华东师范大学学报(自然科学版) ›› 2013, Vol. 2013 ›› Issue (1): 91-103, 114.

• 应用数学与基础数学 • 上一篇    下一篇

加权的~Coxeter~群~$\widetilde{\bm C}_{\bm n}$~的左胞腔

黄 谦   

  1. 华东师范大学 数学系, 上海 200241
  • 收稿日期:2012-02-01 修回日期:2012-05-01 出版日期:2013-01-25 发布日期:2013-01-18

Left cells in the weighted Coxeter group ${\bm {\widetilde C}_{\bm n}}$

HUANG Qian   

  1. Department of Mathematics, East China Normal University,Shanghai 200241, China
  • Received:2012-02-01 Revised:2012-05-01 Online:2013-01-25 Published:2013-01-18

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摘要： 仿射~Weyl~群~($\widetilde{A}_{2n},\widetilde{S}$)
在某个群同构~$\alpha$~(其中~$\alpha(\widetilde{S}) =
\widetilde{S}$)~下的固定点集合
能被看作是仿射~Weyl~群~($\widetilde{C}_n,S$). 那么加权的~Coxeter~群\
($\widetilde{C}_n,\widetilde{\ell}$)的左和双边胞腔($\widetilde{\ell}$
是仿射~Weyl~群~$\widetilde{A}_{2n}$~的长度函数),
就能通过研究仿射~Weyl~群~($\widetilde{A}_{2n},\widetilde{S}$)
在群同构~$\alpha$~下的固定点集合而给出一个清晰的划分.
因此给出了加权的~Coxeter~群~($\widetilde{C}_n,\widetilde{\ell}$)
对应于划分\ $\textbf{k}\textbf{1}^{\textbf{2n+1-k}}$~和~$(2n-1,2)$
的所有左胞腔的清晰刻画, 这里对所有的~$1\leqslant k \leqslant 2n+1$.

关键词: 仿射~Weyl~群, 左胞腔, 拟分裂, 加权的~Coxeter~群

Abstract: The fixed point set of the affine Weyl group
($\widetilde{A}_{2n},\widetilde{S}$) under a certain group
automorphism $\alpha$ with $\alpha\,(\widetilde{S}) = \widetilde{S}$
can be considered as the affine Weyl group ($\widetilde{C}_n,S$).
Then the left and two-sided cells of the weighted Coxeter group
($\widetilde{C}_n,\widetilde{\ell}$), where $\widetilde{\ell}$ is
the length function of $\widetilde{A}_{2n}$, can be given an
explicit description  by studying the fixed point set of the affine
Weyl group ($\widetilde{A}_{2n},\widetilde{S}$) under $\alpha$. We
describe the cells of ($\widetilde{C}_n,\widetilde{\ell}$)
corresponding to the partitions
$\textbf{k}\textbf{1}^{\textbf{2n+1-k}}$ with $1\leqslant k
\leqslant 2n+1$ and $(2n-1,2)$.

Key words: affine Weyl groups, left cells, quasi-split case, weighted Coxeter group

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