Order 16: 77–87, 1999.© 2000Kluwer Academic Publishers. Printed in the Netherlands.

77

Upper Bounds in Affine Weyl Groups under theWeak Order?

DEBRA J. WAUGHDepartment of Science and Mathematics, University of Texas of the Permian Basin, Odessa, TX,U.S.A. E-mail: waugh−d@utpb.edu

(Received: 7 July 1998; accepted: 8 September 1999)

Abstract. Björner and Wachs proved that under the weak order every quotient of a Coxeter group isa meet semi-lattice, and in the finite case is a lattice. In this paper, we examine the case of an affineWeyl groupW with corresponding finite Weyl groupW0. In particular, we show that the quotient ofW byW0 is a lattice and that up to isomorphism this is the only quotient ofW which is a lattice. Wealso determine that the question of which pairs of elements ofW have upper bounds can be reducedto the analogous question within a particular finite subposet.

Mathematics Subject Classifications (1991):06A07, 05E99, 20F55.

Key words: affine Weyl groups, parabolic quotients, upper bounds, weak ordering.

1. Introduction

Let (W, S) be a finitely generatedCoxeter system, that is, letS be a finite set andletW be the group generated byS subject to the relations that each generator hasorder two and possibly additional relations specifying the orders of the productsof some or all pairs of distinct generators. We callW a Coxeter group. Forw =s1 . . . sk ∈ W , with si ∈ S for all i, andk minimal, we says1 . . . sk is a reducedexpressionforw andl(w) = k is thelengthofw. Forx, y ∈ W , we writex <L y ifthere is a reduced expression fory which ends with a reduced expression forx; thispartial ordering onW is called the (left)weak order. Similarly we writex <R y ifthere is a reduced expression fory which begins with a reduced expression forx;this is known as theright weak order. The two weak orders are isomorphic underthe mapw 7→ w−1, and we will restrict our attention to<L.

ForJ ⊂ S, the subgroup ofW generated byJ is denotedWJ and is said to be aparabolic subgroup. Consider the corresponding (parabolic)quotientW/WJ . It iswell-known [3] that

WJ := {w ∈ W : l(ws) > l(w) for all s ∈ J }is the set ofminimal coset representativesfor W/WJ , that is, each element ofWJ

is the unique element of minimal length within its coset. IfJ = S − {s} for some? The results in this paper are part of the author’s doctoral dissertation.

78 DEBRA J. WAUGH

s ∈ S, then the quotientW/WJ is known as themaximal quotientofW with respectto s, and we writeWJ = W 〈s〉. By popular abuse of language, we will refer toWJ

as a quotient, and in particular when we make statements about quotients under theweak order, we are actually referring to the corresponding sets of minimal cosetrepresentatives as induced subposets(WJ ,<L) of (W,<L).

We assume the reader is familiar with several definitions related to partiallyordered sets as developed in Chapter 3 of [6].

THEOREM 1 (Björner and Wachs [1]).All quotients of Coxeter groups are meetsemi-lattices under the weak order.

We say a partially ordered set isdirected if every pair of elements has an upperbound. It is well-known that every finite Coxeter group has a unique longest el-ement which is a maximum under the weak order. In an infinite Coxeter group,however, there is no common upper bound for all of the generators; one way to seethis is to note that such an element would have to send all simple roots and henceall positive roots negative, but there are infinitely many positive roots, so this isimpossible. Therefore a Coxeter group is directed if and only if it is finite.

Since the weak order is graded by the length function and a Coxeter grouphas only finitely many elements of a given length, Coxeter groups and hence theirquotients are locally finite under the weak order.

PROPOSITION 2. A locally finite meet semi-lattice is a lattice if and only if it isdirected.

The proof of Proposition 2 is straightforward and very similar to that of Proposi-tion 3.3.1 in [6].

By Proposition 2, a quotient of a Coxeter group is a lattice if and only if itis directed. Thus Theorem 1 implies that all quotients of finite Coxeter groups arelattices since they have maximum elements and are therefore directed. On the otherhand, no infinite Coxeter group is a lattice, but we shall see that certain quotientsof infinite Coxeter groups are.

