Ž .Journal of Algebra 228, 107]118 2000doi:10.1006rjabr.1999.8276, available online at http:rrwww.idealibrary.com on

T-Increasing Paths on the Bruhat Graph of Affine WeylGroups are Self-Avoiding

Paola Cellini

Department of Mathematics, Uni ersity of Padua, Via Belzoni 7, Padua 35131, Italy

Communicated by Corrado de Concini

Received July 7, 1999

Ž . � y1 4Let W, S be a Coxeter system, T s wsw ¬ s g S, w g W its set of reflec-tions, $ any total reflection order, and G the undirected Bruhat graph. We

� 4consider the natural labeling of the edges of G which assigns to the edge ¨ , w thereflection ¨wy1 . A path on G, i.e., a sequence ¨ , . . . , ¨ such that ¨ ¨y1 g T for1 k i iq1i s 1, . . . , k y 1, is called T-increasing if ¨ ¨y1 $ ??? $ ¨ ¨y1 . T-increasing1 2 ky1 kpaths play an important role in the computation of both the Kazhdan]Lusztig andthe R-polynomials of W. We prove that if W is finite or is an affine Weyl group,then any T-increasing path is self-avoiding, i.e., it has no self-intersection points.Q 2000 Academic Press

INTRODUCTION

Ž . � y1Let W, S be a Coxeter system, ll its length function, and T s wsw ¬4s g S, w g W its set of reflections. We denote by - the Bruhat order on

W. We recall that - is the partial order defined as follows: for each¨ , w g W, ¨ - w if and only if there exist t , . . . , t g T such that w s t1 k k

Ž . Ž . Ž .??? t ¨ and ll ¨ - ll t ??? t ¨ - ll t ??? t ¨ for 1 F i - k. Moreover,1 i 1 iq1 1we recall that ¨ - w if and only if for some reduced expression w s s ??? s1 kthere exist 1 F i - ??? - i F k such that ¨ is obtained from s ??? s by1 h 1 kremoving the generators s , . . . , s . The undirected Bruhat graph of W isi i1 h

W y1�� 4 Ž . 4the graph G having W as the set of vertices and ¨ , w g ¬ ¨w g T2

WŽ .as the set of edges, where is the set of all subsets of W of cardinality 2.2

� 4 y1We label the edge ¨ , w with the reflection ¨w . A path in G is asequence ¨ , . . . , ¨ g W, such that ¨ ¨y1 g T for 1 F i F k y 1.1 k i iq1

Žw xLet us fix any total reflection order $ on T D ; see Section 2 for the.definition . Then we say that a path ¨ , . . . , ¨ in G is label-increasing, or1 k

1070021-8693r00 $35.00

Copyright Q 2000 by Academic PressAll rights of reproduction in any form reserved.

PAOLA CELLINI108

T-increasing, if ¨ ¨y1 $ ??? $ ¨ ¨y1. T-increasing chains, i.e., paths1 2 ky1 k¨ , . . . , ¨ with ¨ - ??? - ¨ , play a crucial role in Dyer’s formula for the1 k 1 k

w xKazhdan]Lusztig R-polynomials D and in Brenti’s formulas for thew xKazhdan]Lusztig polynomials B1, B2 . From these formulas it also follows

Ž .that if ¨ - w ¨ , w g W , then there exists exactly one T-increasingw xmaximal chain from ¨ to w D . The same result has recently been proved,

w xand generalized, in a different context in BFP .In this paper we consider T-increasing paths on the undirected Bruhat

graph. We study whether a T-increasing path may have self-intersectionpoints. It is very easy to see that this cannot happen for finite Coxetergroups, as is proved in Lemma 15 and Proposition 16. The main result ofthis paper is that this also cannot happen for affine Weyl groups. Webriefly state this as follows.

THEOREM 1. If W is finite or an affine Weyl group, then any T-increasingpath on G is self-a¨oiding.

