Schur positivity of difference of products of derived Schur polynomials

Abstract

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

20 pages, 18 figures