Fully commutative elements in the Weyl and affine Weyl groups

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Journal of Algebra 284 (2005) 13–36

www.elsevier.com/locate/jalgebr

Fully commutative elementsin the Weyl and affine Weyl groups

Jian-yi Shi1

Department of Mathematics, East China Normal University, Shanghai 200062, PR ChinaCenter for Combinatorics, Nankai University, Tianjin, 300071, PR China

Received 6 April 2004

Available online 13 December 2004

Communicated by Gordon James

Abstract

LetW be a Weyl or an affine Weyl group and letWc be the set of fully commutative elements inW .We associate eachw ∈ Wc to a digraphG(w). By usingG(w), we give a graph-theoretic descriptiofor Lusztig’sa-function onWc and describe explicitly all thedistinguished involutions ofWc. Theresults verify two conjectures in our case: one was proposed by myself in [Adv. Sci. China M(1990) 79–98, Conjecture 8.10] and the other by Lusztig in [T. Asai et al., Open problems in alggroups, in: R. Hotta (Ed.), Problems from the conference on Algebraic Groups and RepresenKatata, August 29–September 3, 1983]. 2004 Elsevier Inc. All rights reserved.

Introduction

Let W = (W,S) be a Coxeter group withS the distinguished generator set. The fucommutative elementsw ∈ W were defined by Stembridge:

E-mail address:[email protected] Supported by Nankai University, the 973 Project of MST of China, the NSF of China, the SF of the Univers

Doctoral Program of ME of China, the Shanghai Priority Academic Discipline, and the CST of Shangh(no. 03JC14027).

0021-8693/$ – see front matter 2004 Elsevier Inc. All rights reserved.doi:10.1016/j.jalgebra.2004.08.036

14 J. Shi / Journal of Algebra 284 (2005) 13–36

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(i) w is fully commutative, if any two reduced expressions ofw can be transformed fromeach other by only applying the relationsst = ts with s, t ∈ S ando(st) = 2 (o(st) theorder ofst), or equivalently,

(ii) w is fully commutative, if w has no reduced expression of the formw = x(sts . . .)y,wherests . . . is a string of lengtho(st) > 2 for somes �= t in S.

The fully commutative elements were studied extensively by a number of peopl[3,8,10,22–24]). LetWc be the set of all fully commutative elements inW .

The present paper is only concerned with the case whereW is a Weyl or an affine Weygroup unless otherwise specified. In [19], we associated any Coxeter element of a Coxegroup to a directed graph (or a digraph in short). In the present paper, we extend thby associating eachw ∈ Wc to a digraphG(w). Then some techniques developed in [2can be applied here for our purpose. In particular, we use the digraphG(w) to define thenumbern(w), which equals the maximum possible cardinality for the node sets ofG(w)

satisfying condition (2.7.1) (see Lemma 2.7).Our first main result is to evaluate the functiona(w) on Wc by establishing the equa

tion a(w) = n(w) for anyw ∈ Wc (see Theorem 3.1). The functiona(w) was defined byLusztig in [12], which is important in the cell representation theory of the groupW andthe associated Hecke algebra. It is usually a difficult task to compute the valuea(w) for anarbitraryw ∈ W . In establishing the equationa(w) = n(w) for w ∈ Wc, we first show thathe setWc and the functionn(w) are invariant under star operations. It is well knownthe invariance of the functiona(w) under star operations. Then we reduce ourselvesspecial subsetFc of Wc. We explicitly describe all the elements ofFc in each case. Thewe show the equationa(w) = n(w) for anyw in Fc and hence inWc.

Lusztig defined distinguished involutions ofW which play an important role in the lecell representations ofW and the associated Hecke algebras (see [13]). However, efor the case of symmetric groups, it is usually very hard to recognize and to descridistinguished involutions among the elements ofW . We proposed a conjecture in [1Conjecture 8.10] to describe the distinguished involutions ofW , which is supported by athe existing data (see [16,21]). Our second main result is to give an explicit descrfor all the distinguished involutions ofW in the setWc, verifying the conjecture in oucase. Denote byD0(Wc) the set of these elements. In order to describe the elemenD0(Wc), we define a subsetF ′

c of Wc (see 3.10). Writew = wJ · y ∈ F ′c with J = L(w)

and somey ∈ W . Then we conclude thatd = y−1 · wJ · y is the unique element inD0(Wc)

with d ∼L w (see Theorem 4.3). By applying this result, we conclude that any leftL of W with L ∩ Wc �= ∅ contains a unique element (saywL) in F ′

c and that anyz ∈ L

has the formz = x · wL for somex ∈ W (see Corollary 4.11). This further implies thL is left connected (see Remark 4.12(3)), verifying a conjecture of Lusztig on thconnectedness of left cells ofW in our case (see [2]).

The contents of the paper are organized asfollows. We collect some notations, termand known results concerning cells of a Coxeter groupW in Section 1. In Section 2, wassociate eachw ∈ Wc to a digraphG(w) and deduce some results on the elements oWc

by usingG(w). Then two main results of the paper are shown in Sections 3–4, one insection.

J. Shi / Journal of Algebra 284 (2005) 13–36 15

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1. Some results on Coxeter groups

Let (W,S) be a Coxeter system. In the Introduction we defined the setWc of all thefully commutative elements ofW . In this section, we collect some notations, terms,known results for later use.

1.1. Let� be the Bruhat–Chevalley order and�(w) the length function onW . GivenJ ⊆ S,let wJ be the longest element in the subgroupWJ of W generated byJ , provided thatWJ

is finite. CallJ fully commutativeif the elementwJ is so.Forw,x, y ∈ W , we use the notationw = x ·y to meanw = xy and�(w) = �(x)+�(y).

In this case, we say thatw is a left (respectivelyright) extensionof y (respectivelyx), andsay thaty (respectivelyx) is a left (respectivelyright) retractionof w. More generally, wesayz is a retractionof w (or w is anextensionof z), if w = x · z · y for somex, y ∈ W .A retractionz of w is proper if �(z) < �(w).

Lemma. Letw = s1s2 . . . sr be a reduced expression ofw ∈ Wc with si ∈ S.

(1) The multi-set{s1, s2, . . . , sr } only depends onw but not on the choice of a reduceexpression.

(2) For any s ∈ S with sw ∈ Wc, the equationsw = ws holds if and only ifssi = sis forany1� i � r.

(3) If s, t ∈ S satisfysw = wt ∈ Wc, thens = t .(4) If w ∈ Wc, then any retraction ofw is also in Wc. In particular, if w ∈ Wc has an

expressionw = x · wJ · y with x, y ∈ W andJ ⊆ S, thenJ is fully commutative.

Proof. (1) and (2) (respectively (4)) follow by the definition (i) (respectively (ii)) of a fucommutative element (see Introduction). Then (3) is an easy consequence of (1).�1.2. Let �L (respectively�R, �LR) be the preorder onW defined as in [11], and let∼L

(respectively∼R, ∼LR) be the equivalence relation onW determined by�L (respectively�R, �LR). The corresponding equivalence classes are calledleft (respectivelyright, two-sided) cells of W . �L (respectively�R, �LR) induces a partial order on the set of le(respectively right, two-sided) cells ofW .

1.3. Lusztig defined a functiona : W → N ∪ {∞} for a Coxeter groupW in [12]. WhenW

is a Weyl or an affine Weyl group, Lusztig proved in [12,13] the following results:

(a) a(wJ ) = �(wJ ) for J ⊆ S with WJ finite (see [12, Proposition 2.4]) and [13, Propsition 1.2]). In particular, whenJ is fully commutative, we havea(wJ ) = |J |, thecardinality of the setJ .

(b) If x �LR y in W , thena(x) � a(y). Sox ∼LR y impliesa(x) = a(y), i.e., the functiona is constant on a two-sided cell ofW (see [12, Theorem 5.4]).

(c) If w = x · y, thenw �L y andw �R x. Hencea(w) � a(x), a(y).(d) If a(x) = a(y) andx �L y, thenx ∼L y (see [13, Corollary 1.9]).


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Note that (d) remains valid in any finite or affine Coxeter group (i.e., any finite Cogroup or any affine Weyl group) if conditiona(x) = a(y) is replaced byx ∼LR y (see [1,Corollary 3.3]).

1.4. Following Lusztig (see [13]), an elementw ∈ W is distinguished, if �(w) − 2δ(w) =a(w), whereδ(w) = degPe,w, e the identity element ofW and Px,y is the celebratedKazhdan–Lusztig polynomial associated to the ordered pair(x, y) in W . WhenW is a Weylor an affine Weyl group, Lusztig showed in [13, Proposition 1.4(a) and Theorem 1.10] ta distinguished elementw of W is always an involution (i.e.,w2 = e) and that any left celof W contains a unique distinguished involution.

1.5. For anyw ∈ W , letL(w) = {s ∈ S | sw < w} andR(w) = {s ∈ S | ws < w}.Assumem = o(st) > 2 for somes, t ∈ S. A sequence of elements

ys, yst, ysts, . . .︸︷︷︸m−1 terms

is called aright {s, t}-string (or just aright string) if y ∈ W satisfiesR(y) ∩ {s, t} = ∅.We say thatz is obtained fromw by a right {s, t}-star operation(or a right star op-

eration for brevity), if z,w are two neighboring terms in a right{s, t}-string. Note that aresulting elementz of a right{s, t}-star operation onw, when it exists, need not be uniqunlessw is a terminal term of the right{s, t}-string containing it.

Similarly, we can define aleft {s, t}-stringand aleft {s, t}-star operationon an elementThe following result follows directly from the definition of the relations∼L and∼R

onW , which is known from [11,12].

Lemma. If x, y ∈ W can be obtained from each other by successively applying left(re-spectively right) star operations, thenx ∼L y (respectivelyx ∼R y).

1.6. By the notationx—y in W , we mean that

max{degPx,y,degPy,x} = 1

2

(∣∣�(x) − �(y)∣∣ − 1

).