The proof of Theorem 1 for affine Weyl groups relies on some refine-w xment of the results of CP on total reflection orders. The main auxiliary

result is the following theorem, which is interesting in itself. We denote by$U the opposite order of $ , defined by x $U y if and only if y $ x. For

� Ž . Ž .4w g W set T s t g T ¬ ll tw - ll t ; it is well known that any reducedwexpression of w induces a total order on T . Then we have the followingwresult.

THEOREM 2. Assume that W is an affine Weyl group and let A : T befinite. Then there exist ¨ , ¨ g W such that A : T j T , and T l T s1 2 ¨ ¨ ¨ ¨1 2 1 2

B; moreo¨er, there exist a reduced expression of ¨ which induces $ on1A l T and a reduced expression of ¨ which induces $U on A l T .¨ 2 ¨1 2

Theorem 2 is likely to be false for general Coxeter groups.The paper is organized as follows. In Section 1 we give definitions and

preliminaries on total reflection orders and affine root systems. In Section2 we prove Theorem 2 and in Section 3 we prove Theorem 1. The proof forthe finite case is given in Section 3, but it is direct and independent of theprevious sections.

1. NOTATION AND PRELIMINARIES

Ž .Let W, S be a Coxeter system with root system F and positive rootsystem Fq.

w x qDEFINITION 1 D . A total order $ on F is a total reflection order ifand only if for each c, d g Rq and a , b g Fq such that ca q db g Fand a $ b we have a $ ca q db $ b.

T-INCREASING PATHS ON THE BRUHAT GRAPH 109

Any order relation $ on Fq induces an order relation on T throughthe natural bijection between Fq and T. We shall use the same symbol $for denoting the order induced on T.

Ž . � q y1Ž . 4For each w g W the negative set of w is N w s a g F ¬ w a - 0 .Ž . Ž . Ž .It is well known that a g N w if and only if ll s w - ll w and thata

< Ž . < Ž . w x � Ž .4N w s ll w Hu 5.6 . Thus we have T s s ¬ a g N w . Moreover, ifw a

w s s ??? s is any reduced expression of w, and t s s ??? s s s ???1 k i 1 iy1 i iy1Ž . � 4 "Ž . Ž .s , for 1 F i F k t s s , then T s t , . . . , t . Let N w s N w j1 1 1 w 1 k

Ž .yN w . Then it is easy to see that for each ¨ , w g W we have

N " ¨w s N " ¨ D ¨ N " w ,Ž . Ž . Ž .where D denotes the symmetric difference. As a direct consequence weobtain the following results.

LEMMA 1. For any ¨ , w g W the following are equi alent:

Ž . Ž . Ž . Ž .1 ll ¨w s ll ¨ q ll w .q qŽ . Ž . Ž . Ž .2 N ¨w s N ¨ j ¨N w , where j denotes the union of disjoint

sets.Ž . Ž . Ž .3 N ¨ : N ¨w .Ž . Ž . q4 ¨N w : F .Ž . Ž y1 . Ž .5 N ¨ l N w s B.

DEFINITION 2. Let w g W, w s s ??? s be a reduced expression of w,1 kand t s s ??? s s s ??? s , for 1 F i F k. The total order t $ ??? $ ti 1 iy1 i iy1 1 1 k

Ž .on T and the corresponding order on N w through the natural bijectionware called the orders induced by the reduced expression s ??? s .1 k

For finite Coxeter groups, the total reflection orders are completelydetermined as follows.

w x qPROPOSITION 2 D . Let F be finite and let $ be a total order on F .Then $ is a total reflection order if and only if it is induced by some reducedexpression of the longest element of W.

Henceforth we restrict ourselves to affine root systems and their Weylw xgroups. We first recall the basic definitions; we follow K, Chap. 6 except

that we admit reducible systems. Let V s V [ Rd be a real vector space0Ž .and ? , ? be a degenerate symmetric bilinear form with kernel Rd ;

Ž .assume moreover that the restriction of ? , ? to V is positive definite.0Then let F be a finite reduced crystallographic root system with basis P0 0in the Euclidean space V and let Fq be the corresponding set of positive0 0roots. The affine root system associated to F is defined as0

