On some extensions of $p$-restricted completely splittable $\mathrm{GL}(n)$-modules

In this paper, we calculate the space $\mathrm{Ext}_{\mathrm{GL}(n)}(L_n(\lambda),L_n(\mu))$, where $\mathrm{GL}(n)$ is the general linear group of degree $n$ over an algebraically closed field of positive characteristic, $L_n(\lambda)$ and $L_n(\mu)$ are rational irreducible $\mathrm{GL}(n)$-modules with highest weights $\lambda$ and $\mu$, respectively, the restriction of $L_n(\lambda)$ to any Levi subgroup of $\mathrm{GL}(n)$ is semisimple, $\lambda$ is a $p$-restricted weight, and $\mu$ does not strictly dominate $\lambda$.