Two elementsx, y ∈ W forma (left) primitive pair, if there exist two sequences of elemex0 = x, x1, . . . , xr andy0 = y, y1, . . . , yr in W satisfying:

(a) xi—yi for all i, 0� i � r.(b) For everyi, 1� i � r, there exist somesi, ti ∈ S such thatxi−1, xi (and alsoyi−1, yi )

are two neighboring terms in some left{si , ti}-string.(c) EitherL(x) �L(y) andL(yr ) � L(xr), orL(y) � L(x) andL(xr) � L(yr) hold.

Lemma (see [16, Subsection 3.3]). If x, y ∈ W form a left primitive pair, thenx ∼L y.


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2. Digraphs associated to elements of Wc

In [19–21], we associated each generalized Coxeter element ofW to a digraph whichmade it possible to use graph theory in the study of generalized Coxeter elements. Ca generalized Coxeter element is fully commutative. In this section, we shall extendan idea to the setWc. Lemmas 2.6, 2.7, 2.9, and Corollary 2.8 are extensions of sresults of [21]. The proofs of these results can proceed by imitating those of thesponding results in [21] and so are omitted. An important property of the setWc is givenin Proposition 2.10, which asserts thatWc is invariant under star operations.

Let us start with some basic definitions of graph theory.

2.1. By a graph, we mean a finite set of nodes together with a finite set of edges. A gis always assumedsimple(i.e., no loop and no multi-edges). Two nodes of a graphadjacentif they are joined by an edge. In a graphG, the degreedG(v) of a nodev isthe number of edges incident onv; v is a branch nodeif dG(v) > 2, and aterminusifdG(v) � 1. A directed graph(or adigraphfor brevity) is a graph with each edge orientatA directed edge(i.e., an edge with orientation) with two incident nodesv,v′ is denotedby an ordered pair(v,v′), if the orientation is fromv to v′. A nodes of G is a source(respectively asink) if (s, s′) (respectively(s′, s)) is a directed edge ofG for any nodes′adjacent tos. An isolatednode is a node which is both a source and a sink. A sourcesink of G is also called anextreme node. A directed pathξ of a digraphG is a sequenceof nodesv0,v1, . . . ,vr in G with r � 0 such that(vi−1,vi ) is a directed edge ofG for1 � i � r. Call r the length ofξ . A path ξ is maximalif ξ is not properly contained inany other directed path ofG. A pathξ is adirected cycle, if v0 = vr . A digraph isacyclicif it contains no directed cycle. Asubdigraphof a digraphG is a digraph which can bobtained fromG by removing some nodes and all the directed edges incident toremoved nodes.

2.2. To an expression

χ : w = s1s2 . . . sr (2.2.1)

(not necessarily reduced) of anyw ∈ W with si ∈ S, we associate a digraphG(χ) as fol-lows. The node setV of G(χ) is {si | 1 � i � r} (note that thesi ’s are boldfaced), and thdirected edge setE of G(χ) consists of all the ordered pairs(si , sj ) satisfying the conditions i < j , sisj �= sj si and that there does not exist anyi = h0 < h1 < · · · < ht = j witht > 1 such thatshp−1shp �= shp shp−1 for every 1� p � t . The digraphG(χ) so obtainedusually depends on the choice of an expressionχ of w. However, if two expressions ofwcan be obtained from each other by only applying the relations of the formst = ts for somes, t ∈ S with o(st) = 2, then their corresponding digraphs should be the same. In partiwhenw is in Wc and an expressionχ of w in (2.2.1) is reduced, the digraphG(χ) onlydepends on the elementw, but not on the particular choice of a reduced expressionχ of w.In this case, it makes sense to denoteG(χ), V, E by G(w), V(w), E(w), respectively. CalG(w) theassociated digraphof w.


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By the above construction of a digraphG(w) for w ∈ Wc, there exists a natural maφ : si �→ si from V(w) to S and henceV(w) can be regarded as a multi-set inS.

Note that the above definition of the digraphG(w) can be regarded as a reformulatiof Viennot’s notion of a heap (see [25]).

2.3. Here and later, we always use the boldfaced letters, sayI,J,V, . . . (respectivelys, t,v, . . .) to denote node sets (respectively nodes) of a digraph and use the ordinaryI, J,V, . . . (respectivelys, t, v, . . .) to denote the corresponding multi-sets (respectivelements) inS. In the subsequent discussion of the paper, for a given expression ofw ∈ W ,we often first mention a multi-setI (respectively an elements) in S and then use the corresponding boldfaced letterI (respectivelys) to denote a node set (respectively a node) ofdigraphG(w) or the other way round; in such a case, the node setI (respectively the nods) is usually a certain specific one withφ(I) = I and|I| = |I | (respectivelyφ(s) = s), andnotφ−1(I) (respectivelyφ−1(s)) in general. This will be unambiguous from the contex

2.4. Consider the following conditions on an expressionχ of w ∈ W in (2.2.1):

(2.4.1) For any pairi < j with si = sj , there exists a directed path inG(χ) connecting thenodessi andsj .

(2.4.2) For any directed pathsi1, si2, . . . , sim in G(χ) with sih = sih+2 for 1 � h � m − 2andm = o(si1si2) > 2, there exists another directed path withsi1, sim two extremenodes.

The following result follows by a result of Stembridge (see [24, Proposition 3.3]).

2.5. Lemma. Let χ be an expression of somew ∈ W of the form(2.2.1). Thenχ satisfiesboth conditions(2.4.1)and (2.4.2)if and only if the elementw is in Wc with χ reduced.

The next two results can be proved by imitating those for [21, Lemmas 2.1 and 2

2.6. Lemma (comparing with [21, Lemma 2.1]). Let G be an acyclic orientation of agraphG. Then

(i) Each terminus ofG is an extreme node ofG.(ii) Each node ofG is contained in some maximal directed path ofG, which starts with a

source and ends with a sink.(iii) Let w ∈ Wc be withG(w) the associated digraph. ThenL(w) (respectivelyR(w))

(see1.5) is exactly the set of alls ∈ S with φ−1(s) containing a source(respectivelya sink) of G(w).

(iv) Keep the assumption of(iii) . Lets ∈L(w) (respectivelys ∈R(w)). ThenL(w)��L(sw)

(respectivelyR(w)��R(ws)) if and only if the removal of the source(respectively

sink) s from G(w) yields a new source(respectively sink) in the resulting digraph(see2.3).


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2.7. Lemma (comparing with [21, Lemma 2.2]). Givenw ∈ Wc with G(w) the associateddigraph. Then there is an expressionw = x · wJ · y for someJ ⊆ S andx, y ∈ W if andonly if there is a node setJ of G(w) with φ(J) = J such that

(2.7.1) for anys �= t in J, there is no directed path connectings andt in G(w).

For anyw ∈ Wc, denote bym(w) the maximum possible value of�(wJ ) in an expres-sionw = x · wJ · y, and denote byn(w) the maximum possible cardinality of a node seJof G(w) satisfying condition (2.7.1). Then Lemma 2.7 tells us the following.

2.8. Corollary (comparing with [21, Corollary 2.3]). m(w) = n(w) for anyw ∈ Wc.

By Corollary 2.8, we shall not distinguish the numbersm(w) andn(w) for anyw ∈ Wcand denoten(w) for both numbers.

The next result asserts that the numbern(w) remains unchanged under a star operaonw ∈ Wc, whose proof imitates that for [21, Lemma 2.4].

2.9. Lemma (comparing with [21, Lemma 2.4]). If w,y ∈ Wc can be obtained from eacother by a star operation, thenn(w) = n(y).

Finally, we show an important property ofWc involving star operations.

2.10. Proposition. The setWc is invariant under star operations.

Proof. Assume thaty ∈ W can be obtained from somew ∈ Wc by a left {s, t}-star oper-ation for somes, t ∈ S with st �= ts. We want to showy ∈ Wc. We may assumey = sw

for the sake of definiteness. The result follows by Lemma 1.1(4) ify < w. Now assumew < y. Let w = s1s2 . . . sr be a reduced expression ofw with si ∈ S and letG(w) bethe associated digraph ofw with V(w) = {si | 1 � i � r} the node set. LetG be the di-graph for the reduced expressiony = ss1s2 . . . sr with V = V(w)∪{s0} the node set, wherφ(s0) = s0 = s. If y is not fully commutative, then by Lemma 2.5, there exists a direpath, sayξ : si1, si2, . . . , sim , in G with m � 3 such thatsih = sih+2 for 1 � h � m − 2,o(si1si2) = m and that there does not exist any other directed path inG with si1, sim twoextreme nodes (hence for any nodes of G not in ξ , if s is adjacent to two nodessij , sik ofξ in G, then(s, sij ) is a directed edge ofG if and only if (s, sik ) is so). Sinces0 is a sourceof G andw ∈ Wc, we must haves0 = si1. So there is a reduced expression

y = p1p2 . . .pa( ss′ss′ . . .︸︷︷︸m factors

)q1q2 . . . qb, (2.10.1)

of y whose corresponding digraph is againG, wherepi, qj ∈ S, a + b + m = r + 1 ands′ = si2. We may assume thata is the smallest number with this property. Hence

p1p2 . . .pa( ss′ss′ . . .︸︷︷︸ ) ∈ Wc

k factors


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o

for anyk < m. Thenw has the reduced expression

w = p1p2 . . .pa( s′ss′s . . .︸︷︷︸m−1 factors

)q1q2 . . . qb. (2.10.2)

Sincem � 3, there exists at least one factors among them − 1 factors in the parenthesof the expression (2.10.2). We havephs = sph for any h by Lemma 1.1(2) and the facthat the leftmost factors in the parentheses of the expression (2.10.1) correspondssource of the digraphG. In particular, this impliest �= ph for anyh. Next we claim thatt = s′. Otherwise, the leftmost factort in the expression (2.10.2) should beqk for somek.Since there exists at least one factors in the parentheses of the expression (2.10.2),contradicts the fact thatqk is a source of the digraphG(w). So (2.10.2) becomes

w = p1p2 . . .pa( tsts . . .︸︷︷︸m−1 factors

)q1q2 . . . qb. (2.10.3)

Since the leftmost factort in the parentheses of (2.10.3) is a source ofG(w), we havepht = tph for anyh. Soy has the reduced expression

y = ( stst . . .︸︷︷︸m factors

)p1p2 · · ·paq1q2 · · ·qb, (2.10.4)

which is impossible sincey is obtained fromw by a left {s, t}-star operation. This showthatWc is invariant under left star operations. By the same argument, we can show thWcis invariant under right star operations. This proves our result.�

We are told that the conclusion of Proposition 2.10 was proved by Graham in the slaced case of finite Coxeter groups (see [9]). By Lemma 2.6(iv) and Proposition 2.1see that Lemma 2.9 implies that the numbern(w) is invariant under the star operationsan elementw in Wc.

