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Why do we care about Schur Positivity

In algebraic combinatorics, conjectures that certain numbers are integers or are positive or are unimodal are really implicit challenges to find the hidden structure. And finding hidden structure is ...
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Generating function for Schur polynomials

This is done in my paper The character generator of SU(n). I believe there was an essentially the same previous MO question, but I am unable to find it.
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Do you know an elegant proof for this expression for a Schur function?

I would like to suggest an interpretation using super symmetric functions. These are symmetric functions that are symmetric in two sets of variables $\{x_i\}$ and $\{y_j\}$ separately. They satisfy ...
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Applying $\sum_i \partial_{x_i}$, $\sum_i x_i \partial_{x_i}$ and $\sum_i x_i^2 \partial_{x_i}$ to Schur polynomials
The answer by Jules Lamers considers $k=1$. In the comments to that answer, people have mentioned that the case $k=0$ has been solved. Namely, for a given partition $\lambda$ let Del($\lambda$) be the ...