# 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.

## Key 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.

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Source: https://tomesphere.com/paper/1904.13040