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

Abstract

Let G G be a connected reductive algebraic group over an algebraically closed field of characteristic p > 0 p>0 , Δ ( λ ) \Delta (\lambda ) denote the Weyl module of G G of highest weight λ \lambda and ι λ , μ : Δ ( λ + μ ) → Δ ( λ ) ⊗ Δ ( μ ) \iota _{\lambda ,\mu }:\Delta (\lambda +\mu )\to \Delta (\lambda )\otimes \Delta (\mu ) be the canonical G G -morphism. We study the split condition for ι λ , μ \iota _{\lambda ,\mu } over Z ( p ) \mathbb {Z}_{(p)} , and apply this as an approach to compare the Jantzen filtrations of the Weyl modules Δ ( λ ) \Delta (\lambda ) and Δ ( λ + μ ) \Delta (\lambda +\mu ) . In the case when G G is of type A 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.