� 4F s a q kd ¬ a g F , k g Z .0

PAOLA CELLINI110

If FŽ1., . . . , FŽh. are the irreducible components of F and u , . . . , u are0 0 0 1 hthe respective highest roots, then

� 4P s P j yu q d , . . . , yu q d0 1 h

Ž .is the set of simple roots of F associated to P . Each root in F can be0uniquely written as a linear combination of the roots in P with allnon-negative or all non-positive coefficients. A root a g F is calledpositive if a s Ý c b , with c G 0. Let Fq denote the set of positiveb g P b b

roots; then we have

Fqs a q kd ¬ a g Fq , k g N j ya q kd ¬ a g Fq , k g Nq ;� 4 � 40 0

q q Ž . qmoreover, F s F jy F . We also write a ) 0 a - 0 for a g FŽ q.a g yF .

For each non-isotropic b g V let s denote the reflection orthogonal tob

Ž . Ž Ž . Ž ..b , s x s x y 2 b , x r b , b b for each x g V, and for X : V letb

Ž . ² :W X s s ¬ b g X . The affine Weyl group associated to F is definedb 0

Ž . � 4as W s W F ; the set S s s ¬ a g P is a set of Coxeter generators fora

� q4W, and T s s ¬ b g F is the set of reflections. We have that F s W P,b

and the map b l s is a bijection between Fq and T. The Weyl group Wb 0Ž .of F is identified with the parabolic subgroup W F of W, generated by0 0

� 4S s s ¬ a g P .0 a 0ˇLet Q denote the co-root lattice of F . Then W is isomorphic to the0

ˇ ˇsemidirect product W h Q; Q corresponds to the normal abelian sub-0ˇ� Ž . 4group of the ‘‘translations’’ of W, t t ¬ t g Q , where the action of the

Ž . Ž .Ž . Ž .‘‘translation’’ of vector t , t t , is given by t t x s x y t , x d for eachx g V.

Ž .We say that R : F is a subsystem of F in R is W R -invariant. If R is0�a subsystem of F , then the affine root system associated to R , R s a0 0

4q kd ¬ a g R , k g Z , is a subsystem of F; in this case we say that R is0an affine subsystem. If R is a parabolic subsystem of F , then we say that0 0

Ž .R is an affine parabolic subsystem of F. We may identify W R with theaffine Weyl group associated to R and the subgroup of translations of0

ˇ ˇŽ . � Ž . Ž .4 Ž .W R with t t ¬ t g Q R , where Q R is the co-root lattice of R .0 0 0For any subsystem R of F we take Rq s Fq l R as the set of0 0 0 0 0

positive roots and, correspondingly, Rqs Fql R as the set of positiveroots of the associated affine subsystem R. The choice of the positive setRq uniquely determines a basis of R and, correspondingly, a set of simple0 0

Ž .roots for R and hence of Coxeter generators for W R , as described above.Ž .We understand that W R is a Coxeter group with this system of genera-

Ž . Ž . Ž .tors. If R is a subsystem of F and w g W R , then N w s N w l R isRŽ .the negative set of w as an element of W R .

T-INCREASING PATHS ON THE BRUHAT GRAPH 111

Let F be the affine root system associated to a finite crystallographicroot system F . We set0

MM s Fq j yFq q d .Ž .0 0

For each a g MM let ays ya q d ; it is clear that ayg MM. For eachb g Fq, b s a q kd , with a g F , we set0

a if a ) 0 y yb# s ; b s b# q Nd ; b s b#q Nd .½ a q d if a - 0

Note that b# g MM.

Remark. It is easily seen that if $ is a total reflection order then forq y Žeach a g F we have either a $ a i.e., x $ y for each x g a and

y. y y Ž .y g a or a $ a . Moreover, if a $ a , then a# q kd $ a# q k q 1 dy Ž . yand a#q k q 1 d $ a#q kd for each k g N.

DEFINITION 3. We say that A : Fq is a compatible set if

Ž . q q1 For each c, d g R and a , b g F such that ca q db g F, ifa , b g A then also ca q db g A, and if ca q db g A and a f A, thenb g A.

Ž . y2 For each a g MM, if a q kd g A for some k g N, then a l As B.

