There is a way to characterize for which $x_1,...,x_d$ a Schur polynomial, that can be defined as $$s_\lambda(x_1,...,x_d)=\sum_{T\in SSYT(\lambda)}x_1^{t_1}...x_d^{t_d}, $$ with the sum running over the semistandard Young tableaux of shape $\lambda$, takes integer values? there is research along those lines?
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asked Oct 21, 2022 at 6:19
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2$\begingroup$ You mean that for all $\lambda$ the value must be integer? For fixed $\lambda$ there hardly can be a characterization other than "this polynomial at this point takes an integer value". $\endgroup$– Fedor PetrovCommented Oct 21, 2022 at 6:36
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$\begingroup$ Thanks for the comment. I Forgot to specify that I want to know when the specialization is an integer for some set of diagrams, but maybe not all of them. $\endgroup$– Nicolas Medina SanchezCommented Oct 21, 2022 at 6:40
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1$\begingroup$ Can you perhaps give an example of what you want? Say, a certain set of diagrams where you expect or already have a reasonable description? $\endgroup$– Vladimir DotsenkoCommented Oct 21, 2022 at 7:55
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1$\begingroup$ If you want for all of them simultaneously, then that's equivalent to integrality in the other symmetric function bases, the m:s, e:s and h:s. $\endgroup$– Per AlexanderssonCommented Oct 21, 2022 at 14:16
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