3. The value a(w) for w ∈ Wc

Assume thatW is a Weyl or an affine Weyl group. Lusztig’sa-value is an important invariant for an element ofW (see 1.3). It is usually difficult to calculatea(z) for an arbitraryz ∈ W . However, the valuen(w) for w ∈ Wc can be computed easily. The main resultthe present section is Theorem 3.1, which equatesa(w) with n(w) for anyw ∈ Wc.

3.1. Theorem. WhenW is a Weyl or an affine Weyl group, we havea(w) = n(w) for anyw ∈ Wc.

By 1.3(a)–(c), the inequalitya(w) � n(w) holds forw ∈ Wc in general. We have tshow the equality holds. We need only show it in the case whereW is irreducible. So fromnow on, assume that we are in such a case.


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Note that the result in the simply-laced casesare known already (see [14, Theorems 1and 17.6] and [18, Theorem 3.1] for the cases ofAn and An, n � 1; and see [5, Theorem 4.1] for an arbitrary simply-laced case).

3.2. Let W beAn or An (n � 1). Then we have the following two results:

(i) An elementw ∈ W is in Wc if and only if w corresponds to a partition of the form2k1n−2k (i.e., a partition ofn with k parts equal to 2 andn − 2k parts equal to 1for some 0� k � n/2 under the map defined in [14, Definition 5.3] (see [14, Threms 17.4 and 17.6] and [18, Theorem 3.1]).

(ii) For anyw ∈ Wc, we havea(w) = k if and only if w corresponds to 2k1n−2k, whichholds if and only ifn(w) = k (see [18, Theorem 3.1] and [16, formula (6.27)]).

Then Theorem 3.1 follows in this case. So in the subsequent discussion, we aW �= An, An (hence the Coxeter graph ofW is a tree).

3.3. By the Cartier–Foata factorizationof w ∈ W , we mean the expression ofw of theform w = wJ1wJ2 . . .wJr , whereJi = L(wJi wJi+1 . . .wJr ) for any 1� i � r (see [7]).

Let Fc be the set of all the elementsw in Wc such thatL(sw) ⊂ L(w) (or equivalently,L(sw) = L(w) \ {s}) for any s ∈ L(w). Then the following result can be shown from tdefinition.

Lemma. LetW be a Weyl or an affine Weyl group.

(1) Anyw ∈ Wc can be transformed to some of its left retractions inFc by left star opera-tions.

(2) If w ∈ Fc, then any right retraction ofw is also inFc.Letw = wJ1wJ2 . . .wJr be the Cartier–Foata factorization ofw ∈ W .

(3) DenoteJ = J1 andI = J2. Then for anys ∈ I , there exist at least twot �= r in J suchthatst �= ts andrs �= sr. In particular, this implies thatI contains no terminal node othe Coxeter graph ofW .

(4) w is in Wc if and only if the following conditions hold:(i) if s ∈ Ji−1 ∩ Ji+1 for some1 < i < r, then there must exist either somet ∈ Ji

with o(st) > 3, or somet �= t ′ in Ji with o(st), o(st ′) > 2; in the former caseif o(st) = 4 and t ∈ Ji−2 (respectivelyt ∈ Ji+2), then there exists either soms′ ∈ Ji−1 \ {s} (respectivelys′ ∈ Ji+1 \ {s}), with o(s′t) > 2 or somet ′ ∈ Ji \ {t}with o(st ′) > 2;

(ii) if there exist some1 � i < r − 5 and s, t ∈ S with o(st) = 6 such thats ∈ Ji ∩Ji+2 ∩ Ji+4 and t ∈ Ji+1 ∩ Ji+3 ∩ Ji+5, then there must exist either somes′ ∈(Ji+2 ∪Ji+4)\ {s} with o(s′t) > 2 or somet ′ ∈ (Ji+1 ∪Ji+3)\ {t} with o(st ′) > 2.

Proof. By applying induction on�(w) � 0, (1) follows directly from the definition of thsetFc. Since no element ofI is in L(w), there exists, for anys ∈ I , at least onet ∈ J suchthatst �= ts. If, for somes ∈ I , t is the only element inJ satisfying the conditionst �= ts,then t ∈ L(w) \ L(tw) and s ∈ L(tw) \ L(w). Sow �→ tw is a left {s, t}-star operation


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r-

ylators

with tw < w, contradicting the assumption ofw ∈ Fc. Hence (3) follows. For (2), we neeonly show that ifw = x · s ∈ Fc ands ∈ S thenx ∈ Fc. We havex ∈ Wc by Lemma 1.1(4)Supposex /∈ Fc. Then there exists somet ∈ L(rx)\L(x) for somer ∈ L(x). SinceL(x) ⊆L(w) andL(rx) ⊆ L(rw), we haver ∈ L(w) andt ∈L(rw). By the assumption ofw ∈ Fc,we havet ∈ L(w). As t /∈ L(x), this impliesx · s = t · x by the exchange condition onW .This impliess = t by Lemma 1.1(3) and the fact thatw ∈ Wc. Moreover,s commuteswith and is not equal to any factorv ∈ S in a reduced expression ofx by the fact thats · x = x · s ∈ Wc and Lemma 1.1(2). This contradicts the fact thats = t ∈ L(rx). Hence(2) is shown. Finally, (4) follows directly by Lemma 2.5.�

For anyw ∈ Wc, there is somey ∈ Fc obtained fromw by left star operations bLemma 3.3(1). We havea(w) = a(y) andn(w) = n(y) by 1.3(b) and Lemmas 1.5, 2.So we need only consider the case ofw ∈ Fc (rather thanw ∈ Wc) in the proof of Theo-rem 3.1.

3.4. In 3.4 and 3.6–3.7, we always assume thatw ∈ Fc andI, J ⊆ S are as in Lemma 3.3(3We may assumeI �= ∅. For otherwise,w = wJ , the equationa(w) = n(w) clearly holds.ThenG(wJ wI ) is a subdigraph ofG(w) with the node setI ∪ J. Let

I ∪ J = K1 ∪ · · · ∪ Ku (3.4.1)

be a partition ofI ∪ J with WI∪J = WK1 × · · · × WKu (direct product), where eachWKi

is an irreducible standard parabolic subgroup ofW . Then eachKi is the node set of aconnected subgraphΓi of the Coxeter graphΓ of W . By Lemma 3.3, we see thatI ∩ Ki ,when it is nonempty (or equivalently|Ki | � 3), is fully commutative and contains no teminus ofΓi . Thus|I ∩ Ki | � 1

2|Ki | for anyi. In particular, whenI ∩ Ki �= ∅, the equation|I∩Ki | = 1

2|Ki | holds only when|Ki | is even and the underlying graph ofG(wKi ) is a cir-cle (i.e.,WKi = A|Ki |−1). The latter case never happens by our assumption onW (see 3.2).

3.5. In the subsequent discussion, the subscriptsfor the generators of the irreducible Weand affine Weyl groups are given as follows (following [4, pp. 250–275]) . The geners1, s2, . . . , s8 of E8 satisfy that

o(s1s3) = o(s3s4) = o(s2s4) = o(s4s5) = o(s5s6) = o(s6s7) = o(s7s8) = 3.

The groupsE6 andE7 can be regarded as the standard parabolic subgroups ofE8 generatedby {s1, . . . , s6} and{s1, . . . , s7} respectively. ThenEm is the extension ofEm with an addi-tional generators0 such thato(s0s2) = 3 if m = 6, o(s0s1) = 3 if m = 7, ando(s0s8) = 3 ifm = 8.

The generator setS = {s0, s1, . . . , sn} of the groupAn (respectivelyBn, Cn, Dn) satisfiesthato(sisi+1) = 3 for 0� i � n with the subscripts modulon+1 (respectivelyo(sisi+1) =o(s0s2) = 3 for 1 � i < n − 1 ando(sn−1sn) = 4; o(sisi+1) = 3 for 1 � i < n − 1 ando(s0s1) = o(sn−1sn) = 4; o(sisi+1) = o(s0s2) = o(sn−2sn) = 3 for 1� i < n − 1).


t

s

The generator setS = {s0, s1, s2, s3, s4} of F4 satisfies thato(s0s1) = o(s1s2) =o(s3s4) = 3 and o(s2s3) = 4. The generator setS = {s0, s1, s2} of G2 satisfies thao(s0s2) = 3 ando(s1s2) = 6.

Then a Weyl groupX ∈ {Ah,Bk,Cl,Dm,F4,G2 | h > 0, k > 2, l > 1, m > 3} canbe regarded as the standard parabolic subgroup of the affine Weyl groupX generated byS \ {s0}.3.6. In 3.6–3.7, we assume the digraphG(wJ wI ) to be connected withΓ ′ the underlyinggraph. First assume thatI ∪ J contains no branch node of the Coxeter graphΓ of W . Thenthe graphΓ ′ is a line, andI ∪ J = {s1, s2, . . . , st }, wheresisi+1 �= si+1si for 1 � i < t

by relabelling if necessary. By Lemma 3.3(3), we see thatt is odd, sayt = 2h + 1 forsomeh � 1, and thatJ = {s1, s3, . . . , s2h+1} andI = {s2, s4, . . . , s2h}. The directed edgeof G(wJ wI ) are(s2i±1, s2i ) for 1 � i � h.