Ž .It is easy to see that the above condition 1 implies that if A iscompatible, a g MM, and a q kd g A for some k g N, then a q hd g Afor each h g N such that h F k. In particular, if a l A is infinite, then

Ž . Ž .a : A. Moreover, it is easy to see that, for a finite A, 1 implies 2 .

DEFINITION 4. Let A : Fq be a compatible set. A total order $ onA is called compatible if for each c, d g Rq and a , b g Fq such thatca q db g F we have

Ž .1 If a , b g A and a $ b then a $ ca q db $ b.Ž .2 If ca q db g A and b f A, then a g A and a $ ca q db.

w x w xThe following result is a direct consequence of D, 2.11 and P .

PROPOSITION 3. Let A : Fq be finite and $ be a total order on A.Then the following are equi alent:

Ž .1 A and $ are compatible.Ž . Ž .2 A s N w for some w g W and $ is induced on A by some

reduced expression of w.Ž . X q Ž .3 There exists a total reflection order $ on F such that A, $ is

Ž q X.an initial section of F , $ .

PAOLA CELLINI112

The next lemma is essentially a restatement of a standard result on weakŽ w x.orders see BB, Prop. 3.1.3 .

Ž . Ž .LEMMA 4. Assume ¨ , w g W and N ¨ : N w . If $ is a compatibleŽ . X Ž .order on N ¨ , then there exists a compatible order $ on N w which has

Ž Ž . .N ¨ , $ as an initial section.y1 Ž . Ž . Ž .Proof. Set u s ¨ w. Since N ¨ : N w , by Lemma 1 we have ll w

Ž . Ž .s ll ¨ q ll u . It follows that if ¨ s s ??? s is a reduced expression1 hŽ . X Xwhich induces $ on N ¨ and u s s ??? s is any reduced expression for1 k

u, then w s s ??? s sX??? sX is a reduced expression. It is clear that1 h 1 kŽ . Ž Ž . .the order induced on N w by such a reduced expression has N ¨ , $ as

an initial section.

2. PROOF OF THEOREM 2

Let F be an affine root system and $ be a total reflection order onqF . Let v be the order type of N and v the corresponding opposite

w x qorder. In CP it is proved that F is the disjoint union of two sets, say S1Ž .and S , such that the order type of S , $ is a finite sum v q ??? qv and2 1

Ž .the order type of S , $ is a finite sum v q ??? qv. This means that S is2 1q q Ž .a disjoint union S j ??? j S with S , $ of type v for 1 F i F h,1, 1 1, h 1, i

q q UŽ .and S $ ??? $ S . Similarly, S s S j ??? j S with S , $ of1, 1 1, h 2 2, 1 2, k 2, i

type v for 1 F i F k, and S $U ??? $US . Moreover, each of the S2, 1 2, k 1, i

and S is a compatible subset of some affine subsystem of F, up to2, j

adding finitely many roots. This motivates the interest for compatible sets.w xSuch sets are algebraically characterized in CP . Our first step here is to

make such a characterization more precise.Ž k . Ž kq1.If t is a translation in W, then we have N t ; N t for each k g N.

Ž . y1Ž . Ž . Ž .In fact, if a g N t , then t a s a y yt , a d s a q t , a d - 0,Ž . kŽ . Ž .which implies t , a - 0, since a ) 0. Thus t a s a y k t , a d ) 0

kq1 k q kŽ . Ž . Ž .and by Lemma 1 N t s N t j t N t . For any translation t g W set

N t` s N t k .Ž . Ž .DkgN

Ž `.By Proposition 3 N t is an increasing union of compatible sets andtherefore is compatible.

ˇ HŽ . Ž .LEMMA 5. Let t s t t , t g Q, and set R t s F l t . Then

Ž . Ž . Ž `. Ž .1 t acts tri ially on R t , hence N t l R t s B.Ž . Ž `. � Ž .4 � Ž . 42 N t s D a ¬ a# g N t s D a ¬ a g MM, t , a - 0 .

