tableaux ?

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