The classification of the quotients which are distributive lattices under the weakorder appears in [7] within the larger context of the following result:

THEOREM 3 (Stembridge).The following are equivalent:

(1) WJ is a lattice under the Bruhat order.(2) WJ is a distributive lattice under the Bruhat order.(3) WJ is a distributive lattice under the weak order.(4) The weak order and Bruhat order coincide onWJ .

The classification of these quotients in the finite case is also given in [4] within thecontext of the first condition and with the observation that the second condition isequivalent.

UPPER BOUNDS IN AFFINE WEYL GROUPS UNDER THE WEAK ORDER 79

The quotients satisfying the conditions of the theorem are certain maximal quo-tients of finite Weyl groups known as minuscule quotients; the maximal quotientsof the infinite dihedral groupI2(∞); and one maximal quotient ofH3.

In light of the equivalent conditions of Theorem 3, it is natural to ask whenWJ is a lattice under the weak order. As we remarked above, all quotients offinite Coxeter groups are lattices. In this paper we will examine only the affinecase. In particular, we will show that for an affine Weyl group, the quotient bythe corresponding finite Weyl group is a lattice, and that up to isomorphism noother quotient is a lattice (Theorem 18, our main result). More generally, in [8], weprove that for infinite indecomposableW, (WJ ,<L) is a lattice if and only ifW isaffine andWJ is isomorphic to the corresponding finite Weyl group; also for anyW, (WJ ,<L) is a lattice if and only ifWJ∩I

I is a lattice for allI ⊂ S such that theinduced subgraph0I is a connected component of0.

Because our methods depend upon a particular description of affine Weyl groups,we will review some of the theory here. The reference for this material is [3].

LetW be an affine Weyl group with corresponding finite Weyl groupW0. It iswell-known that there is a Coxeter system(W0, S0) in which the generating setS0

is a set of reflections across hyperplanes through the origin inRn, wheren = |S0|.We sayH is agenerating hyperplaneif somes ∈ S0 is reflection acrossH .

Let 〈·, ·〉 be the standard Euclidean inner product onV = Rn. In this setting, aroot systemforW is a set8 ⊂ V , which has a linearly independent subset1 calledthebasewith 8 = W1, such that for eachα in 8, the only other scalar multipleof α in 8 is−α, and eitherα or−α is in the nonnegative linear span of1.

A root system8 is said to becrystallographicif for all α, β ∈ 8 we have

2〈α, β〉/〈α, α〉 ∈ Z.

In this case we also say the corresponding Coxeter group is crystallographic. Thefinite crystallographic Coxeter groups are precisely the finite Weyl groups.

We can choose a crystallographic root system8 and base1 for W0 so that1consists of one vector orthogonal to each of the generating hyperplanes.

Let 1 = {α1, . . . , αn}. Every rootα ∈ 8 can be writtenα = ∑i ciαi where

either noci is negative, in which case we callα apositive root, or noci is positive,in which case we callα a negative root. Theheightof α ∈ 8 with respect to1 isdefined to be ht(α) = ∑i ci . We denote the set of positive roots by8+. SinceW0

is finite, it is well-known that8 contains a unique rootα of maximal height, calledthe highest root.

The arrangement of hyperplanes obtained by reflecting the generating hyper-planes across each other and across the resulting images divides the underlyingvector spaceV into |W | isomorphic cones calledWeyl chambers; the chamberbounded by the generating hyperplanes is called thefundamental chamber.

The affine Weyl groupW = W0 corresponding toW0 can be considered geomet-rically as the group generated by the setS0 of original finite Weyl group generatorstogether with the affine reflections0 across the hyperplane{λ ∈ V : 〈λ, α〉 = 1}.

80 DEBRA J. WAUGH

The set of hyperplanes fixed by the reflections inS = S0 ∪ {s0} bounds a simplexA0, thefundamental alcove; the complement of the setH of hyperplanes obtainedby reflecting these hyperplanes across each other is a set of alcoves upon whichthe groupW acts simply transitively. Thus each alcoveA can naturally be labeledby the group elementw ∈ W such thatA = w−1A0. We will frequently identifya group element with the corresponding alcove. We say two alcoves areadjacentif there is exactly one hyperplane that separates them. Apath to A is a sequenceof adjacent alcoves(A0, A1, . . . , Ak = A). It is well-known that paths toA cor-respond to expressions forw and in particular shortest paths toA correspond toreduced expressions forw, and we havex <L w if and only if there is a shortestpath toA through the alcoveB corresponding tox. In this case we will also writeB <L A. Each hyperplaneH(α, k) := {λ ∈ V : 〈λ, α〉 = k} in H is indexed by apositive rootα ∈ 8+ of the original finite Weyl group together with an integerk.The termsk-hyperplane andα-hyperplane are used to specify only one index.