T-INCREASING PATHS ON THE BRUHAT GRAPH 113

Ž . Ž . Ž `. y Ž `.3 For each a g MM _ R t either a : N t or a : N t , and a :Ž `. y Ž `.N t if and only if a l N t s B.

y1 Ž . Ž `.Proof. It is clear that t and t act trivially on R t , hence N t lŽ . q Ž . Ž .R t s B. By definition, for each a g F _ R t we have t , a / 0. If0

Ž . y1Ž . Ž . Ž .t , a - 0, then t a s a q t , a - 0, hence a g N t . Moreover,kŽ . Ž Ž .. Ž .t a q hd s a q h q k t , a d - 0 for 0 F h - yk t , a ; hence a :Ž `. Ž . Ž .N t . On the other hand, if t , ya - 0, then, since t , ya is an integer,

y1Ž . Ž Ž .we have t ya q d s ya q 1 q t , ya d - 0 and we obtain simi-y `Ž .larly that a : N t .

Ž . Ž . Ž .If t is a translation in W and w g W is such that ll wt s ll w q ll t ,Ž k . Ž . Ž .then we have ll wt s ll w q k ll t . In fact, if a g MM and a q kd g

Ž y1 . Ž y1 . Ž y1 .N w for some k g N, then a g N w , since N w is compatible. IfŽ k . Ž .moreover a q kd g N t , then, by Lemma 5, a g N t . Therefore, since

Ž y1 . Ž . Ž y1 . Ž k .by assumption N w l N t s B, we have N w l N t s B, foreach k g N.

Ž . Ž . Ž .If t is a translation of W and w g W is such that ll wt s ll w q ll t ,then we set

N wt` s N wt k .Ž . Ž .DkgN

Ž `.N wt is an increasing union of compatible sets and therefore is compati-ble. The following result holds:

w x qPROPOSITION 6 CP, 3.12 . If F is an affine root system and A : F isan infinite compatible set, then there exist t, w g W such that t is a translation,Ž . Ž . Ž . Ž `.ll wt s ll w q ll t , and A s N wt .

Ž . Ž .LEMMA 7. Assume that w, t g W, t is a translation, and ll wt s ll wy1 ` ` qŽ . Ž . Ž . Ž .q ll t . Set r s wtw . Then N wt s N r j N w .RŽ r .

` q `Ž . Ž . Ž . Ž . �Proof. Let t s t t . Then we have N wt s N w j wN t s w a ¬ aŽ . 4 Ž . Ž .g MM, t , a - 0 . For each a g MM and k g N we have w a q kd s w a

Ž . Ž .q kd , hence, if w a ) 0, w a is equal to w a minus a finite subset.Ž .Ž `. � Ž . 4Therefore wN t is equal to w a ¬ t , a - 0 minus a finite subset.Ž .

Ž `. � Ž . 4 Ž `.Since N wt is compatible, it follows that w a ¬ t , a - 0 : N wt . ItŽ .Ž `. � Ž . 4 Ž .is clear that N wt _ w a ¬ t , a - 0 is included in N w ; moreover,Ž .

Ž `. Ž . Ž `.since N wt is compatible, it is included in R t . It follows that N wt sq� Ž . 4 Ž . Ž . Ž .w a ¬ t , a - 0 j N w . Now set r s w t , so that r s t r . WeŽ . RŽ t .

Ž `. � Ž . 4 Ž . Ž .have N r s a ¬ r, a - 0 . Since r s w t and , is W-invariant, weŽ . Ž y1 . Ž `. � Ž . 4have r, a s t , w a , hence N r s w a ¬ t , a - 0 , which con-Ž .

cludes the proof.

PAOLA CELLINI114

For any affine root system F and any total reflection order $ onFq let

C F , $ s a g Fq¬ $ a is finite ,� 4Ž . Ž .

Ž . � q 4 Ž .where $ a s b g F ¬ b $ a . It is clear that C F, $ is an initialsection of Fq. Then let

R F , $ s a , ya ¬ a g Fq , a­ C F , $ , ay ­ C F , $ ,� 4Ž . Ž . Ž .0 0

R F , $ s a q kd ¬ a g R F , $ , k g Z .� 4Ž . Ž .0

w x Ž . Ž .PROPOSITION 8 CP, 4.3 . 1 C F, $ is an infinite compatible set wellŽ . Ž `.ordered by $ ; in particular, C F, $ s N wt for some translation t and

Ž . Ž . Ž .w in W such that ll wt s ll w q ll t .