(i) If o(s2i±1s2i ) = 3 for all 1 � i � h, then let z = s1s3 . . . s2h+1 · s2s4 . . . s2h ·s3s5 . . . s2h−1 · · · · · shsh+2 · sh+1 (here and later we express the elementsz, zk , etc., inthe form of Cartier–Foata factorizations, see 3.3).

(ii) If there exists exactly one pair (says, t) in I∪J, satisfyingo(st) = 4 and if one ofs, tis a terminus inΓ ′ (says2h+1 ∈ {s, t} for the sake of definiteness), thenWI∪J = B2h+1 forsomeh � 1. Letz be the elements1s3 . . . s2h+1 · s2s4 . . . s2h · s3s5 . . . s2h+1 · s4s6 . . . s2h · · · · ·s2h−1s2h+1 · s2h · s2h+1 if W ∈ {Bl, Bl, Cl} for somel � 2h + 1 (note that the subscriptsiof thesi ’s here and in (i) are not those described in 3.5),s2s4 · s3 · s2 · s1 or s1s3 · s2 · s3 · s4 ifW = F4, ands2s4 · s3 · s2 · s1 · s0 or s1s3 · s2 · s3 · s4 if W = F4 (here and later the subscriptsi

of thesi ’s are given as in 3.5).(iii) If there exists exactly one pair (says, t) in I ∪ J, satisfyingo(st) = 4 and none of

s, t is a terminus ofΓ ′, thenWI∪J = F4. Let z = s0s2s4 · s1s3 · s2 · s3 · s4.Then in any of the cases (i)–(iii), the elementw is a right retraction ofz (see 1.1) by

Lemmas 2.5, 3.3 and the assumption ofw ∈ Fc.(iv) Suppose that there exists exactly one pair (says, t) in I ∪ J with o(st) = 6. Then

WI∪J = G2. Let x = s0s1 · s2 · s1 · s2 and then letzk = xk for anyk � 1.(v) If there exist two pairs (say{s, t}, {s′, t′}) in I ∪ J with o(st) = o(s′t ′) = 4, then

WI∪J = C2h for someh � 1 and(s, t, s′, t′) = (s0, s1, s2h−1, s2h). Let x = s0s2 . . . s2h ·s1s3 . . . s2h−1 and then letzk = xk for k � 1.

Then in any of the cases (iv)–(v), the elementw is a right retraction ofzk with somek � 1 by Lemmas 2.5, 3.3 and the assumption ofw ∈ Fc.

3.7. Next assume thatI ∪ J contains a branch node, says, of the Coxeter graphΓ of W .If s is a terminus in the underlying graphΓ ′ of the digraphG(wJ wI ), then the situation isthe same as that in 3.6. Now assume that we are not in such a case.

(i) First assumes ∈ J. ThenW ∈ {Ei, Ei | i = 6,7,8} ands = s4. WhenW is E6, E7 orE7, let z = s1s4s6 · s3s5 · s4 · s2; whenW = E6, let z be one of the elements

s1s4s6 · s3s5 · s4 · s2 · s0, s1s4s0 · s3s2 · s4 · s5 · s6, s0s4s6 · s2s5 · s4 · s3 · s1, and

s0s1s4s6 · s2s3s5 · s4;whenW is E8 or E8, let z = s1s4s6s8 · s3s5s7 · s4s6 · s2s5 · s4 · s3 · s1.


(ii) Next assumes ∈ I. ThenW is Dn, Dn, Bm, Ei or Ei (n � 4, m � 3 andi = 6,7,8).WhenW = Dn with s = s2, sn−2 two branch nodes (hencen > 4), let x = s0s1 · s2 · s3 ·· · · · sn−2, y = sn−1sn · sn−2 · sn−3 · · · · · s2, then letzk = xyx . . . andz′

k = yxy . . . (k factorseach) fork � 1; whenn = 4, the branch nodes is s2, let xij = sisj · s2 for any i �= j in

{0,1,3,4} and letxij = xlm be with {i, j, l,m} = {0,1,3,4}, then letz(ij)

k = xij xij xij . . .

(k factors) for anyk � 1, let z(m) = sisj sl · s2 · sm for {i, j, l,m} = {0,1,3,4}, and letz0 = s0s1s3s4 · s2. WhenW = Bm is with s = s2 the branch node, letx = s0s1 · s2 · s3 · · · · ·sm−1 · sm · sm−1 · · · · · s2, then letzk = xk for k � 1; in particular, whenW = B3, we furtherlet y0 = s0s3 · s2 andy1 = s1s3 · s2, then letz′

k = y0y1y0 . . . andz′′k = y1y0y1 . . . (k factors

each) fork � 1. Also, letz = s0s1s3 · s2 · s3. WhenW is Ei or Ei , the branch nodes isalwayss4. Thenz0 = s2s3s5 · s4 is always inFc. Now we consider the other elements ofFcin Ei or Ei . WhenW = E6, let

z0 ∈ {s3s5 · s4 · s2 · s0, s2s5 · s4 · s3 · s1, s2s3 · s4 · s5 · s6}.

WhenW = E7, let

x = s2s5s7 · s4s6 · s3s5 · s1s4, y = s0s2s3 · s1s4 · s3s5 · s4s6,

then letzk = xyx . . . andz′k = yxy . . . (k factors each) fork � 1; let

w1 = s0s3s5s7 · s1s4s6 · u

with

u ∈ {s2s3s5 · s4, s2s3 · s4 · s5, s2s5 · s4 · s3, s3s5 · s4 · s2},

let

w2 = s0s2s3s5s7 · s1s4s6 · s3s5 · s4 · s2, w3 = s0s3s5 · s1s4 · s2s3 · s4 · s5 · s6 · s7 and

w4 = s3s5s7 · s4s6 · s2s5 · s4 · s3 · s1 · s0.

WhenW = E8, let

z1 = s3s2s5s7s0 · s4s6s8 · s5s7 · s6, z2 = s2s5s7s0 · s4s6s8 · s3s5s7 · s1s4s6 · u

with

u ∈ {s3s2s5 · s4, s3s5 · s4 · s2, s3s2 · s4 · s5, s2s5 · s4 · s3},z3 = s3s5s7s0 · s4s6s8 · s2s5s7 · s4s6 · s3s5 · s1s4 · s2s3 · s4 · s5 · s6 · s7 · s8 · s0,

z4 = s2s5s7 · s4s6 · s3s5 · s1s4 · s2s3 · s4 · s5 · s6 · s7 · s8 · s0, and

z5 = s2s3 · s4 · s5 · s6 · s7 · s8 · s0.


ytt

e

e

erving

Note thatEi is a standard parabolic subgroup ofEi for i = 6,7,8. We see that in anof the above affine Weyl groups and of the corresponding Weyl groups, an elemenw ofFc with I ∪ J containing a branch node ofΓ and withWI∪J irreducible must be a righretraction of somez, zk, z′

k , z′′k , z

(ij)k , z(m) or wk whenever it is applicable.

Lemmas 3.8 and 3.9 below can be obtained by the list of elements ofFc in 3.6–3.7.

3.8. Lemma. LetW be an irreducible Weyl or affine Weyl group. Letw ∈ Fc andI, J ⊆ S

be as in Lemma3.3 with I ∪ J containing no branch node of the Coxeter graphΓ of W .Then the set{s ∈ S | s � w} is contained inI ∪ J except for the case whereW ∈ {F4, F4}andw is a right extension ofs1s3 · s2 · s3 · s4 or s2s4 · s3 · s2 · s1. In this case(i.e., {s ∈ S |s � w} ⊆ I ∪ J ), let u be the number of parts in the partition(3.4.1)of I ∪ J , then thereexists a decompositionw = w1 ·w2 · · · · ·wu with wh ∈ WKh ∩Fc, where eachwh is a right

retraction of some suitablez, zk , z′k , z′′

k , z(ij)k , z(m) or wk.

3.9. Lemma. LetW be an irreducible Weyl or affine Weyl group. For anyw ∈ Fc in 3.6–3.7with WI∪J irreducible, we haven(sw) � n(w) = |L(w)| for anys ∈ L(w). More precisely,we haven(sw) < n(w) = |L(w)| for anys ∈ L(w), unlessw is a right extension of somelementw′ defined below:

(1) W = Dn. Whenn > 4, let u = s0s1 · s2 · s3 · · · · · sn−2 · sn−1sn, then letw′ ∈ {u,u−1};whenn = 4, let w′ ∈ {sisj · s2 · slsm | {i, j, l,m} = {0,1,3,4}}.

(2) W = Bm. Whenm > 3, let w′ = s0s1 · s2 · s3 · · · · · sm−1 · sm · sm−1 · · · · · s2 · s1s0; whenm = 3, let w′ ∈ {s0s1 · s2 · s3 · s2 · s0s1, s0s3 · s2 · s1s3, s1s3 · s2 · s0s3}.

(3) W = Cl for some evenl � 2. Letw′ = s0s2s4 . . . sl · s1s3 . . . sl−1 · s0s2s4 . . . sl .(4) W = E7. Letu = s2s5s7 · s4s6 · s3s5 · s1s4 · s0s3s2, then letw′ ∈ {u,u−1}.(5) W = G2. Letw′ = s0s1 · s2 · s1 · s2 · s1s0.

3.10. Let F ′c be the set of all the elementsw in Fc with n(sw) < n(w) for any s ∈ L(w).

Let F ′′c = Fc \ F ′

c. We record a simple fact onF ′′c for later use.

