On integral $\mathrm{E}\mathrm{x}{\mathrm{t}}^2$ between certain Weyl modules of $\mathrm{G}\mathrm{L}\mathrm{n}$

Consider partitions of the form $\lambda =(a,1^b)$ and $\mu =(a+1,b-1)$,\ where $a+1>b-1$. In this paper, we determine the extension groups ${\mathrm{E}\mathrm{x}\mathrm{t}}_A^2(K_{\lambda }F,K_{\mu }F)$, where $F$ is a free $\mathbb{Z}-$module of finite rank $n$, $K_{\lambda }F$ and $K_{\mu }F$ are the Weyl modules of the general linear group $G{L}_n(\mathbb{Z})$ corresponding to $\lambda$ and $\mu$, respectively, $A=S_{\mathbb{Z}}(n,r)$ is the integral Schur algebra and $r=a+b$.