TheShi arrangementor sandwich arrangementof hyperplanes is

S = {H(α, k) : α ∈ 8+, k ∈ {0,1}}.Theregionsof S are the connected components of the complement,V −⋃S.

PROPOSITION 4 (Shi [5]).Each regionR of S contains a minimum element withrespect to<L.

Remark 5.Shi’s proof of the existence of a unique element of minimum lengthwithin each region actually establishes the stronger result stated in the above propo-sition.

In Figure 1, the thick lines represent the Shi arrangementS for (A2,<L) andthe shaded alcoves are the minimal elements of the regions ofS.

Figure 1. Minimal elements of regions ofS for (A2, <L).

UPPER BOUNDS IN AFFINE WEYL GROUPS UNDER THE WEAK ORDER 81

LetM be the set of minimal elements of regions ofS. Letx, y ∈ W . Let x0 andy0 be the minimal elements of the regions containingx andy, respectively. We willshow thatx andy have an upper bound in(W,<L) if and only if x0 andy0 have anupper bound in the finite subposet(M,<L) (Theorem 24).

2. Classification of Quotients

PROPOSITION 6 (Eriksson [2]).A shortest path toA cannot cross anyH(α, k)more than once.

Proof. Suppose(A0, A1, . . . , Am = A) crossesH(α, k) at least twice, sayH(α, k) is the hyperplane separating bothAi andAi+1 andAj andAj+1, 0 ≤i < j ≤ m. Then it is easy to check that the sequence of alcoves

(A0, A1, . . . , Ai, A′i+2, . . . , A

′j−1, Aj+1, . . . , Am = A),

whereA′r is the image ofAr under reflection acrossH(α, k), is a shorter pathtoA. 2LetA be the alcove labeled byw, and letλ be a vector whose terminal point is in theinterior ofA. We writeA ∈ H(α, k)+ (respectivelyA ∈ H(α, k)−) if 〈λ, α〉 > k

(respectively,〈λ, α〉 < k). We call {λ : 〈λ, α〉 > k} the positive side ofH(α, k)and{λ : 〈λ, α〉 < k} the negative side.

PROPOSITION 7. If A1 andA2 are alcoves such thatA1 lies on the positive sideof H(α,1) andA2 lies on the negative side ofH(α,0), then the elements ofWcorresponding toA1 andA2 have no upper bound under the weak order.

Proof. Assume toward a contradiction that an upper bound exists; letA be thecorresponding alcove. There is a shortest path toA throughA1, and sinceH(α,1)separatesA1 andA0 this path must crossH(α,1). By Proposition 6,H(α,1)separatesA from A0. Similarly,H(α,0) separatesA from A0, but thenA lies inH(α,0)− ∩H(α,1)+ = ∅, which is absurd. 2Let C be a Weyl chamber ofW0. Let 2 be the set of positive rootsα such thatH(α,0) is a wall of C, that is,H(α,0) intersectsC in a region of codimensionone. Let2+ (respectively,2−) be the set of those elementsα of 2 for whichC ⊂ H(α,0)+ (respectively,H(α,0)−). We have

C :=⋂α∈2+

H(α,0)+⋂α∈2−

H(α,0)−.

Define

Ca :=⋂α∈2+

H(α,1)+⋂α∈2−

H(α,0)−.

82 DEBRA J. WAUGH

We can visualize the process of obtainingCa fromC as the removal of the slice ofC betweenH(α,0) andH(α,1) for eachα ∈ 2+. Note thatCa is a translation ofC; we will refer toCa as atranslated chamber. Since the Weyl chambers are non-empty, the corresponding translated chambers are nonempty, and since the walls ofthe translated chambers are inH , the translated chambers contain alcoves.

PROPOSITION 8. For each positive rootα of W0, Ca either lies inH(α,0)− orin H(α,1)+.

