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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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  • $\begingroup$ I think "plethysm" is the keyword you want to google, but someone more expert will probably have something more precise to say. (The other thing is that if you get non-semistandard tableaux in your formula, you can convert them to SSYT using a straightening law, right?) $\endgroup$ Aug 23, 2021 at 13:01
  • $\begingroup$ Thank you very much for your answer (I do not know much about tableaux, I am just trying to use them to do some computations). Isn't plethysm the decomposition of products or composition of representations in irreducible ones? $\endgroup$
    – pi_1
    Aug 23, 2021 at 13:25
  • $\begingroup$ But, I have had a look at "straightening laws"; it seems to be an interesting operation. I will try to read about it to make sure that it is what I need. Thank you. $\endgroup$
    – pi_1
    Aug 23, 2021 at 13:27
  • $\begingroup$ Hmm, yes, you are right about plethysm, but my guess is that there is still some way of looking at your question from perspective of plethysm. $\endgroup$ Aug 23, 2021 at 13:28
  • $\begingroup$ I think, one can express those modules (in the nice cases) in terms of kernel or quotients of "usual" tensors (made of composition of exterior and symmetric powers) but the problem is then (if somebody wants to compute, for example, the rank of $\varphi$) to find a basis... $\endgroup$
    – pi_1
    Aug 23, 2021 at 13:33

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