The relation $ac{LR}$ on some elements of the affine Weyl group $widetilde{C}_n$
The relation $ac{LR}$ on some elements of the affine Weyl group $widetilde{C}_n$
复制标题
仿射 Weyl 群 $widetilde{C}_n$ 的某些元素上的关系 $ac{LR}$
DOI：
10.1016/j.jalgebra
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发表时间：
2017
影响因子：
0.9
通讯作者：
时俭益
中科院分区：
文献类型：
--
作者：
时俭益
设$(W,S)$为类型为$\widetilde{C}_n$的仿射Weyl群，$S$为其Coxeter发电机组。设$\overline{\Lambda}_{2n+1}$为$2n+1$的所有分区$\lambda=(\lambda_1,...,\lambda_r)$的集合，这样$\sum^{2k+1}_{j=1}\lambda_j$对于含有$2k+1\leqslant r$的任何$k\in\Bbb{N}$来说都是奇数。对于任意$J\subsetneq S$，设$w_J$为$J$生成的$W$的抛物子群中最长的元素。我们定义了一个映射$\overline{\phi}:\{w_J\mid J\subsetneq S\}\longrightarrow\overline{\Lambda}_{2n+1}$，并研究了集合$\{w_J\mid J\subsetneq S\}$上的预定顺序$\ac{LR}$及其关系。对于集$\{\overline{\phi}(w_J)\mid J\subsetneq S\}$上的偏序$\leqslant$，其中迭代星型操作和原语对起着重要作用。
Let $(W,S)$ be the affine Weyl group of type $\widetilde{C}_n$ with $S$ its Coxeter generator set. Let $\overline{\Lambda}_{2n+1}$ be the set of all partitions $\lambda=(\lambda_1,...,\lambda_r)$ of $2n+1$ such that $\sum^{2k+1}_{j=1}\lambda_j$ is odd for any $k\in\Bbb{N}$ with $2k+1\leqslant r$. For any $J\subsetneq S$, let $w_J$ be the longest element in the parabolic subgroup of $W$ generated by $J$. We define a map $\overline{\phi}:\{w_J\mid J\subsetneq S\}\longrightarrow\overline{\Lambda}_{2n+1}$ and study the preorder $\ac{LR}$ on the set $\{w_J\mid J\subsetneq S\}$ and its relation.with the partial order $\leqslant$ on the set $\{\overline{\phi}(w_J)\mid J\subsetneq S\}$, where iterating star operations and primitive pairs play an important role.