Cells of the weighted Coxeter group

YUE Ming-shi

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

Cells of the weighted Coxeter group

YUE Ming-shi

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

Letαbe a group automorphism of the affine WeylgroupwithAffine Weylgroupcan be seen as the fixed point set of the affine Weylgroupunder its group automorphism α.The restriction toof the length functiononcan be seen as a weight function on.In this paper,we give the description for all the left and two-sided cells of the specific weighted Coxeter groupand p rove that each left cell inis left-connected.

affine Wey l group;weighted Coxeter group;quasi-sp lit case;partitions of n; left cells

0 Introduction

A weighted Coxeter group(W,L)is,by definition,a Coxeter group W together With a weight function L:on it.Suppose that(W,L)is in the quasi-sp lit case,W can beseen as a fixed point set of some Coxeter systemunder its group automorphism(denote byα)withα.Letbe the Coxeter generator set of the affine Weylgroupfor any i,j in［0,2n］(w rite s2n+1=s0) with〈j〉/=〈i±1〉.Letαbe the group automorphism offor any i∈［0,2n］.Affine Weylgroupcan be seen as a fixed point set ofunderα.We can see the restriction toof the length functiononasa weight function on.The cells of the weighted Coxeter groupshaVe been described in［2］,［3］,［4］.In this paper,we giVe the description for all the cells of the specific weighted Coxeter group.

1 Cell theories of a weigh ted Coxeter group

Some concepts and results concerning a weighted Coxeter group w ill be introduced in this section.All follow from Lusztig in［1］but Lemma 1.1 follows from Shi in［2］.

Let(W,S)be a Coxeter System and l be the length function on W.An expression w=s1s2···sr∈W With si∈S is called reduced if l(w)=r.A weight f unction on W is, by definition,a map L:With L(w)=L(s1)+L(s2)+···+L(sr)for any reduced expression w=s1s2···sr∈W and L(s)=L(t)for any conjugated s,t in S.A weighted Coxeter group(W,L)is a Coxeter group W together With a weight function L on it.In the case L=l,we call(W,L)in the split case.

Suppose thatαis a group automorphism of W withα(S)=S.Let Wαbe the set of all w∈W satisfyingα(w)=w.For anyα-orbit J on S,let wJbe the longest elem ent in the subgroup WJof W generated by J.Let Sαbe the set of all elements wJWith J ranging oVer allα-orbits on S.Then(Wα,Sα)can be seen as a Coxeter System and the restriction of l to Wαcan be seen as a weight function on Wα.The weighted Coxeter group(Wα,l)is called in the quasi-split case.

Denote by a the a-function on W defined by Lusztig.We have the following result.

Lemma 1.1(see［2,Lemma 1.7］)Suppose that W is either a finite or an affine Coxeter group and that(W,L)is either in the split case or in the quasi-split case.A non-empty subset E o?W is a union o? some two-sided cells o?W i? the ?ollowing conditions(a)-(b)hold:

(b)The set E with E={x|x-1∈E}is a union o?some le?t cells o?W.

2 Specific weighted Coxeter groups

Lemma 2.2 Let w=［a1,a2,···,an］andinwith i∈［0,n］.For any j∈［n］

(i)The Vertex set of the graph M(w)is M(w),each Vertex x∈M(w)With label R(x);

(ii)Let x,y in M(w).We draw a solid edge from x to y if x-1y∈S and x can be obtained from y by one or two right star operations.

A path in M(w)is,by definition,a sequence x0,x1,···,xrin M(w)With r＞0 such that xi-1and xiare joined by a solid edge for any i∈［r］.Let x,y∈x,y are called have the same generalized τ-invariants if for any path x=x0,x1,···,xrin M(x),there existsa corresponding path y=y0,y1,···,yrin M(y)With R(xi)=R(yi)for any i∈［r］and the condition still holds if the roles of x and y are interchanged.The graph M(w)can be used in proVing that two elements ofhave the different generalized τ-inVariants(see［2］).Let x,y inWe draw a dashed edge from x to y if M(x)∩M(y)=?and x-1y∈S.

For exam p le.In Fig.1,the Vertex［5,4,6］With label{t2,t3},the Vertex［6,4,5］With label {t1,t3}and the Vertex［8,4,5］With label{t0,t3}.The notiondenotes the set{t2,t3}and the notiondenotes two Vertices［5,4,6］and［6,4,5］are joined by a solid edge.There isa dashed edge joined the Vertices［6,4,5］and［8,4,5］since［6,4,5］=［8,4,5］t0and M(［8,4,5］)∩M(［6,4,5］)=?.We see that no two elements in Fig.2 have the same generalizedτ-inVariants.