Proof. SinceCa ⊂ C, we have eitherCa ⊂ H(α,0)− or Ca ⊂ H(α,0)+. IfCa ⊂ H(α,0)+, then by construction,Ca ⊂ C ∩H(α,1)+, soCa ⊂ H(α,1)+. 2PROPOSITION 9.Any two elements from distinct translated chambers ofW0 haveno upper bound inW .

Proof. Let C 6= C′ be two Weyl chambers withA ∈ Ca,A′ ∈ C′a. Every

pair of distinct Weyl chambers is separated from each other by at least one 0-hyperplane, and without loss of generality we may assumeC lies on the negativeside ofH(α,0)whileC′ lies on the positive side ofH(α,0). By definition,Ca ⊂ CandC′a ⊂ C′, so we haveA ∈ H(α,0)− andA′ ∈ H(α,0)+. By Proposition 8,A′ ∈ H(α,1)+, and therefore by Proposition 7, the elements ofW correspondingtoA andA′ have no upper bound under the weak order. 2

Remark 10.Proposition 9 follows from Lemma 3 in [2] because each translatedchamber is contained in exactly one of Eriksson’s truncated cones.

We omit the proof of the following standard observation.

PROPOSITION 11. Let v be an arbitrary vertex ofA0. Let s be the generatorcorresponding to reflection across the unique wall ofA0 not containingv. Thearrangement of hyperplanes passing throughv is isomorphic to the arrangementof hyperplanes corresponding to the finite Weyl groupWS−{s}.LetW,v, s be as in Proposition 11. Note thatW 〈s〉 is the region formed by remov-ing the wall ofA0 not containingv; we will call this thequotient cone. If s is theadded generators0, the quotient cone is the fundamental chamber for the originalfinite Weyl groupW0; we will examine this case in more detail later.

Otherwise the quotient cone can be viewed as a chamber for the correspondingWeyl groupWS−{s}, with the origin translated tov, and all walls of the quotientcone except forH(α,1) are 0-hyperplanes. Consider the corresponding cone withthe wallH(α,0) instead ofH(α,1); this cone is obtained from the quotient coneby removing the strip between these two parallel hyperplanes and we will refer tothis cone as thetranslated quotient coneand denote it byQs.

We will call s a specialgenerator ifs is the image of the added generators0under a weighted graph automorphism of the Coxeter graph0 ofW , or equivalentlyif WS−{s} ' W0.

PROPOSITION 12.If s is not special, thenW 〈s〉 contains a pair of elements withno upper bound inW and in particular is not a lattice.

UPPER BOUNDS IN AFFINE WEYL GROUPS UNDER THE WEAK ORDER 83

Proof. Note thatQs is a nondegenerate cone all of whose walls are 0-hyper-planes, so its vertex must be at the origin and therefore the translated quotientcone is a union of Weyl chambers ofW0. Since the walls of the translated quo-tient cone are in the same configuration as the walls of the quotient cone, and byProposition 11 this is the configuration of the walls of the fundamental chamberof WS−{s}, it follows thatWS−{s} is isomorphic to a subgroup of the original finiteWeyl groupW0 and furthermore that the translated quotient cone, and hence thequotient cone, contains[W0 : WS−{s}] of our translated chambers. In particular,sinces is not special,WS−{s} 6' W0, so the quotient contains at least two of ourtranslated chambers and consequently by Proposition 9 contains a pair of elementswith no upper bound inW and in particular is not a lattice. 2The following proposition was inspired by a conjecture of Curtis Bennett (personalcommunication) in which convexity under the weak order is replaced by geometricconvexity.

PROPOSITION 13.For any Coxeter group, a subsetU which contains the identityand is convex in the weak order is a lattice if and only if it is an order ideal gen-erated either by a single element or by a series of elements which is monotonicallystrictly increasing under the weak order.

Proof.LetU ⊂ W be convex under the weak order and contain the identity. IfU is finite, the result is clear, so we may assumeU is infinite.

First supposeU is the order ideal generated by a strictly increasing chain ofelementsu1 <L u2 <L · · · . Letw, v ∈ U , sayw <L ui, v <L uj . Without loss ofgenerality, we may assumei ≤ j , and then we havew <L ui <L uj , souj is anupper bound forw andv and thereforeU is directed. SinceU is an order ideal ofa locally finite meet semi-lattice,U is itself a locally finite meet semi-lattice, so byProposition 2,U is a lattice.

