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

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Jantzen filtration of Weyl modules, product of Young symmetrizers and denominator of Young’s seminormal basis
Representation Theory ( IF 0.7 ) Pub Date : 2020-10-29 , DOI: 10.1090/ert/553
Ming Fang , Kay Jin Lim , Kai Meng Tan

Abstract:Let

be a connected reductive algebraic group over an algebraically closed field of characteristic

, $\Delta (\lambda )$ denote the Weyl module of

of highest weight $\lambda$ and $\iota _{\lambda ,\mu }:\Delta (\lambda +\mu )\to \Delta (\lambda )\otimes \Delta (\mu )$ be the canonical

-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

is of type

, 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.

中文翻译：

Weyl组件的Jantzen过滤，Young对称子的乘积和Young半标准的分母

摘要：

在特征的代数封闭域上

，使之成为连通的还原代数群，表示权重最大的Weyl模，并且是典范态。我们研究了拆分条件了，并应用此作为一种方法来比较魏尔模块的詹特伦的过滤和。在类型为的情况下，我们表明分裂条件与某些Young对称器的乘积紧密相关，并且在某些温和条件下，其特征还在于某些Young半准基向量的分母。在某些情况下，我们获得了拆分条件的明确公式。 $\ Delta（\ lambda）$

$\ lambda$ $\ iota _ {\ lambda，\ mu}：\ Delta（\ lambda + \ mu）\ to \ Delta（\ lambda）\ otimes \ Delta（\ mu）$

$\ iota _ {\ lambda，\ mu}$ $\ mathbb {Z} _ {{p）}$ $\ Delta（\ lambda）$ $\ Delta（\ lambda + \ mu）$

更新日期：2020-10-30

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