<s>[INST] Is the polynomial $x^4+2x^3 y+4x^2 y^2+2 x y^3+y^4$ Schur positive? I tried to check that whether the symmetric polynomial $p=x^4+2x^3 y+4x^2 y^2+2 x y^3+y^4$ is Schur positive.
We have Schur polynomials $s_0 = 1$, $s_1 = x+y$, $s_2=x^2 + xy + y^2$, $s_3 = x^3 + x^2y + xy^2 + y^3$, $s_4 = x^4 + x^3y + x^2y^2 + xy^3 + y^4$.
Therefore 
$$
p=s_4 + xys_2 + 2x^2y^2. 
$$ 
Here the coefficients involve $x, y$. To express $p$ in terms of a linear combination of Schur polynomials, do we allow coefficients involve $x,y$? Is $p$ Schur positive? Thank you very much.
 [/INST] The Jacobi-Trudi identity gives
$$
s_{(3,1)}=\left|\begin{array}\mbox{s}_3&s_4\\s_0&s_1\end{array}\right|=x^3y+x^2y^2+xy^3
$$
and
$$
s_{(2,2)}=\left|\begin{array}\mbox{s}_2&s_3\\s_1&s_2\end{array}\right|=x^2y^2
$$
So $p= s_4+s_{(3,1)}+2s_{(2,2)}$.
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