关于加权Coxeter群的胞腔理论的综述

doi:10.3969/j.issn.1000-5641.2023.06.001

摘要/Abstract

摘要：

介绍了在加权Coxeter群的胞腔理论方面所取得的成果, 详细描述了拟分裂情形下仿射Weyl群 $ \widetilde{C}_n $ 的胞腔分解, 简要描述了拟分裂情形下仿射Weyl群 $ \widetilde{B}_n $ 和一般情形下加权泛Coxeter群的胞腔分解¹.

关键词: 仿射Weyl群, 加权Coxeter群, 拟分裂情形, 胞腔, 划分, 表

Abstract:

We give a survey on the contribution of our research group to the cell theory of weighted Coxeter groups. We present some detailed account for the description of cells of the affine Weyl group $ \widetilde{C}_n $ in the quasi-split case and a brief account for that of the affine Weyl group $ \widetilde{B}_n $ in the quasi-split case and of the weighted universal Coxeter group in general case.

Key words: affine Weyl group, weighted Coxeter group, quasi-split case, cells, partitions, tabloids

中图分类号:

时俭益, 黄谦. 关于加权Coxeter群的胞腔理论的综述[J]. 华东师范大学学报（自然科学版）, 2023, 2023(6): 1-13.

Jianyi SHI, Qian HUANG. A survey on the cell theory of weighted Coxeter groups[J]. Journal of East China Normal University(Natural Science), 2023, 2023(6): 1-13.