Ž . Ž . Ž .2 R F, $ is a proper parabolic subsystem of F , hence R F, $ is0 0an affine parabolic subsystem of F.

Ž . Ž . Ž q .We call R F, $ the first residual system of F , $ .Ž .q Ž . qThe restriction of $ to R F, $ s R F, $ l F is clearly a total

reflection order. We inductively define

FŽ1. s F , FŽ iq1. s R FŽ i. , $ , C s C FŽ i. , $Ž . Ž .i

for each i g Nq. We recall that if $ is a total reflection order then $U

UŽ1. Ž iq1. Ž i. Ž i.Ž . Žalso is one. We also set F s F, F s R F , $ , and C s C F ,iU .$ .The results of Proposition 8 apply to C and FŽ iq1. relative to FŽ i., andi

Ž iq1. Ž i. Ž iq1. Ž iq1.Ž .similarly to C , F relative to F . In particular, F F is aiŽ i. Ž i.Ž .proper affine parabolic subsystem of F F , if it is not empty. We

Žkq1. Žhq1. Žk . Žh.assume F s B, F s B, and F / B, F / B. Finally, we set

C s C j ??? j C , C s C j ??? j C .1 k 1 h

We have:

w x Ž . Ž .PROPOSITION 9 CP, 5.1 . 1 C l F C l F is finite for 1 F i - j F ki j i jŽ .1 F i - j F h .

qŽ .2 F s C j C and C l C s B.Ž .3 If for some x g C, y g C we ha¨e y $ x, then x and y are

orthogonal.

If A is a finite subset of C, then it is obvious that A l C is included inisome finite initial section of C , say A . We shall prove that with a suitablei ichoice of the A ’s we can make the union D A be a compatible subset ofi i i

T-INCREASING PATHS ON THE BRUHAT GRAPH 115

Fq. Moreover, we can find a compatible order $X on D A which isi iequal to $ when restricted to A. Theorem 2 will follow easily.

PROPOSITION 10. Assume 0 F i F k. Then

Ž . Ž `. Ž `. Ž i.Ž i.1 C s N wt s N wt l F for a suitable translation t ini F

Ž Ž i.. Ž Ž i.. Ž . Ž .Ž i. Ž i.W F and a w g W F such that N w : N wt .F F

Ž . y1 Ž iq1. Ž .2 If we set r s wtw , then F s R r ; in particular, r actsŽ iq1. Ž `. Ž Ž ..Ž i. Ž iq1.tri ially on F . Moreo¨er, C s N r j N w .i F F

Ž . Ž . Ž .Proof. Assertion 1 follows from Proposition 8 1 . Assertion 2 followsŽ .from 1 and Lemma 7.

LEMMA 11. For 0 F i F k there exists a translation r in W such thatŽ . Ž iq1. Ž `. Ž iq1. Ž .C j ??? j C _F s N r and F s R r . In particular, r acts0 i

tri ially on FŽ iq1..

Proof. By Proposition 10, for each 0 F i F k there exists a translation rŽ Ž i.. Ž iq1. Ž `. Ž iq1. Ž .Ž i.in W F such that C _F s N r and F s R r . In particu-i F

lar, we obtain the result for C _FŽ1.. We assume by induction that there0Ž . Ž i. Ž `.exists a translation u in W such that C j ??? j C _F s N u and0 iy1

Ž i. Ž . Ž Ž i.. Ž iq1.F s R u . Then we take a translation ¨ in W F such that C _FiŽ `. Ž iq1. Ž . w xŽ i.s N ¨ and F s R ¨ . By CP, 3.3 there exists an n g N suchF

Ž nq1. Ž n. Ž .that ll ¨u s ll ¨u q ll u ; since translations commute, by Lemma 1Ž . Ž nq1. nq1this is equivalent to N u : N ¨u . We set r s ¨u . Then clearly

Ž `. Ž `. Ž i. Ž . Ž `. Ž `. Ž i.N u : N r ; moreover, since F s R u , N r _ N u : F . Now"Ž . "Ž nq1. nq1 "Ž . Ž . Ž i. Ž . Ž i.N r s N u q u N ¨ , hence N r l F s N ¨ l F sŽ . Ž `. Ž `. Ž `. Ž iq1.

