# An Ehrhart positivity question related to Schur polynomials

Consider the Schur polynomial $$s_\lambda(x_1,\dotsc,x_k)$$. It is easy to see from the hook-content formula for counting the number of semi-standard tableaux, that the function $$n \to s_{n \lambda}(1,1,\dotsc,1) = |SSYT(n \lambda,k)|$$ is a polynomial in $$n$$ with non-negative coefficients. Here, $$SSYT(\lambda,k)$$ denotes the set of semi-standard tableaux of shape $$\lambda$$, and no entry exceeding $$k$$. The notation $$n\lambda$$ denotes element-wise multiplication.

Now, what about the map $$n \to s_{n \lambda/n\mu}(1,1,\dotsc,1) = |SSYT(n \lambda/n\mu,k)|$$ where we now consider skew Schur polynomials?

It is easy to prove that this is again a polynomial (it is the Ehrhart polynomial of certain Gelfand-Tsetlin polytopes), but there is a notorious lack of a hook-content formula for counting skew SSYTs.

Is there some argument that proves that the above map is a polynomial with non-negative coefficients?

• Would it be more precise to say "nλ denotes the partition where each part of λ is multiplied by n" instead of "The notation nλ denotes element-wise multiplication"? – Wolfgang Dec 8 '18 at 21:37
• @Wolfgang Sure, that is another way to phrase it. – Per Alexandersson Dec 8 '18 at 21:58
• To make sure I understand, $k$ is fixed throughout, yes? – David E Speyer Dec 9 '18 at 15:25
• @DavidESpeyer: Yes, exactly. – Per Alexandersson Dec 9 '18 at 15:32