3.11. Lemma. Let W be an irreducible Weyl or affine Weyl group. Thens � w for anyw ∈ F ′′

c ands ∈ S.

3.12. In the above discussion onw ∈ Fc, we always assumeWI∪J irreducible. Now assumthatI ∪J is as in (3.4.1) withu > 1. For eachi, let Vi be the set of all the nodess in G(w)

such that there exists a directed path connectings with some node inKi . Let Gi be thesubdigraph ofG(w) with Vi its node set. ThenGi will be the associated digraph of somelementwi of Fc in one of the cases discussed in 3.5–3.9. By Lemma 3.8 and by obsethe cases ofW = F4, F4, we see that there exist some 1� i < j � u with Vi ∩Vj �= ∅ onlywhenKi ∪ Kj contains some branch node of the Coxeter graph ofW . By the definition ofthe setF ′′

c , we see thatw ∈ F ′′c if and only if there exist some 1� k � u with wk ∈ F ′′

c .Then we have the following lemma.

3.13. Lemma. LetW be an irreducible Weyl or affine Weyl group. Ifw ∈ F ′′c thenWI∪J is

irreducible, whereI ∪ J is determined byw as in Lemma3.3(3).


phy

oo

sy

Proof. Write w = wJ ·x with J = L(w) and somex ∈ Wc. Let I = L(x). ThenI ∪J has apartition (3.4.1) for someu � 1. We must showu = 1. Recall the notationsKi , Ki , Vi andwi (1 � i � u) in (3.4.1) and in 3.12. By 3.12, there exists some 1� k � u with wk ∈ F ′′

c .We may assumek = 1 by relabelling theKi ’s if necessary. By Lemma 3.9, we see thatw1 ∈F ′′

c only if W is Dn (n � 4), Bm (m � 3), Cl (evenl � 2), E7 or G2. Write w1 = wK · ywith K = L(w1) and somey ∈ Wc. Let H = L(y). ThenK ∪ H = K1 ⊆ I ∪ J . WhenW

is Cl or G2, we haveK ∪ H = S by Lemma 3.9(3), (5). This impliesK1 = I ∪ J , i.e.,u = 1. WhenW = E7, K1 is equal to either{s0, s1, s3, s4, s2} or {s2, s4, s5, s6, s7}. AssumeK1 = {s0, s1, s3, s4, s2}. Thenw1 is a right extension ofz = s0s3s2 · s1s4 · s3s5 · s4s6 · s2s5s7.Let V′ = V(w) \ V1 and letG′ be the subdigraph ofG(w) with V′ its node set. ThenG′ will be the associated digraph of some right retraction (writtenw′) of w. We havew = w′ · w1 by the construction ofV1. If u > 1 thenw′ �= e. Sincew = w′ · w1 ∈ Wc andsince the sourcess0, s3, s2 of the subdigraphG(w1) are also the sources of the digraG(w), the elementw′ contains no factorssi with 0 � i � 4 in its reduced expression bLemma 1.1(2). SoR(w′) ⊆ {s5, s6, s7}. It can be checked easily thatsj z /∈ Wc for anyj = 5,6,7. Sow′ �= e would imply w /∈ Wc, a contradiction. Hence we again getu = 1.Similarly for the case ofK1 = {s2, s4, s5, s6, s7}. The arguments for the remaining twcases (i.e.,W = Dn, Bm) are similar to that for the case ofW = E7 and hence are left tthe readers. �3.14. Now we consider the setF ′

c. For anyz = wK · z′ ∈ Wc with K = L(z) andz′ ∈ Wc,let n′(z) be the maximum possible cardinality for a node setV in the digraphG(z) whichsatisfies conditions (2.7.1) andV �= K (K being the set of sources in the digraphG(z),see 2.3). Clearly, the inequalityn′(z) � n(z) holds in general.

The following is concerned with the properties of the setF ′c.

Lemma. LetW be a Weyl or an affine Weyl group.

(1) The following statements on an elementw ∈ Wc are equivalent:(a) w ∈ F ′

c;(b) n(sw) < n(w) for anys ∈ L(w);(c) a(sw) < a(w) for anys ∈ L(w);(d) w <L sw for anys ∈L(w);(e) n′(w) < n(w);(f) n′(w) < |L(w)|.

(2) If w ∈ F ′c, then any right retraction ofw is also inF ′

c.

Proof. (1) Write w = wJ · x with J = L(w) and somex ∈ Wc.(b) ⇔ (c). This follows by Theorem 3.1.(c) ⇔ (d). We havew �L sw for anys ∈ L(w) in general. Hence the result is an ea

consequence of 1.3(b), (d).(b) ⇔ (e)⇔ (f). This can be shown by the facts thatn′(w) = max{n(sw) | s ∈ J } and

n(w) = max{n′(w), |J |}.(a)⇒ (b). This follows by the definition of the setF ′

c.


at ifph

t.

d

.9 (see

)

we

d

(b) ⇒ (a). Condition (b) ensures that ifV is a node set ofG(w) satisfying conditions(2.7.1) and|V| = n(w) thenV = J (J being the set of sources ofG(w), correspondingto J = L(w)). This implies that|L(sw)| < |L(w)| for anys ∈ L(w). In general, we haveL(sw) ⊇ L(w) \ {s} for any s ∈ L(w). Hence the inequality|L(sw)| < |L(w)| impliesL(sw) = L(w)\{s} ⊂ L(w) for anys ∈L(w). Sow ∈ Fc. Thusw is in F ′

c by condition (b)and by the definition of the setF ′

c.(2) According to the transitivity of taking right retraction, we need only show th

w = z · s ∈ F ′c ands ∈ S thenz ∈ F ′

c. Sincez ∈ Wc, we may consider the associated digraG(z). Clearly,n′(z) � n′(w) < |L(w)| by the equivalence of (a) and (f) in (1), andL(z) ⊆L(w). Again by the equivalence of (a) and (f) in (1), it suffices to shown′(z) < |L(z)|.The result is obvious in the case ofL(z) = L(w). Now assumeL(z) � L(w) = J . Thuss ∈ J (s being the node ofG(w) corresponding to the rightmost factors in the expressionw = z · s) andL(z) = L(w) \ {s} by the exchange condition onW and the fact ofw ∈ Wc.Hences commutes with anyt ∈ S satisfyingt � z by Lemma 1.1(2). This implies thathe nodes is contained in any maximal node set ofG(w) satisfying condition (2.7.1)So n′(w) − 1 is the maximum possible cardinality for a node setV of the digraphG(z)

(regarded as a subdigraph ofG(w)) satisfying conditions (2.7.1) andV �= J\{s}. Thereforen′(z) = n′(w) − 1 < |L(w)| − 1 = |L(z)|. This showsz ∈ F ′

c by the equivalence of (a) an(f) in (1). �

We have the following important properties for the elements inFc.

3.15. Lemma. LetW be a Weyl or an affine Weyl group.

(1) For any w ∈ F ′c, there exists a sequence of elementsx0 = w,x1, . . . , xr = wK in F ′

cwith K = L(w) such thatxi can be obtained fromxi−1 by a right star operation andxi < xi−1 for every1 � i � r. In particular, we haven(w) = |L(w)|.

(2) For anyw ∈ F ′′c , there exists somes ∈ L(w) such thatn(sw) = n(w) = |L(w)| and

that {w, sw} is a primitive pair(see 1.6).(3) For any w ∈ Wc, there exists somey ∈ F ′

c such thaty is a left retraction ofw withy ∼L w andn(y) = n(w).

Proof. For w ∈ Fc, write w = wJ · x with J = L(w) and somex ∈ Wc. Let I = L(x).We may assumeI �= ∅, for otherwise, the results are trivial. WhenWI∪J is irreducible,results (1)–(2) can be shown by a close observation of all the cases listed in 3.6–3Examples 3.19 for illustration). Lemma 3.13 tells us thatWI∪J is always irreducible forw ∈ F ′′

c wheneverW is irreducible. So (2) follows.Now we want to prove (1). Suppose thatw ∈ F ′

c and thatI ∪ J has a partition (3.4.1with u > 1. Keep the notationswi , Vi , 1 � i � u, in 3.12. If anywi , 1 � i � u, has theform wHi for someHi ⊆ S, then so does the elementw and hence the result is true. Noassume that there exists some 1� k � u with wk �= wH for anyH ⊆ S. We may assumk = 1 by relabelling theKi ’s in (3.4.1) if necessary. LetV′ = V(w) \ V1 and letG′ be thesubdigraph ofG(w) with the node setV′. Then it is easily seen thatG′ is the associatedigraph of some right retraction (writtenw′) of w. Moreover, we havew = w′ · w1. By thelast sentence of 3.12,w1 is in F ′

c. Write w1 = wK · y with K = L(w1) and somey ∈ Wc.


.e

o

g

e,

st

-er

le:

only(f) in4(2).

so-ed

Let H = L(y). ThenWK∪H is irreducible (noteK ∪ H = K1 in the notation of (3.4.1))Hencew1 can be transformed towK by a sequence of right staroperations according to thlist in 3.6–3.7 for the elements ofF ′

c in the irreducible case ofWI∪J . By Lemma 2.6(iv)and by the construction of the elementsw1, w′, this implies thatw can be transformed tw′′ = w′ · wK by the same sequence of right star operations aswK obtained fromw1. Wesee thatw′′ is a proper right retraction ofw. Hence by Lemma 3.14(2),w′′ is in F ′

c as sois w. Therefore, (1) follows by induction on�(w) − |L(w)| � 0.

