华东师范大学学报（自然科学版） >

2016 , Vol. 2016 >Issue 4: 1 - 10

DOI: https://doi.org/10.3969/j.issn.1000-5641.2016.04.001

数学

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
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  • 临沂大学 物流学院, 山东 临沂 276000

收稿日期: 2015-05-27

  网络出版日期: 2016-09-29

基金资助

国家自然科学基金(11071073)

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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
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  • School of Logistics, Linyi University, Linyi Shandong 276000, China

Received date: 2015-05-27

  Online 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

本文引用格式

岳明仕 . 某种拟分裂情形下加权Coxeter群(\widetilde C_n,\tilde l_{2n})的胞腔 （英）[J]. 华东师范大学学报（自然科学版）, 2016 , 2016(4) : 1 -10 . DOI: 10.3969/j.issn.1000-5641.2016.04.001

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

参考文献

[ 1 ] LUSZTIG G. Hecke algebra with unequal parameters [M]. Providence: American Mathmatical Society, 2003.
[ 2 ] SHI J Y. The Kazhdan-Lusztig Cells in Certain Affine Weyl Groups [M]. Berlin: Springer-Verlag, 1986.
[ 3 ] HUANG Q. Left cells in the weighted Coxeter group e Cn [J]. J East China Norm Univ, 2013(1): 91-103.
[ 4 ] YUE M S. Cells of the affine Weyl group e Cn in quasi-split case [J]. J East China Norm Univ, 2014(3): 77-92.
[ 5 ] YUE M S. Cells of the weighted Coxeter group e Cn [J]. J East China Norm Univ, 2015(3): 38-46.
[ 6 ] YUE M S. Left cells of weighted Coxeter group ( e Cn,el2n) [J]. Advances in Mathematics (China), 2015, 44(4): 505-518.
[ 7 ] YUE M S. Left cells of the weighted Coxeter group e Cn [J]. Advances in Mathematics (China), 2016, 45(1): 67- 79.
[ 8 ] SHI J Y. The cells of the affine Weyl group e Cn in a certain quasi-split case [J]. J Algebra, 2015, 422: 697-729.
[ 9 ] GREENE C. Some partitions associated with a partially ordered set [J]. J Comb Theory(A), 1976, 20(1): 69-79.

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