Ž i. Ž i.N ¨ . It follows that N r s N u j N ¨ s C j ??? j C _FF F 0 iŽ iq1.Ž .and that, moreover, R r s F .

LEMMA 12. Let A ; C be finite and let b g Fq be such that A $ b.Ž . Ž .Then there exists w g W such that A : N w ; C and b f N w .

Ž . Ž . Ž .Proof. Set a s max A, $ . If A ; C , then U a s N w for some0Ž .w g W and b f N w ; C : C. Assume a g C _C , i ) 0. Then there0 i iy1

Ž Ž i.. Ž . Ž i. Ž .Ž i.exists u g W F such that U a l F s N u . By Lemma 11, thereFŽ i. Ž `.exists a translation r in W such that C j ??? j C _F s N r and0 iy1

Ž i. Ž . y1 w xF s R r . Set t s u ru. By CP, 3.3 there exists an n g N such thatn Ž . Ž . Ž . Ž .Ž i.for ¨ s ut we have ll ¨t s ll ¨ q ll t . It is easy to see that N ¨ sF

n ` ` qŽ . Ž . Ž . Ž .Ž i. Ž i.N r u s N u , thus by Lemma 7 we obtain N ¨t s N r jF F

Ž . Ž `. Ž `.Ž i.N u . In particular, this implies that N ¨t : C and that b f N ¨t .FŽ i. Ž i. Ž `. Ž i. Ž .Ž i.Since A_F ; C j ??? j C _F s N r and A j F : N u ,0 iy1 F

` kŽ . Ž .we obtain that A : N ¨t , hence A : N ¨t for some k g N.

LEMMA 13. Let A : C be finite. Then there exists a w g W and aX Ž . Ž .compatible order $ on N w such that A : N w ; C and the order

induced by $X on A is the same as that induced by $ .

PAOLA CELLINI116

� 4Proof. Let A s a $ ??? $ a . By Lemma 12 there exists a ¨ g W1 h 1Ž . Ž .such that a g N ¨ ; C and a f N ¨ . It is obvious that any compati-1 1 2 1Ž . � 4ble order on N ¨ induces $ on a . By repeated applications of1 1

Lemma 12 we inductively define ¨ , . . . , ¨ as follows: once we define ¨2 h iŽ . � 4 Ž . Ž .0 F i - h such that a , . . . , a : N ¨ ; C and a f N ¨ , we take1 i i iq1 i

Ž . Ž . Ž . Ž .¨ such that N ¨ : N ¨ ; C and a f N ¨ if i q 1 - h . Byiq1 i iq1 iq2 iq1X Ž .induction, we assume that we can find a compatible order $ on N ¨ i

� 4which induces $ on a , . . . , a , 1 F i - h. Then by Lemma 4 we can find1 iŽ . Ž Ž . X.a compatible order $ 0 on N ¨ which has N ¨ , $ as an initialiq1 i

Ž . Y � 4section; since a f N ¨ it is clear that $ induces $ on a , . . . , a .iq1 i 1 iqiThus the claim is proved.

Now we can prove the following result, which is clearly equivalent toTheorem 2.

THEOREM 14. Let $ be a total reflection order and A ; Fq be finite.Then there exist A , A ; Fq, ¨ , ¨ g W, such that1 2 1 2

Ž . Ž .1 A s A j A ; A l A s B; A : N ¨ for i s 1, 2.1 2 1 2 i i

Ž . Ž y1 . Ž . Ž .2 ll ¨ ¨ s ll ¨ q ll ¨ .1 2 1 2

Ž . Ž .3 There exist a compatible order on N ¨ which induces $ on A1 1Ž . Uand a compatible order on N ¨ which induces $ on A .2 2

Ž . Ž .Proof. Let A s A l C and A s A l C. Assertions 1 and 3 follow1 2directly from Lemma 13, since $U is still a total reflection order.

