Jantzen filtration of Weyl modules, product of Young symmetrizers and denominator of Young’s seminormal basis

Let $G$ be a connected reductive algebraic group over an algebraically closed field of characteristic $p>0$, $\Delta (\lambda )$ denote the Weyl module of $G$ of highest weight $\lambda$ and $\iota _{\lambda ,\mu }:\Delta (\lambda +\mu )\to \Delta (\lambda )\otimes \Delta (\mu )$ be the canonical $G$-morphism. We study the split condition for $\iota _{\lambda ,\mu }$ over $\mathbb {Z}_{(p)}$, and apply this as an approach to compare the Jantzen filtrations of the Weyl modules $\Delta (\lambda )$ and $\Delta (\lambda +\mu )$. In the case when $G$ is of type $A$, we show that the split condition is closely related to the product of certain Young symmetrizers and, under some mild conditions, is further characterized by the denominator of a certain Young’s seminormal basis vector. We obtain explicit formulas for the split condition in some cases.