Now supposeU is a lattice. Note thatU is countable since it contains finitelymany elements of each length and there are countably many lengths; therefore wecan index the elements ofU by the natural numbers, sayU = {uk : k ∈ N}. Weconstruct an increasing chainC = c1 <L c2 <L · · · of elements ofU as follows.Definec1 := u1 andck := ck−1 ∨ uk . Note that the chainC cannot be eventuallyconstant, as in this caseU would be the order ideal generated by that constant andin particular would be finite. Thus we can discard repetitions inC to obtain thedesired strictly increasing chain which generatesU as an order ideal. 2COROLLARY 14. If WJ is a lattice, then there is an increasing series

w1 <L w2 <L · · ·of elements ofWJ such that for everyw ∈ WJ , we havew <L wj for somej .

In the proof of Proposition 15, we will provide an algorithm to construct such aseries forW 〈s0〉.

84 DEBRA J. WAUGH

Forw ∈ W , defineH(w) to be the set of hyperplanes separatingw−1A0 fromA0. It is well-known [3] that ifw = si1 · · · sik is reduced, then

H(w) = {Hik , sik (Hik−1), . . . , sik · · · si2(Hi1)},whereH0 = H(α,1) andHj = H(αj ,0), j 6= 0. Therefore we havex <L y ifand only ifH(x) ⊂ H(y). Note thatw ∈ W 〈s0〉 if and only if

H(w) ⊂ {H(α, k) : α ∈ 8+, k ∈ Z+}.

PROPOSITION 15.There exists a series of elements{wi} ⊂ W 〈s0〉 such that

(1) w1 <L w2 <L · · ·(2) H(wi) ⊃ {H(α, k) : α ∈ 8+,1≤ k ≤ i}.

Proof.Fix γ ∈ A0. Let1 = {αi : 1 ≤ i ≤ n}, and let1∗ = {ωi : 1 ≤ i ≤ n}be the corresponding dual basis, so that〈αi, ωj 〉 = δij . Defineλ = ∑n

i=1ωi. Ifα ∈ 8+, then〈α, λ〉 = ht(α) ≥ 1. Also 0< 〈α, γ 〉 < 1, so we have

k ht(α) < 〈kλ+ γ, α〉 < k ht(α)+ 1

for all k. In particular, the vectorkλ+ γ crosses the hyperplanes

{H(α, j) : 1≤ j ≤ k},and thereforeλ is not contained withinH(α,0) for any α. Now if hλ + γ liesstrictly in the interior of an alcove, we takewk to be the element corresponding tothis alcove and defineεk = 0. Otherwise, for sufficiently smallε > 0, we have(k + ε)λ+ γ strictly in the interior of an alcove such that every smaller choice ofepsilon would result in the same choice of alcove, and we can definewk to be theelement corresponding to this unique minimal alcove; without loss of generalitywe may assume thatε < 1, and defineεk = ε. To see that forj > k, we havewj > wk, note thatH(wi) is the set of hyperplanesH(v) in H crossed by anyvectorv in the interior of the corresponding alcove. Thus we have

H(wk) = H((k + εk)λ+ γ ).Now for fixedα ∈ 8+, we knowH(wk) containsH(α, i) if and only if

i < (k + εk)ht(α).

Fork < j , we have

(k + εk)ht(α) < (k + 1)ht(α) ≤ j ht(α) ≤ (j + εj )ht(α).

ThereforeH(wj) contains allα-hyperplanes inH(wk) for eachα ∈ 8+, andconsequentlywj > wk, as required. 2COROLLARY 16. For any nonmaximal quotient ofW,WJ is not a lattice.

UPPER BOUNDS IN AFFINE WEYL GROUPS UNDER THE WEAK ORDER 85

Proof. Case 1.W 〈s〉 ⊂ WJ for some special generators.Without loss of generality, we may assumeW 〈s0〉 ⊂ WJ . Since the quotient

is nonmaximal,WJ must also containW 〈si 〉 for some simplei 6= 0. The elementsi ∈ W 〈si 〉 is separated fromA0 by H(αi,0) while the elementw1 ∈ W 〈s0〉 con-structed above is separated fromA0 byH(αi,1); therefore by Proposition 7 thesetwo elements have no upper bound soWJ is not a lattice.