Ž .Moreover, we can choose ¨ and ¨ in such a way that N ¨ ; C and1 2 1

Ž . Ž .N ¨ ; C. Since C l C s B, we obtain 2 using Lemma 1.2

3. PROOF OF THEOREM 1

Ž . XLEMMA 15. Let W, S be any Coxeter system. Let m g W and $ bethe order induced on T by some reduced expression of w. Assume r , . . . , rw 1 kg T and r $X

??? $X r . Then r ??? r w - w in the Bruhat order. Inw 1 k 1 kparticular, r ??? r / e, where e denotes the identity of W.1 k

Proof. There exists a reduced expression w s s ??? s such that, if we1 hŽ . � 4set t s s ??? s s s ??? s for i s 1, . . . , h, then N w s t , . . . , ti 1 iy1 i iy1 1 1 h

and t $X??? $X t . Let t s r for 1 F j F k. The claim is obvious if1 h i jj$

Ž .Ž .k s 1, since t s s ??? s ??? s s ??? s . Assume k ) 1. If j - m, theni 1 i h h 1j 1 $t ? s ??? s s s ??? s s s ??? s s ??? s s s ??? s ??? s . Itj 1 my1 1 jy1 j jy1 1 1 my1 1 j my1$ $

Ž .Ž .follows that t ??? t s s ??? s ??? s ??? s s ??? s , which is thei i 1 i i i i 11 k 1 k h h

claimed result.

T-INCREASING PATHS ON THE BRUHAT GRAPH 117

As direct consequence of Lemma 15 and Proposition 2 we obtain:

PROPOSITION 16. Let W be a finite Coxeter group and $ be a totalreflection order for W. The for any set of reflections r , . . . , r such that1 kr $ ??? $ r , we ha¨e r ??? r / e.1 k 1 k

It is clear that Proposition 16 says exactly that T-increasing paths areself-avoiding for finite Coxeter systems. Now we use the results of theprevious section to prove that the assertion of Proposition 16 also holds foraffine Weyl groups.

THEOREM 17. Let W be an affine Weyl group and $ be a total reflection� 4order for W. Let t , . . . , t : T and t $ ??? $ t . Then t ??? t / e.1 l 1 l 1 l

q Ž .Proof. Let t s s , a g F , for 1 F i F l. By Proposition 9 3 , if fori a ii

some i we have a g C and a g C, then t and t commute. Hence,i iq1 i iq1� 4we can reorder the t in such a way that, for some m g 1, . . . , l , we havei

a , . . . , a g C and a , . . . , a g C. By Lemma 13 there exists a w g W1 m mq1 l� 4 Ž . Ž .such that a , . . . , a : N w : C and a compatible order on N w1 m

� 4which induces $ on a , . . . , a . Hence, by Lemma 15 we have that1 mt ??? t s ¨wy1, where ¨ g W is less than w in the Bruhat order. If1 mm s l we are done. Otherwise, again by Theorem 14 and Lemma 15 we

y1 Ž .have that t ??? t s xy , with x, y g W, N y : C, and x - y. Itl mq1y1 y1 Ž .follows that t ??? t s ¨w yx . Now we observe that, since N y : C,1 l

y1 qŽ . Ž .N w : C, and, by Proposition 9, C l C s B, we have w N y : F .Ž y1 . Ž y1 . Ž . y1Therefore, by Lemma 1, ll w y s ll w q ll y . Then we have x -

y1 Ž y1 . Ž y1 . Ž . y1 y1y , ¨ - w, and ll y w s ll y q ll w , which implies x ¨ - y wand therefore xy1 ¨wy1 y / e. Since ¨wy1 yxy1 s e if and only if xy1 ¨wy1 ys e, the claim is proved.

Proposition 16 and Theorem 17 together give Theorem 1.The author conjectures that the result of Theorem 1 holds for all

Coxeter groups.

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