论文标题
在良好特征的古典群体中某些不可还原表示中的nilpotent元素的约旦块
Jordan blocks of nilpotent elements in some irreducible representations of classical groups in good characteristic
论文作者
论文摘要
让$ g $是一个具有天然模块$ v $的古典群体，而lie代数$ \ mathfrak {g} $在代数封闭的字段$ k $的良好特征上。对于理性的不可约表示$ f：g \ rightarrow \ permatorAname {gl}（w）$作为$ v \ otimes v^*$，$ \ wedge^2（v）$的组合因子，以及$ s^2（v）$ \ Mathfrak {G} $。该描述是根据$ v \ otimes v^*$，$ \ wedge^2（v）$和$ s^2（v）$的Jordan块大小的$ e $的块大小给出的。我们的结果类似于早期工作（Proc。Amer。Math。Soc。，147（2019）4205-4219），我们考虑了这些相同的表示，并描述了G $中每个单位元素$ u \ $ f（u）$的Jordan正常形式。
Let $G$ be a classical group with natural module $V$ and Lie algebra $\mathfrak{g}$ over an algebraically closed field $K$ of good characteristic. For rational irreducible representations $f: G \rightarrow \operatorname{GL}(W)$ occurring as composition factors of $V \otimes V^*$, $\wedge^2(V)$, and $S^2(V)$, we describe the Jordan normal form of $\mathrm{d} f(e)$ for all nilpotent elements $e \in \mathfrak{g}$. The description is given in terms of the Jordan block sizes of the action of $e$ on $V \otimes V^*$, $\wedge^2(V)$, and $S^2(V)$, for which recursive formulae are known. Our results are in analogue to earlier work (Proc. Amer. Math. Soc., 147 (2019) 4205-4219), where we considered these same representations and described the Jordan normal form of $f(u)$ for every unipotent element $u \in G$.
