YUE Ming-shi

(School of Logistics，Linyi University，Linyi Shandong 276000，China)



Article ID: 1000-5641(2016)04-0001-10

YUE Ming-shi

(School of Logistics，Linyi University，Linyi Shandong 276000，China)

affine Weyl group；weighted Coxeter group；left cells；quasi-split case；partitions of n

0　 Introduction

In this paper，we discuss some results about cells of the weighted Coxeter group(W，L)，which is，by definition，a Coxeter group W together with a weight function L on it. Lusztig in [1，Chapter 8]introduced the concepts of cells of the weighted Coxeter group(W，L). Each leftand two-sided cell of(W，L)provides a representation of the associated Iwahori-Hecke algebra H(W). Hence it is a big project to describe all the cells of any weighted Coxeter group(W，L).

收稿日期：2015-05
基金项目：国家自然科学基金(11071073)
作者简介：岳明仕，男，讲师，研究方向为Heck代数及表示理论. E-mail：lymsyue@gmail.com.

1　 Preliminaries

We introduce some results from Lusztig in [1]concerning the weighted Coxeter group.

2　 Two weighted Coxeter groups

3　 Partial order≺won the set[2n + 1]

Suppose that j /= k in [2n]are both w-wild heads and that i∈[2n]is w-tame head，then

(iv)i≺wk if and only if i，k are w-comparable；

(i)i1

(iii)There exists some l∈N∗such that(i)w -(j)w>l(2n + 1)for any i∈E1and j∈E2∪{2n + 1}.

(1)l(w′)= l(w)- l(ww′-1)；

(4)00；

(5)w′，w are in the same left-connected component of Eλwith ψ(w′)=ψ(w).

We can denote the generalized tabloid T determined by w∈Ω by T(w). The map T：Ω-→C2n+1is surjective by a result in [2].

4　　 The set

Theorem 4.1 will be proved in Proposition 4.2 and Propositions 4.10—4.12.

(a)There are some pairwise not 2n-dual i1，i2，···，it+m+h，k in [2n]with h∈N，t，m in N∗and t + m + h = n - 1，where k is a w-tame head，satisfying：

(a1)0

(b)There are some pairwise not 2n-dual i1，i2，···，it+m+h，k in [2n]with h∈N，t，m in N∗and t + m + h = n - 1，where k is a w-tame head，satisfying：

(b1)0

(b3)There exactly one of the following conditions holds：

(a′2)0<(it+m+h)w<···<(it+1)w<2n+1<(it)w<···<(i1)w<(it+m+h)w+2n+1 and n<(k)w<2n + 1；

Fig. 1 The matrix form of some

Figure 1 displays the corresponding part for the matrix form of some

the symbolp

(b′2)0<(it+m)w<···<(it+1)w<2n + 1<(it)w<···<(i1)w<(it+m+h)w<···<(it+m+1)w<(it+m)w + 2n + 1 and n<(k)w<2n + 1；

(b′3)There exactly one of the following conditions holds for some r∈[t]：

Fig. 2 The matrix form of some

Let w0= w′and w1= tn-atn-a+1···tn-2tn-1w0，

w2= tn···tn-a+2tn-a+1)······tntn-1tn-2tn-3tntn-1tn-2tntn-1tnw1，

w3= ta···tn-2tn-1······t3···tn-a+1tn-a+2t2···tn-atn-a+1t1···tn-a-1tn-aw2，

w2= tn···tn-h+2tn-h+1······tntn-1tn-2tn-3tntn-1tn-2tntn-1tnw1，

w3= th···tn-2tn-1······t3···tn-h+1tn-h+2t2···tn-htn-h+1t1···tn-h-1tn-hw2，w4= t0t1t0t2t1t0······th-3···t1t0th-2···t1t0th-1···t1t0w3.

w2=tn-(m+h)···tn-2tn-1······t3···tm+h+1tm+h+2t2···tm+htm+h+1t1···tm+h-1tm+hw1，w3=tn···tn-(m+h)+2tn-(m+h)+1······tntn-1tn-2tn-3tntn-1tn-2tntn-1tnw2，

w4=tn-1tn-2···tn-(m+h)+1tn-(m+h)w3.

Let α：=(n - 1，3，n - 1)∈eΛ2n+1and Cα：={T =(T1，T2，T3)|T∈C2n+1，ξ(T)=α}. We give the following result the generalized tabloids of all the elements in F332n-4.

We see from Lemma 4.7 that by setting T(w)= T(w)for any w∈F332n-4we can define a map T from the set F332n-4to the set Cα.

Lemma 4.8 Let T=(T1，T2，T3)∈Cα，T∈T(F332n-4)if and only if T satisfies the following condition(d)：

(d)There exist some pairwise not 2n-dual p1，q1，···，qn-1in [2n]with qn-1<···

Proof For any w∈F332n-4，let(p1，q1，q2，···，qn-1)=((k)w，(i1)w-2n-1，(i2)w-2n - 1，···，(in-1)w - 2n - 1)，we see from condition(c)in Lemmas 4.5 and 4.7 that the integers p1，q1，q2，···，qn-1satisfy condition(d)above. Conversely，suppose that T=(T1，T2，T3)∈Cαsatisfies condition(d). Let(k)w = p1and(ij)w = qj+ 2n + 1 for any j∈[n - 1]. The result follows from Lemmas 4.5 and 4.7.

We see from Lemma 4.8 that |T(F332n-4)| is the number of the choice for the integers p1，q1，q2，···，qn-1satisfying condition(d)above.

Lemma 4.9 No two elements in F332n-4have the same generalized tabloids.

Proof The result follows from Lemma 4.7 and the fact that〈(2n + 1 - i)w〉= 2n + 1 -〈(i)w〉for any i∈[2n]and w∈F332n-4. Let F332n-4= F3

332n-4. By Lemmas 2.5 and 4.6 we see that the set F332n-4contains the

By Lemma 4.9 we see that |F332n-4| = |T(F332n-4)|.

Proof The result follows from Lemmas 2.5，3.3，4.6 and 4.9.

Proof We consider the choice for the integers p1，q1，q2，···，qn-1satisfying condition(d)in Lemma 4.8. Clearly，the choice for the integer p1in [2n]with¯p1

Case I(ia)w -(ia+1)w = 1 for any a∈[n - 2].

Case II(ia)w -(ia+1)w>1 for some a∈[n - 2].

Acknowledgements The author thanks Professor J. Y. Shi of Department of Mathematics in East China Normal University for his hospitality and crucial guidance.

[References]

[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.

[9]GREENE C. Some partitions associated with a partially ordered set[J]. J Comb Theory(A)，1976，20(1)：69-79.

(责任编辑：林磊)

10.3969/j.issn.1000-5641.2016.04.001

岳明仕

(临沂大学物流学院，山东临沂276000)

仿射Weyl群；加权Coxeter群；左胞腔；拟分裂情形；整数n的划分

O152 Document code：A