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Let $1 \leq i_1 < i_2 < i_3 \leq n$. I know that there is an injective map from $V(\omega_{i_1}+\omega_{{i_2} -1})\otimes V(\omega_{{i_3}+1})$ to $V(\omega_{i_1}+\omega_{i_2})\otimes V(\omega_{i_3})$. Can anyone tell me how to realize $V(\omega_{i_1}+\omega_{{i_2} -1})\otimes V(\omega_{{i_3}+1})$ inside $V(\omega_{i_1}+\omega_{i_2})\otimes V(\omega_{i_3})$? In my notation $\omega_i$'s are the fundamental weights of the simple Lie algebra $\mathfrak{sl}_{n+1}$ and $V(\lambda)$'s are finite dimensional irreducible modules for $\mathfrak{sl_{n+1}}$. I basically want to know a combinatorial interpretation of the numbers $\text{dim}V(\omega_{i_1}+\omega_{{i_2} -1})\otimes V(\omega_{{i_3}+1})-\text{dim}V(\omega_{i_1}+\omega_{i_2})\otimes V(\omega_{i_3})$ in terms of young tableaux.

P.S. In my notation , I also assume that $\omega_0=\omega_{n+1}=0$

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  • $\begingroup$ Do you know how to phrase this in a more combinatorial way, say, as coefficients in products of Schur polynomials or similar? $\endgroup$
    – Per Alexandersson
    Oct 29, 2019 at 19:44
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    $\begingroup$ Yes, what I am asking is the following. what is the combinatorial interpretation of the following difference of product of two Schur functions $s_{2^{i_1}1^{i_2-i_1}}s_{1^{i_3}}-s_{2^{i_1}1^{i_2-i_1-1}}s_{1^{i_3 +1}}$ in the case of $\mathfrak{sl}_{n+1}$ Where I have written partitions in exponential form? $\endgroup$
    – Rekha Biswal
    Oct 29, 2019 at 19:56
  • $\begingroup$ Ah, hm, so product can be encoded as taking union of the corresponding Young diagrams, forming a skew shape. You can then try to apply Jeu-de-taqin on the corresponding sets of skew semi-standard Young tableaux. Since the shapes are so special, it might be possible to track what it is, or use the Littlewood Richardson rule to expand both products in terms of Schur functions.. $\endgroup$
    – Per Alexandersson
    Oct 29, 2019 at 20:01
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    $\begingroup$ @PerAlexandersson It may be easier than that - since the second Schur function in each term is a column Schur function, one can use Pieri's rule. Let $\lambda=(\lambda_1,\lambda_2)$ be a partition with two columns, and let $\mu=(\lambda_1+1,\lambda_2)$. One needs to construct an injective map from the set of partitions obtained by adding a vertical strip of length $k$ to $\mu$ into the set of partitions obtained by adding a vertical strip of length $k+1$ to $\lambda$. $\endgroup$
    – Amritanshu Prasad
    Oct 30, 2019 at 3:27
  • $\begingroup$ @Per and Amri, thank you very much for your reply. Both of your comments are very helpful. $\endgroup$
    – Rekha Biswal
    Oct 30, 2019 at 7:44

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