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

TL;DR

This paper investigates the conditions under which certain morphisms between Weyl modules split over integers localized at a prime, connecting these conditions to Young symmetrizers and seminormal basis denominators, especially for type A groups.

Contribution

It introduces a new approach to compare Jantzen filtrations via split conditions and relates these to products of Young symmetrizers and seminormal basis denominators in type A.

Findings

Established criteria for split conditions over \\mathbb{Z}_{(p)}.

Linked split conditions to Young symmetrizer products.

Derived explicit formulas for specific cases.

Abstract

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.