华东师范大学学报(自然科学版)

• 数学 •    下一篇

某种拟分裂情形下加权Coxeter群(\widetilde C_n,\tilde l_{2n})的胞腔 （英）

岳明仕   

  1. 临沂大学 物流学院, 山东 临沂 276000
  • 收稿日期:2015-05-27 出版日期:2016-07-25 发布日期:2016-09-29
  • 通讯作者: 岳明仕, 男, 讲师, 研究方向为 Heck 代数及表示理论. E-mail: lymsyue@gmail.com.
  • 基金资助:

    国家自然科学基金(11071073)

Cells of the weighted Coxeter group\\ $\textbf{(}\widetilde{\bm C}_{\bm n},\widetilde{\bm l}_{\textbf{2}\bm n}\textbf{)}$ in a certain quasi-split case

YUE Ming-shi   

  1. School of Logistics, Linyi University, Linyi Shandong 276000, China
  • Received:2015-05-27 Online:2016-07-25 Published:2016-09-29

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摘要：

仿射Weyl群$(\widetilde{C}_n,S)$可以看作仿射Weyl群$(\widetilde{A}_{2n},\widetilde{S})$ 在其某个满足$\alpha(\widetilde{S})=\widetilde{S}$的群自同构$\alpha$下的固定点集合. $\widetilde{A}_{2n}$上的长度函数$\widetilde{l}_{2n}$在$\widetilde{C}_{n}$上的限制可以看做$\widetilde{C}_{n}$上的权函数. 通过研究$(\widetilde{A}_{2n},\widetilde{S})$在$\alpha$下的固定点集合,本文刻画了加权Coxeter群$(\widetilde{C}_n,\widetilde{l}_{2n})$对应于划分$\bf{3^32^{n-4}}$的所有胞腔. 证明了文中左胞腔的左连通性,从而验证了Lusztig提出的一个猜想.

关键词: 仿射Weyl群, 加权Coxeter群, 左胞腔, 拟分裂情形, 整数n的划分

Abstract:

The fixed point set of the affine Weyl group $(\widetilde{A}_{2n},\widetilde{S})$ under its group automorphism $\alpha$ with $\alpha(\widetilde{S})=\widetilde{S}$ can be seen as the affine Weyl group $(\widetilde{C}_n,S)$. The restriction to $\widetilde{C}_{n}$ of the length function $\widetilde{l}_{2n}$ on $\widetilde{A}_{2n}$ can be seen as a weight function on $\widetilde{C}_{n}$. In the present paper, by studying the fixed point set of the affine Weyl group $(\widetilde{A}_{2n},\widetilde{S})$ under $\alpha$, we give the description for all the cells of the weighted Coxeter group $(\widetilde{C}_{n},\widetilde{l}_{2n})$ corresponding to the specific partition $\bf{3^32^{n-4}}$. We also prove that each left cell we considered in this paper is left-connected, verifying a conjecture of Lusztig in our case.

Key words: affine Weyl group, weighted Coxeter group , left cells , quasi-split case , partitions of n