Permutation modules of the walled Brauer algebras

In this article, we study the permutation modules and Young modules of the group algebras of the direct product of symmetric groups $K{\mathfrak{S}}_{a,b}$, and the walled Brauer algebras ${\text{B}}_{r,t}(\delta )$. In the category of dual Specht-filtered modules, if the characteristic of the field is neither $2$ nor $3$, then the permutation modules are dual Specht filtered, and the Young modules are relative projective cover of the dual Specht modules. We prove that the restriction of the cell modules of ${\text{B}}_{r,t}(\delta )$ to the group algebras of the direct product of the symmetric groups is dual Specht filtered, and the Young modules act as the relative projective cover of the cell modules of ${\text{B}}_{r,t}(\delta )$. Finally, we prove that if $\mathrm{c}\mathrm{h}\mathrm{a}\mathrm{r}K\mathrm{\ne }2,3$, then the permutation module of ${\text{B}}_{r,t}(\delta )$ can be written as a direct sum of indecomposable Young modules.