For (3), if w /∈ Fc, then by the definition of the setFc, there exists somes ∈ L(w)

such thatw′ = sw can be obtained fromw by a left star operation. Clearly,w′ is a properleft retraction ofw with w′ ∼L w andn(w′) = n(w) by Lemmas 1.5 and 2.9. Applyininduction on�(w) � 0, we can show that there exists somey ∈ Fc such thaty is a leftretraction ofw with y ∼L w andn(y) = n(w). If y ∈ F ′

c then we are done. Otherwisby (2), there exists somes ∈ L(y) such that{y, sy} is a primitive pair andn(sy) = n(y).Hencey ′ = sy is a proper left retraction ofy with y ′ ∼L y by Lemma 1.6. Sincey ′ is in Wc,we can find a left retractiony1 of y ′ in Fc with y1 ∼L y ′ andn(y1) = n(y ′). Continue theprocess. Sincey1 is a proper left retraction ofy and since�(y) < ∞, such a process mustop after a finite number of steps. So we can eventually find a required element ofF ′

c. �3.16. Proof of Theorem 3.1. By Lemma 3.15(3) and 1.3(b), anyw ∈ Wc can be transformed to somey ∈ F ′

c with a(y) = a(w) andn(y) = n(w). Then by Lemma 3.15(1), whavey ∼R wJ with J = L(y), wherewJ is obtained fromy by a sequence of right staoperations. Thenn(y) = n(wJ ) = |J | by Lemma 2.9. Also,a(y) = a(wJ ) = |J | by 1.3(a),(b). This impliesa(y) = n(y) and hencea(w) = n(w). �3.17. Remark. The careful reader may suspect that the above arguments proceed in circ

Theorem 3.1 �⇒ Lemma 3.14(1) �⇒ Lemma 3.14(2)�⇒ Lemma 3.15(1) �⇒ Theorem 3.1.

Now we would like to explain that this is not the case. Theorem 3.1 is appliedin the proof for the equivalence between (c) and (d), but not between (a) andLemma 3.14(1); only the latter equivalence is applied in the proof of Lemma 3.1Thus the validity of Lemma 3.14(2)does not depend on Theorem 3.1.

3.18. Corollary. Let W be a Weyl or an affine Weyl group. Thena(w) = |L(w)| for anyw ∈ Fc.

Proof. Let w = wJ1 · · · · · wJr be the Cartier–Foata factorization ofw. The result is obvi-ous if r = 1. Now assumer > 1. If the groupWJ1∪J2 is irreducible then the result followby Theorem 3.1 and Lemma 3.9. WhenWJ1∪J2 is reducible, we can make the decompsition (3.4.1) withJ1 ∪ J2 in the place ofI ∪ J . Then our proof in this case can procesimilar to that for Lemma 3.15(1).�3.19. Example. (1) Let W = E8 and letw = s2s5s7s0 · s4s6s8 · s3s5s7 · s1s4s6 · s3s2s5 · s4.Thenw is in F ′

c with n(w) = |L(w)| = 4. The required right star operations onw are just


et

nce

e-dof

rs

to remove the factorss4, s3, s5, s2, s6, s4, s1, s7, s5, s3, s4, s6, s8 in turn on the right-sideof w. Then the resulting element isw0257= s2s5s7s0. So by 1.3(a) and Lemma 1.5, we ga(w) = a(w0257) = 4.

(2) Let W = E7 and letw = s2s5s7 · s4s6 · s3s5 · s1s4 · s0s3s2 · z (somez ∈ Wc) be aright retraction of the elementzk but not that ofzk−1 for somek � 2 (see 3.7 forzk in E7).Thenw ∈ F ′′

c . Takey = s2w. We haven(y) = n(w) = |L(w)| = 3. We claim that{w,y} isa primitive pair. Letw0 = w,w1, . . . ,w8 be such thatw1 = s4w, w2 = s6w1, w3 = s3w2,w4 = s5w3, w5 = s1w4, w6 = s4w5, w7 = s0w6, w8 = s3w7. Also, lety0 = y, y1, . . . , y8 besuch thaty1 = s5y, y2 = s7y1, y3 = s4y2, y4 = s6y3, y5 = s3y4, y6 = s5y5, y7 = s1y6, y8 =s4y7 = s0s3s2 · z. Then we see thatL(w0) = {s2, s5, s7} � {s5, s7} = L(y0), L(wi) = L(yi)

for 1 � i < 8, L(w8) = {s0, s3} � {s0, s3, s2} = L(y8) and thatwj is obtained fromwj−1by the same left star operation asyj from yj−1 for any 1� j � 8. This implies that{w,y}is a primitive pair and hencew ∼L y ∼L y8 by Lemmas 1.6 and 1.5. Soa(w) = a(y) =a(y8) andn(w) = n(y) = n(y8) = 3 by 1.3(a), (b) and Proposition 2.10. The elementy8 isin Fc with L(y8) = {s0, s3, s2} which is a right retraction ofz′

k−1 but not that ofz′k−2 (see

3.7 forz′k in E7). Applying induction onk � 1, we can eventually find somey ′ in F ′

c withL(y ′) ∈ {{s0, s3, s2}, {s2, s5, s7}} andw ∼L y ′ which is a right retraction ofz1 or z′

1. Theny ′ can be transformed tow257 or w023 by a sequence of right star operations and hea(y ′) = 3 = n(y ′). This impliesa(w) = a(y ′) = 3 by 1.3(a), (b) and Lemma 1.6.

4. Distinguished involutions in Wc

Again assume thatW is a Weyl or an affine Weyl group in this section. In [21], we dscribed all the distinguished involutions in the left cells ofW containing some generalizeCoxeter elements. In this section, we shall describe all the distinguished involutionsW

in the setWc. The main result is Theorem 4.3.

4.1. By Lemma 3.15, we see that for anyz ∈ Wc, there exists somew ∈ F ′c which is a left

retraction ofz and satisfiesw ∼L z. By Lemma 3.14, we also see that for anyw ∈ F ′c and

anys ∈L(w), the inequalitiesn(sw) < n(w) and hencea(sw) < a(w) hold, which impliesw <L sw again by Lemma 3.14. So we can say that anyw ∈ F ′

c is a minimal element (withrespect to left retraction) in the left cell ofW containing it.

4.2. For w ∈ F ′c, write w = wJ · x with J = L(w) and x ∈ Wc. Let I = L(x). By

Lemma 3.15(1), there is a reduced expressionx = s1s2 . . . sa with si ∈ S such that, letwk = wJ s1s2 . . . sk (0 � k � a), thenwk can be obtained fromwk−1 by a right{sk, rk}-staroperation for somerk ∈ S with skrk �= rksk . Given a reduced expressionwJ = t1t2 . . . tbwith tj ∈ S, let G be the digraph determined by the expression

d = x−1wJ x = sa . . . s2s1t1t2 . . . tbs1s2 . . . sa (4.2.1)

with s′a, . . . , s′

2, s′1, t1, t2, . . . , tb, s1, s2, . . . , sa the nodes corresponding to the facto

sa, . . . , s2, s1, t1, t2, . . . , tb, s1, s2, . . . , sa in (4.2.1) respectively (hence the node setJ of


t

ions.1).to

in

G corresponding toJ is {t1, t2, . . . , tb}, see 2.3). Two facts concerning the digraphsG(w)

andG can be seen easily:

(i) A node ofG(w) is adjacent to some node inJ if and only if it is in I (I being the seof sources in the subdigraphG(x) of G(w)).

(ii) G has no directed edge of the form(s′i , sj ) for anyi, j � 1.

Now we state the main result of the section.

4.3. Theorem. Assume thatW is a Weyl or an affine Weyl group. Letw = wJ · x ∈ F ′c be

as above. Then we have

(1) The elementd = x−1wJ x satisfies�(d) = �(wJ ) + 2�(x);(2) d ∈ Wc;(3) d ∼L w;(4) d is a distinguished involution ofW .

To show Theorem 4.3, we need first prove some lemmas.

4.4. Lemma. In the setup of4.2, let

dk = sk . . . s2s1t1t2 . . . tbs1s2 . . . sk (4.4.1)

for 0 � k � a with the convention thatd0 = t1t2 . . . tb = wJ .

(1) dk is an involution ofW in Wc for any0 � k � a.(2) The expression(4.4.1)of dk is reduced for any0 � k � a.

In particular, (1)–(2)hold ford = da .(3) For any1 � k � a, we havedk = sk · dk−1 · sk , which can be obtained fromdk−1 by a

left {sk, rk}-star operation followed by a right{sk, rk}-star operation.

Proof. That dk is an involution follows by noting in (4.4.1) thatwJ = t1t2 . . . tb is aninvolution. To showdk ∈ Wc, we first claim that the expression (4.4.1) satisfies condit(2.4.1) and (2.4.2) for any 0� k � a. First we show that (4.4.1) satisfies condition (2.4We know thatsk . . . s2s1t1t2 . . . tb andt1t2 . . . tbs1s2 . . . sk are reduced expressions. Thenshow the claim, we need only prove that for any nodess′

i , sj (1� i, j � k) of G with si = sj(keep the notations in 4.2), there exists a directed path ofG connectings′

i andsj . Let h

be the smallest number withsh = sj . Then there exists a directed pathξ ′ (respectivelyξ )of G connecting the nodess′

i , s′h (respectivelysh, sj ) (we allow a directed path to conta

only a single node; see 2.1). There also exists a directed pathζ : tl, spc , spc−1, . . . , sp0 = sh

of G(w) (and hence ofG) connecting the nodestl , sh for somel, c � 1 by Lemma 2.6(ii).Henceζ ′: s′

p0= s′

h, s′p1

, . . . , s′pc

, tl is a directed path ofG connecting the nodess′h, tl by

symmetry. Letλ be obtained by concatenating the directed pathsξ ′, ζ ′, ζ, ξ . Thenλ is adirected path ofG connecting the nodess′ , sj .
i


n

path

of

ts

t

ble

ely

g

Next show that (4.4.1) satisfies condition (2.4.2) for any 0� k � a. Suppose not. Theby Lemma 2.5, there exists some directed pathµ: v1,v2, . . . ,vm of G with vh = vh+2 for1 � h � m − 2 andm = o(v1v2) such that there does not exist any other directedof G connectingv1 andvm. Since bothsk . . . s2s1t1t2 . . . tb andt1t2 . . . tbs1s2 . . . sk are re-duced expressions of some elements ofWc, the directed pathµ is neither a subsequences′k, . . . , s′

2, s′1, t1, t2, . . . , tb, nor a subsequence oft1, t2, . . . , tb, s1, s2, . . . , sk . So by 4.2(ii),

there exists some 1< h < m such thatvh−1 = s′p , vh = tq , vh+1 = sr for somep,q, r � 1.