图/表 8

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参考文献 34

1	KAZHDAN D, LUSZTIG G.. Representations of Coxeter groups and Hecke algebras. Inventiones Mathematicae, 1979, 53, 165- 184.
2	LUSZTIG G. Hecke algebras with unequal parameters [M]. Providence: American Mathematical Society, 2003.
3	GECK M.. On Iwahori-Hecke algebras with unequal parameters and Lusztig’s isomorphism theorem. Pure and Applied Mathematics Quarterly, 2011, 7 (3): 587- 620.
4	GUILHOT J, PARKINSON J.. A proof of Lusztig’s conjectures for affine type $G_2$ with arbitrary parameters . Proceedings of the London Mathematical Society, 2019, 118 (5): 1188- 1244.
5	GUILHOT J, PARKINSON J.. Balanced representations, the asymptotic Plancherel formula, and Lusztig’s conjectures for $C_2$ . Journal of Algebraic Combinatorics, 2019, 2 (5): 969- 1031.
6	BONNAFÉ C.. On Kazhdan-Lusztig cells in type B . Journal of Algebraic Combinatorics, 2010, 31, 53- 82.
7	BONNAFÉ C, GECK M, IANCU L, et al. On domino insertion and Kazhdan-Lusztig cells in type $B_n$ [C]// GYOTA A, NAKAJIMA H, SHINODA K, et al. Representation theory of algebraic groups and quantum groups. New York: Birkhäuser/Springer, 2010: 33-54.
8	LUSZTIG G. Left cells in Weyl groups [C]// HERB R, LIPSMAN R, ROSENBERG J. Lie Group Representation I. Berlin: Springer-Verlag, 1984: 99-111.
9	MI Q Q, SHI J Y.. Left cells of the weighted Coxeter group $(\widetilde{B}_n, \widetilde{\ell})$ . Communications in Algebra, 2015, 43 (4): 1487- 1508.
10	米倩倩, 时俭益.. 加权Coxeter群 $(\widetilde{B}_3, \widetilde{\ell})$ 的胞腔(英) . 华东师范大学学报(自然科学版), 2015, (1): 27- 41.
11	HUANG Q.. Some cells in the weighted Coxeter group $(\widetilde C_n, \widetilde{\ell}_{2n})$ . Journal of Algebra and Its Applications, 2018, 17 (8): 1850154.
12	HUANG Q, SHI J Y.. Some cells in the weighted Coxeter group $(\widetilde{C}_n, \widetilde{\ell}_{2n+1})$ . Journal of Algebra, 2013, 395, 63- 81.
13	SHI J Y.. The cells of the affine Weyl group $\widetilde{C}_n$ in a certain quasi-split case, Ⅱ . Journal of Algebra, 2014, 404, 31- 59.
14	SHI J Y.. The cells of the affine Weyl group $\widetilde{C}_n$ in a certain quasi-split case . Journal of Algebra, 2015, 422, 697- 729.
15	SHI J Y.. The cells in the weighted Coxeter group $(\widetilde{C}_n, \widetilde{\ell}_m)$ . Journal of Algebra, 2015, 443, 13- 32.
16	LUSZTIG G. Cells in affine Weyl groups [C]// HOTTA R. Algebraic Groups and Related Topics. Kinokunia North Holland: Advanced Studies in Pure Mathematics, 1985: 255-287.
17	LUSZTIG G.. Cells in affine Weyl groups, Ⅱ. Journal of Algebra, 1987, 109, 536- 548.
18	SHI J Y. The Kazhdan-Lusztig cells in certain affine Weyl groups [M]. Berlin: Springer-Verlag, 1986.
19	GREENE C.. Some partitions associated with a partially ordered set. Journal of Combinatorial Theory Series A, 1976, 20, 69- 79.
20	岳明仕.. 拟分裂情形下仿射Weyl群 $\widetilde{C}_4$ 的胞腔 . 华东师范大学学报(自然科学版), 2013, (1): 61- 75.
21	岳明仕. 拟分裂情形下仿射群${\widetilde C_n} $的胞腔(英) [J]. 华东师范大学学报(自然科学版), 2014(3): 77-92.
22	岳明仕. 带有权函数的Coxeter群${\widetilde C_n} $的胞腔(英) [J]. 华东师范大学学报(自然科学版), 2015(3): 38-46.
23	YUE M S. Left cells of weighted Coxeter group $({\widetilde C_n},{\widetilde l_{2n}}) $ [J]. Advances in Mathematics (China), 2015, 44: 505-518.
24	YUE M S. Left cells of the weighted Coxeter group ${\widetilde C_n} $ [J]. Advances in Mathematics (China), 2016, 45(1): 67-79.
25	岳明仕.. 加权Coxeter群 $(\widetilde{C}_3, \widetilde{l}_6)$ 的胞腔(英) . 华东师范大学学报(自然科学版), 2016, (1): 27- 38.
26	岳明仕. 某种拟分裂情形下加权Coxeter群$({\widetilde C_n},{\widetilde l_{2n}}) $的胞腔(英) [J]. 华东师范大学学报(自然科学版), 2016(4): 1-10.
27	ASAI T. Open problems in algebraic groups [C]// Problems from the conference on algebraic groups and representations. Katata, August 29-September 3, 1983.
28	SHI J Y.. Left-connectedness of some left cells in certain Coxeter groups of simply-laced type. Journal of Algebra, 2008, 319, 2410- 2433.
29	SHI J Y, YANG G.. The weighted universal Coxeter group and some related conjectures of Lusztig. Journal of Algebra, 2015, 441, 678- 694.
30	SHI J Y, YANG G. The boundedness of the weighted Coxeter group with complete graph [J]. Proceedings of The American Mathematical Society, 2016, 144(11): 4573-4581.
31	LI Y, SHI J Y.. The boundedness of a weighted Coxeter group with non-3-edge-labeling graph. Journal of Algebra and Its Applications, 2019, 18 (5): 1950085.
32	GAO J W, XIE X.. Conjectures P1-P15 for hyperbolic Coxeter groups of rank 3. Journal of Algebra, 2021, 567, 139- 195.
33	XIE X.. Conjectures P1-P15 for Coxeter groups with complete graph. Advances in Mathematics, 2021, 379, 107565.
34	SHI J Y, YANG G.. Kazhdan-Lusztig cells in some weighted Coxeter group. Science China Mathematics, 2018, 61 (2): 325- 352.

