Schur positivity of difference of products of derived Schur polynomials

Abstract

To any Schur polynomial $s_λ$ one can associated its derived polynomials $s_λ{(i)}$ $i=0,\ldots,|λ|$ by the rule $$s_λ(x_1+t,\ldots,x_n+t) = \sum_i s_λ^{(i)}(x_1,\ldots,x_n) t^i.$$ We conjecture that $$(s_λ^{(i)})^2 - s_λ^{(i-1)} s_λ^{(i+1)}$$ is always Schur positive and prove this when $i=1$ for rectangles $λ= (k^\ell)$, for hooks $λ= (k, 1^{\ell -1})$, and when $λ= (k,k,1)$ or $λ= (3,2^{k-1})$.