By 4.2(i) and the facts oftq ∈ J, sp = vh−1 = vh+1 = sr , we havep = r, sr ∈ I ands′r ∈ I′

(I′ being the set of sinks in the subdigraphG(x−1) of G). By Lemma 3.3(3), there exissomeq ′ �= q with tq ′sr �= sr tq ′ . Let µ′ be obtained fromµ by replacingtq by tq ′ . Thenµ′is another directed path ofG connectingv1 andvm, contradicting our assumption. Sodk isin Wc with (4.4.1) reduced by Lemma 2.5. We get (1) and (2).

In particular, we havedk = sk · dk−1 · sk for 1 � k � a. It is known that the elementwk

can be obtained fromwk−1 by a right{sk, rk}-star operation withsk ∈R(wk) for 1 � k � a

(see 4.2). By Lemma 2.6(iii) and the fact thatdk, dk−1 ∈ Wc, we haveR(wk) =R(dk) for0 � k � a by comparing the sinks in the digraphsG(wk) andG(dk). This implies thatR(dk) ∩ {sk, rk} = {sk} andR(dk−1) ∩ {sk, rk} = {rk}. So (3) follows by (1) and the facthatdk = sk · dk−1 · sk . �

By Lemma 4.4, we can use the notationG(dk) for any 0� k � a.For 1� k � a, letyk be the shortest element in the double coset〈sk, rk〉dk〈sk, rk〉, where

sk �= rk in S are given in 4.2, and〈sk, rk〉 is the subgroup ofW generated bysk, rk .

4.5. Lemma. Letyk be as above for1 � k � a.

(1) The elementyk is an involution inWc.(2) skyk �= ykrk .(3) There exists at least one, sayt , of sk, rk satisfyingtyk �= ykt .

Proof. (1) Sincedk is an involution, bothyk andy−1k are the shortest element in the dou

coset〈sk, rk〉dk〈sk, rk〉, which must be equal by [6, Proposition 2.7.3]. Soyk is an involu-tion. We know thatdk is in Wc and thatyk is a retraction ofdk. Henceyk is also inWc.

(2) Sincedk is an involution andR(dk)∩{sk, rk} �= ∅, at least one (sayz) of the elementssk ·yk andyk · rk is a retraction ofdk . Thenz is in Wc by the fact thatdk ∈ Wc. This impliesskyk �= ykrk by Lemma 1.1(3) and the fact ofsk �= rk .

(3) The elementdk has an expression (4.4.1). LetIk = L(s1s2 . . . sk). Denote byIk

(respectivelyI′k) the node set ofG(dk) corresponding to the set of sources (respectiv

sinks) of the subdigraphG(s1s2 . . . sk) (respectivelyG(sk . . . s2s1)). We can writedk =y · f1f2 . . . fc with fh = sk if h ≡ c (mod 2) andfh = rk if h ≡ c − 1 (mod 2), where1 < c < m = o(skrk), andy ∈ Wc satisfiesR(y) ∩ {sk, rk} = ∅. Then the correspondindirected pathξ : f1, f2, . . . , fc of the digraphG(dk) satisfies

(4.5.1) for any 1� h � c, there does not exist any nodev of G(dk) outsideξ with (fh,v)

a directed edge ofG(dk).


t

e

ing

h

e

de

ve

We claim thatξ is a subsequence oft1, t2, . . . , tb, s1, s2, . . . , sk (note thatξ contains at mosone node inJ = {t1, t2, . . . , tb}). For otherwise, there would exist some 1< h � c such thatfh−1 = s′

p , fh = tq for somep,q � 1 by 4.2(ii) (note that there is no sink ofG(dk) amongs′k, . . . , s′

2, s′1). Thenfh−1 ∈ I′

k and fh ∈ J by 4.2(i). By Lemma 3.3(3), there exists somnodetq ′ ∈ J with q ′ �= q such that(fh−1, tq ′) is a directed edge ofG(dk), contradictingcondition (4.5.1).

The elementy has an expressiony = gc′ . . . g2g1 · yk with c′ � c such thatgh = sk ifh ≡ c′ (mod 2) andgh = rk if h ≡ c′ −1 (mod 2). We can also show that the corresponddirected pathζ : gc′ , . . . ,g2,g1 of G(dk) is a subsequence ofs′

k, . . . , s′2, s′

1, t1, t2, . . . , tbby the same argument as above. This implies that among the nodesgc′ , . . . ,g2,g1, f1, f2,

. . . , fc, only two nodesf1,g1 could be possibly inJ.Recall the notationwk = wJ · s1s2 . . . sk in 4.2.First assumef1,g1 /∈ J. Then by symmetry for the factorss1, . . . , sk occurring in the

expression (4.4.1) ofdk, we see thatc = c′ � 1 and that for any 1� h � c, the equa-tion (fh,gh) = (sjh , s′

jh) holds for some 1� jh � k. Hencefh = gh for any h. In par-

ticular, f1 = g1. There exists a directed pathth, sm1, sm2, . . . , smp = f1 of the digraphG(wk) with p � 1 for some 1� h � b and some 1� m1 < m2 < · · · < mp � k. Thens′mp

= g1, . . . , s′m2

, s′m1

, th, sm1, sm2, . . . , smp = f1 is a directed path of the digraphG(dk)

by symmetry, wheres′mp−1

, . . . , s′m2

, s′m1

, th, sm1, sm2, . . . , smp−1 also form a directed patin G(yk). If p = 1, thenthf1 �= f1th. If p > 1, thensmp−1f1 �= f1smp−1. Clearly, we haveth � yk whenp = 1 andsmp−1 � yk whenp > 1. In either case, we havef1yk �= ykf1 bythe fact ofykf1 ∈ Wc and Lemma 1.1(2).

Next assumef1 ∈ J, sayf1 = th for some 1� h � b. Hencec > 1 by the fact thatfc =sk ∈ R(wk) andrk ∈ R(wksk). By condition (4.5.1) on the directed pathξ : f1, f2, . . . , fcin G(dk), the facts thatwk ∈ Wc and thatf1 is a source ofG(wk), we havef1v = vf1

for any nodev ∈ V(wk) \ {fl | 1 � l � c} by Lemma 1.1(2). This impliesf1yk = ykf1.Similarly, we can show that ifg1 ∈ J theng1yk = ykg1. We claim that we cannot havboth f1,g1 in J. Otherwise, we would have both equationsf1yk = ykf1 andg1yk = ykg1.Thusf1 �= g1 asykg1f1 = g1ykf1 = g1 · yk · f1. This implies{f1, g1} = {sk, rk} by thefact thatf1, g1 ∈ {sk, rk}. Hencef1g1 �= g1f1, contradicting condition (2.7.1) on the noset J. By the construction off1, g1, we see that if{f1,g1} ∩ J �= ∅ then we must havef1 ∈ J andg1 /∈ J. In this case, we havec = c′ + 1 and that for any 1� h � c′, we have(fh+1,gh) = (sjh , s′

jh) for some 1� jh � k by symmetry. On the other hand, we ha

f2 = sq ∈ I for some 1� q � k by 4.2(i). So there exists someh′ �= h with (th′ , f2) adirected edge of the digraphG(wk) by Lemma 3.3(3). This implies

(a) th′f2 �= f2th′ .

By the claim just shown, we have

(b) th′ � yk.

We know thatykf1f2 is a retraction ofdk and thatykf2 is a retraction off1ykf2 = ykf1f2.Soykf2 is a retraction ofdk. Sincedk ∈ Wc by Lemma 4.4(1), we get


s

e

4.2

ts

4.5n

t they

by

(c) ykf2 ∈ Wc.

Hencef2yk �= ykf2 by (a)–(c) and Lemma 1.1(2). So (3) follows.�Let D0 be the set of all the distinguished involutions ofW . We record a known result a

follows.

4.6. Lemma (see [17, Proposition 5.12(a), (b)]). Lety be an involution ofW and lets, t ∈ S

satisfyo(st) ∈ {3,4,6} ands, t /∈L(y). Define a subset ofD0 as follows:

G(y, s, t) = {z ∈ D0

∣∣ z ∈ 〈s, t〉y〈s, t〉, |L(z) ∩ {s, t}| = 1}.

SupposeG(y, s, t) �= ∅.

(a) If ry �= yr for r = s, t , and sy �= yt , thenG(y, s, t) is the set of all elements of thform zyz−1, wherez runs over one or two{s, t}-strings in〈s, t〉.

(b) Suppose thatry = yr for exactly oner ∈ {s, t}. Let

G1 = {zryz−1

∣∣ z ∈ 〈s, t〉, r /∈ R(z), and0 � �(z) < o(st) − 1}

and

G2 = {zyz−1

∣∣ z ∈ 〈s, t〉, r /∈ R(z), and 0< �(z) � o(st) − 1}.

ThenG(y, s, t) = G1, G2 or G1 ∪ G2.

Now we are ready to prove the main result of the section.

4.7. Proof of Theorem 4.3. (1) and (2) follow by Lemma 4.4. Keep the notations inand 4.4, in particular, the expressions ofd , wk , dk . Applying induction onk � 0, we wantto show, for any 0� k � a, the following two results:

(a) dk ∼L wk ;(b) dk is a distinguished involution ofW .

