A survey on the cell theory of weighted Coxeter groups

doi:10.3969/j.issn.1000-5641.2023.06.001

Abstract

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

CLC Number:

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.

Figures/Tables 8

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References 34

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$\lambda$	${\bf{6}}$	$ {\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