$\lambda$	${\bf{6}}$	$ {\bf{5}}{\bf{1}}$	${\bf{4}}{\bf{2}}$	${\bf{4}}{\bf{1}}^{\bf{2}}$	${\bf{3}}^{\bf{2}}$	${\bf{3}}{\bf{2}}{\bf{1}}$	${\bf{3}}{\bf{1}}^{\bf{3}}$	${\bf{2}}^{\bf{3}}$	${\bf{2}}^{\bf{2}}{\bf{1}}^{\bf{2}}$	${\bf{2}}{\bf{1}}^{\bf{4}}$	$ {\bf{1}}^{\bf{6}} $
$n(\lambda)$	48	24	24	12	12	6	6	8	5	2	1

$\lambda$	${\bf{7}}{\bf{1}}$	$ {\bf{5}}{\bf{3}}$	${\bf{5}}{\bf{2}}{\bf{1}}$	${\bf{5}}{\bf{1}}^{\bf{3}}$	${\bf{4}}{\bf{2}}{\bf{1}}^{\bf{2}}$	${\bf{3}}^{\bf{2}}{\bf{2}}$	${\bf{3}}^{\bf{2}}{\bf{1}}^{\bf{2}}$	${\bf{3}}{\bf{2}}^{\bf{2}}{\bf{1}}$	${\bf{3}}{\bf{2}}{\bf{1}}^{\bf{3}}$	${\bf{3}}{\bf{1}}^{\bf{5}}$	${\bf{2}}^{\bf{2}}{\bf{1}}^{\bf{4}}$	$ {\bf{1}}^{\bf{8}} $
$n(\lambda)$	48	24	12	24	6	8	12	6	2	6	2	1

$\lambda$	${\bf{7}}$	${\bf{6}}{\bf{1}} $	${\bf{5}}{\bf{2}}$	${\bf{5}}{\bf{1}}^{\bf{2}}$	${\bf{4}}{\bf{3}}$	${\bf{4}}{\bf{2}}{\bf{1}}$	${\bf{4}}{\bf{1}}^{\bf{3}}$	${\bf{3}}^{\bf{2}}{\bf{1}}$	${\bf{3}}{\bf{2}}^{\bf{2}}$	${\bf{3}}{\bf{2}}{\bf{1}}^{\bf{2}}$	${\bf{3}}{\bf{1}}^{\bf{4}}$	${\bf{2}}^{\bf{3}}{\bf{1}}$	${\bf{2}}^{\bf{2}}{\bf{1}}^{\bf{3}}$	$ {\bf{2}}{\bf{1}}^{\bf{5}} $
$n(\lambda)$	48	24	24	24	12	12	12	12	8	5	6	3	3	1

$k$	$m_{2,1}(n,k)$ , $n > 6$	$m_{2,1}(6,k)$	$m_{2,1}(5,k)$	$m_{2,1}(4,k)$	$m_{2,1}(3,k)$
6	$\binom{n+2}3+(n+1)^2$	127	45	30	18
5	$n^2+n+1$	$6^2+6+1$	47	10	3
4	$\binom{n+1}2+2n+1$	$\binom{6+1}2+2\cdot6+1$	$\binom{5+1}2+2\cdot5+1$	24	6
3	$2n+1$	$2\cdot6+1$	$2\cdot5+1$	$2\cdot4+1$	11
2	$n+1$	$6+1$	$5+1$	$4+1$	$3+1$
1	1	1	1	1	1

$k$	$m_{2,3}(n,k)$ , $n > 5$	$m_{2,3}(5,k)$	$m_{2,3}(4,k)$	$m_{2,3}(3,k)$
6	$\binom{n+1}3+n^2+1$	36	34	9
5	$\binom{n+1}2+1$	$\binom 62+1$	$\binom 52+1$	8
4	$\binom{n+2}2$	$\binom 72$	$\binom 62$	7
3	$n+1$	6	5	4
2	$n$	5	4	3
1	0	0	0	0