The results obviously hold whenk = 0. Now let 0< k � a. Suppose that the resulhave been shown for all the smallerk. Let yk be as in Lemma 4.5. Then bothdk−1 anddk are in the double coset〈sk, rk〉yk〈sk, rk〉 with dk−1 a distinguished involution ofW asassumed. Thusdk−1 ∈ G(yk, sk, rk) (see Lemma 4.6 for the notation). By Lemmasand 4.6, we conclude thatdk is in G(yk, sk, rk) and hence is a distinguished involutioof W , too. We havedk ∼LR dk−1 ∼L wk−1 ∼R wk , where the relationdk−1 ∼L wk−1 fol-lows by the inductive hypothesis, and the other two relations follow by the fact thaconcerned pair of elementscan be obtained from each other either by star operations or ba right star operation. Soa(dk) = a(wk) by 1.3(b). Sincewk is a left retraction ofdk, thisimpliesdk �L wk by 1.3(c). Thendk ∼L wk by 1.3(d). So the assertions (a)–(b) followinduction. In particular,d = da is a distinguished involution ofW with d = da ∼L wa = w.Our proof is completed. �


ed

ses

hat

ss

y.

4.8. Keep the notations in 4.2. Takew = wJ · x ∈ F ′c and the corresponding distinguish

involutiond = x−1 · wJ · x. ThenG(d) is symmetric with respect to the node setJ in thefollowing sense:

(i) for any 1� p < q � a, (sp, sq) is a directed edge ofG(d) if and only if (s′q, s′

p) is so;(ii) for any 1� p � a and 1� h � b, (th, sp) is a directed edge ofG(d) if and only if

(s′p, th) is so;

(iii) J satisfies condition (2.7.1).

We also have

(iv) neither (s′p, sq) nor (sq, s′

p) is a directed edge ofG(d) for any 1� p,q � a

(see 4.2(ii));(v) for any 1� p � a, there exists a directed pathtk, sm1, sm2, . . . , smr = sp in G(w) for

some 1� k � b and some 1� m1 < m2 < · · · < mr = p (see Lemma 2.6(ii)).

Given a nodeu of G(d), letVu be the set of all the nodesv of G(d) such that there exista directed pathξ in G(d) with u,v two extreme nodes, whereu could be either a sourcor a sink inξ . Let Gu be the subdigraph ofG(d) with Vu its node set. Then by condition(i)–(v) on G(d), we see that a nodeu of G(d) is in J if and only if Gu is symmetric withrespect tou (i.e.,Gu satisfies conditions (i)–(iii) above withu in the place ofJ). Hence thenode setJ is entirely determined by the digraphG(d). Then the subdigraphG(w) is alsodetermined byG(d): G(w) can be obtained fromG(d) by removing all such nodesv /∈ Jthat there exists some directed path ofG(d) from v to some node inJ. So we have

4.9. Lemma. ψ : wJ ·x �→ x−1 ·wJ ·x gives an injective map from the setF ′c to D0, where

J = L(wJ x).

4.10. Corollary. For w,y ∈ F ′c , we havew ∼L y if and only ifw = y.

Proof. The implication “⇐” is obvious. To show the other implication, we assume tw,y ∈ F ′

c satisfyw ∼L y. Write w = wJ · x andy = wI · z with J = L(w), I = L(y) andsomex, z ∈ Wc. Thend = x−1 · wJ · x andd ′ = z−1 · wI · z are distinguished involutionof W by Theorem 4.3. We haved ∼L w ∼L y ∼L d ′ again by Theorem 4.3. This implied = d ′ since each left cell ofW contains a unique distinguished involution ofW (see [13,Theorem 1.10]). Hencew = y by Lemma 4.9. �

By Lemma 3.15(3) and Corollary 4.10, it is immediate to get the following corollar

4.11. Corollary. If a left cell L of W satisfiesL ∩ Wc �= ∅, then the intersectionL ∩ F ′c

contains a unique element, saywL. Any elementz of L has the formz = x · wL for somex ∈ W .


cellt

t0],

of

ll[14–

on

re

on

-

ppl.

n Math.,

98.of

51

979)

d-

9,

4.12. Remark. (1) By Theorem 4.3, we can get the distinguished involution in any leftL of W , provided thatL contains some element ofWc. Then Corollary 4.11 tells us thathe setF ′

c forms a representative set for all these left cells ofW .(2) LetH be the Hecke algebra ofW overA = Z[q−1, q] with q an indeterminate. Le

{Tw | w ∈ W } be the standardA-basis ofH (in the sense of [11]). In [16, Conjecture 8.1we proposed a conjecture that any distinguished involutiond of W should have the formλ(x−1, x), wherex is a shortest element in the left cell ofW containingd , andλ(x−1, x)

is the unique maximal elementy (under the Bruhat–Chevalley order) withfy �= 0 in theproductTx−1Tx = ∑

z fzTz, fz ∈ A. By the description of the elementsλ(x, y) in [15,Proposition 2.3], we haveλ(w−1,w) = x−1 · wJ · x for any w = wJ · x ∈ F ′

c with J =L(w). So Theorem 4.3 verifies this conjecture for any distinguished involutions inWc.

(3) A subsetK of W is left connected, if for any x, y ∈ K, there exists a sequenceelementsx0 = x, x1, . . . , xr = y in K with somer � 0 such thatxi−1x

−1i ∈ S for every

1 � i � r. Lusztig conjectured in [2] that ifW is an affine Weyl group then any left ceL of W is left connected. The conjecture is supported by all the existing data (see16,21]). Now letW be a Weyl or an affine Weyl group. Assume thatL is a left cell ofWwith L∩Wc �= ∅. Then by Corollary 4.11, there exists a unique element, saywL, in L∩F ′

csuch that anyz ∈ L has the formz = x · wL for somex ∈ W . Take any reduced expressix = s1s2 . . . sr of x with si ∈ S. Denote bywi = sisi+1 . . . sr ·wL for 1 � i � r +1 with theconvention thatwr+1 = wL. Thenz = w1 �L w2 �L · · · �L wr+1 = wL ∼L z by 1.3(c).Hence all thewi ’s, 1� i � r + 1, are inL. SoL is left connected, verifying the conjectuin the case whereL contains some element ofWc.

References

[1] D. Alvis, The left cells of the Coxeter group of typeH4, J. Algebra 107 (1987) 160–168.[2] T. Asai, et al., Open problems in algebraic groups, in: R. Hotta (Ed.), Problems from the conference

Algebraic Groups and Representations, Katata, August 29–September 3, 1983.[3] S.C. Billey, G.S. Warrington, Kazhdan–Lusztig polynomials for 321-hexagon-avoiding permutations, J. Al

gebraic Combin. 13 (2001) 111–136.[4] N. Bourbaki, Groupes et algèbres de Lie, Chapters 4–6, Hermann, Paris, 1968.[5] K. Bremke, C.K. Fan, Comparison ofa-functions, J. Algebra 203 (1998) 355–360.[6] R.W. Carter, Finite Groups of Lie Type: Conjugacy Classes and Complex Characters, Wiley Ser. Pure A

Math., Wiley, London, 1985.[7] P. Cartier, D. Foata, Problèmes combinatoires de commutation et réarrangements, Lecture Notes i

vol. 85, Springer-Verlag, New York, 1969.[8] C.K. Fan, J.R. Stembridge, Nilpotent orbits and commutative elements, J. Algebra 196 (1997) 490–4[9] J.J. Graham, Modular representations of Hecke algebras and related algebras, PhD thesis, University

Sydney, 1995.[10] R.M. Green, J. Losonczy, Fully commutative Kazhdan–Lusztig cells, Ann. Inst. Fourier (Grenoble)

(2001) 1025–1045.[11] D. Kazhdan, G. Lusztig, Representations of Coxeter groups and Hecke algebras, Invent. Math. 53 (1

165–184.[12] G. Lusztig, Cells in affine Weyl groups, in: R. Hotta(Ed.), Algebraic Groups and Related Topics, in: A

vanced Stud. Pure Math., Kinokuniya and North-Holland, 1985, pp. 255–287.[13] G. Lusztig, Cells in affine Weyl groups, II, J. Algebra 109 (1987) 536–548.[14] J. Shi, The Kazhdan–Lusztig Cells in Certain Affine Weyl Groups, Lecture Notes in Math., vol. 117

Springer-Verlag, Berlin, 1986.


89.4)

oc.

II,

96)

ux

[15] J. Shi, A two-sided cell in an affine Weyl group, II, J. London Math. Soc. 37 (1988) 253–264.[16] J. Shi, A survey on the cell theory of affine Weyl groups, Adv. Sci. China Math. 3 (1990) 79–98.[17] J. Shi, The joint relations and the setD1 in certain crystallographic groups, Adv. Math. 81 (1) (1990) 66–[18] J. Shi, Some results relating two presentationsof certain affine Weyl groups, J. Algebra 163 (1) (199

235–257.[19] J. Shi, The enumeration of Coxeter elements, J. Algebraic Combin. 6 (2) (1997) 161–171.[20] J. Shi, Conjugacy relations on Coxeter elements, Adv. Math. 161 (1) (2001) 1–19.[21] J. Shi, Coxeter elements and Kazhdan–Lusztig cells, J. Algebra 250 (2002) 229–251.[22] J. Shi, Fully commutative elements and Kazhdan–Lusztig cells in the finite and affine Coxeter groups, Pr

Amer. Math. Soc. 131 (2003) 3371–3378.[23] J. Shi, Fully commutative elements and Kazhdan–Lusztig cells in the finite and affine Coxeter groups,

Proc. Amer. Math. Soc., in press.[24] J.R. Stembridge, On the fully commutative elements of Coxeter groups, J. Algebraic Combin. 5 (19

353–385.[25] G.X. Viennot, Heaps of pieces, I: basic definitionsand combinational lemmas, in: G. Labelle, P. Lero

(Eds.), Combinatoire Énumerative, Springer-Verlag, Berlin, 1986, pp. 321–350.

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Fully commutative elements in the Weyl and affine Weyl groups