Lemma 2.4(see［1,Lemma 16.14］)Let x,y in

3 Partial order≤w on［2n+1]determined by

Two technical tools from Shi in［2,Section 3］w ill be introduced in this section.One is used in proVing the left-connectedness of a left cell of(see Theorem 3.2).The other is used in checking whether two elements ofare in the same left cell(see Lemma 3.3).

Lemma 3.1(see［7,Lemma 3.2］)Fix w∈.

(i)Let i/=j in［2n］,then j?wk i?and only i?

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

(iii)i,k are w-uncom parable i?and on ly i?i/=k;

Theorem 3.2(see［7,3.3］)For any k∈,leti?〈k〉∈［n］,i?〈k〉∈［n+1,2n］andi?otherwise.Let?or any i∈and j∈*. Suppose that x∈and i∈satis?y(i)x-2n-1＞(j)x?or any j∈［i+1,i+a］with some a∈［2n］.Let x′=ti,ax,we have l(x′)=l(x)-l(ti,a)andψ(x)=ψ(x′).In addition,suppose that(i)x-2n-1＞(j)x?or any j∈［i+1,i+2n］.Let x′′=ti,2n+1,we have l(x′′)=l(x)-2n-1 andψ(x)=ψ(x′′).Moreover,?or any m∈Z,we have

4 The m ain resu lts

Tab.1 The numbers n(λ)for allλ∈Λ7

Theorem 4.1 w ill be proved in section 5 by case-by-case argument.LetΔ:={421,321, 3212}.For anyλ∈Λ7,the set Eλw ithλ/∈Δhas been described in［7-11］.We need only to consider the sets E421,E321and E3212.

Letλ∈Δ.We will find a subset Fλof Eλsuch that the set Fλhas a non-empty intersection With each left-connected component of Eλ(by Theorem 3.2 and Various left star operations) and that no two elements in Fλare in the same left cell of(by Lemma 2.4 and Lemma 3.3). Then by Lemma 2.5 and Lemma 3.3 we see that the set Fλcan be seen as a representatiVe set for the left cells ofin Eλ.Then the number n(λ)is just the cardinal of the set Fλ.We usually prove that the set Eλform s a single two-sided cell ofby proVing that the set Eλis two-sided-connected.Lemma 1.1 w ill be used in proVing that Eλis a union of two two-sided cells of.

5 The p roof of Theorem 4.1

Theorem 4.1 w ill be proved by case-by-case argument in the following part of this section (see Proposition 5.3,Proposition 5.6 and Proposition 5.10).

Case 1 The set E421

By Lemma 3.1 we see that for any w∈,w∈E421if and only if w satisfies one of the following conditions(a)-(c):

(a)There exist some pairwise not 6-dual i,j,k in［6］with i,j are both w-tame heads and k is w-wild head,satisfying

(b)There exist some pairwise not 6-dual i,j,k in［6］With i is w-tame head and j,k are both w-wild heads,satisfying 3＜(j)w＜7 and either

(c)There exist some pairwise not 6-dual i,j,k in［6］With i,j,k are all w-wild heads, satisfying:

(c1)j?wk but i,j,k is not a w-chain;

(c2)3＜(i)w＜7 and 3＜(j)w＜7;

Proposition 5.1 The set E421is infinite.

Fig.1 The right-connectedness of the set F421

Fig.2 The right-connectedness of the set F321

Fig.3 The right-connectedness of the set

So far we have proved all the assertions in Theorem 4.1.

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

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(責任編輯林磊)

岳明仕
(臨沂大學物流學院,山東臨沂276000)

取α是仿射W eyl群上某個滿足的群自同構.仿射Wey l群可以看做仿射Wey l群在其群自同構α下的固定點集合.上的長度函數(shù)在上的限制可以看做上的某個權函數(shù).本文給出了加權的Coxeter群中所有左胞腔以及雙邊胞腔的清晰刻畫并且證明中的每個左胞腔都是左連通的.

仿射Wey l群;加權Coxeter群;擬分裂情形;整數(shù)n的劃分;左胞腔

2014-12

國家自然科學基金(11071073)

岳明仕,男,講師,研究方向為Heck代數(shù)及表示理論.E-mail:lym syue@gmail.com.

O152 Document code:A

10.3969/j.issn.1000-5641.2016.01.004

1000-5641(2016)01-0027-12