Case 2.W 〈s〉 6⊂ WJ for any specials.In this case,WJ must containW 〈s〉 for somes which is not special.By Proposition 12,W 〈s〉 contains elements with no upper bound inW under the

weak order. HenceWJ contains these elements and is not a lattice. 2PROPOSITION 17.If s is a special generator, thenW 〈s〉 is a lattice.

Proof. By symmetry it suffices to proveW 〈s0〉 is a lattice. Let{wi} be as inProposition 15. Letx ∈ W 〈s0〉, say

H(x) =⋃α

{H(α,1), (α,2), . . . ,H(α,mα)}.

If m = max{mα : α ∈ 8+}, thenx ≤ wm. ThusW 〈s0〉 is contained in the orderideal generated by thewi ; sinceW 〈s0〉 is an order ideal and contains{wi}, it followsthatW 〈s0〉 is the order ideal generated by thewi. Thus by Proposition 13,W 〈s0〉 is alattice. 2THEOREM 18. For affineW,WJ is a lattice if and only ifJ = S − {s}, wheresis a special generator.

Proof.Combine Propositions 17 and 12, and Corollary 16. 2

3. Join Sets

PROPOSITION 19. If W is an affine Weyl group withx, y ∈ W , thenx and yhave an upper bound if and only if the minimal elementsx0 andy0 of the regionsof S containingx andy, respectively, have an upper bound.

Proof.First supposex andy have an upper boundz. In this case we havex0 ≤x ≤ z andy0 ≤ y ≤ z, soz is also an upper bound forx0 andy0.

Now let z0 be an upper bound forx0 andy0. We need to show that there is azwith H(z) ⊃ H(x)∪H(y). ConsiderH(x)−H(x0); this is the set of hyperplanesseparatingx from x0 and cannot contain any of the hyperplanes ofS sincex andx0 are in the same region. So ifH(α, k) is in this set, we must havek /∈ {0,1}.Thus eitherk > 1 andH(α,1) ∈ H(x0) or k < 0 andH(α,0) ∈ H(x0). Inparticular there is already anα-hyperplane on the same side ofA0 asH(α, k) inthe setH(x0) ⊂ H(z0). Fix an interior pointv ∈ A0 and consider the directedline segmentL(v, z0v) from v to z0v. This segment must cross anα-hyperplane onthe same side ofA0 asH(α, k), so for sufficiently large numbersm, the directed

86 DEBRA J. WAUGH

line segmentmL(v, z0v) from v throughz0v with lengthm times the length ofL(v, z0v) must crossH(α, k). Therefore for sufficiently largem0, the directedline segmentm0L(v, z0v) will cross all hyperplanes inH(x) ∪H(y), so the wordcorresponding to the alcove containing the terminal point ofm0L(v, z0v) will bean upper bound forx andy; in case this is not an interior point, continue in thedirection ofL(v, z0v) from this point to choose an alcove – noteL(v, z0v) doesnot lie along a hyperplane since it contains interior points ofA0 andz0A0. 2Forx ∈ W , thejoin setof x is {y ∈ W : x ∨ y exists}.COROLLARY 20. The join set ofx depends only on which region ofS containsx, and is a union of regions ofS.

COROLLARY 21. Let R be a region ofS such that for eachα ∈ 8+, eitherR ⊂ H(α,0)− or R ⊂ H(α,1)+. For all x, y ∈ R, x andy have an upper boundin R; furthermore all upper bounds forx andy lie in R.

Proof. By Proposition 19,x andy have an upper boundz. Supposez /∈ R. Inthis casez must be separated fromR by one of the walls ofR. Either this is a0-hyperplane withR on the negative side andz on the positive side, or it is a 1-hyperplane withR on the positive side andz on the negative side. In any case, thisis a hyperplane which separates all elements ofR from A0 and so must separateany element greater than an element ofR fromA0 but does not separatez fromA0,which is absurd. 2LetA be an alcove. Thesign typeof A is a sequence indexed by the positive rootsof the finite Weyl group with entries in{+,0,−} where the entry corresponding toα is defined to be+ if A ∈ H(α,1)+,− if A ∈ H(α,0)−, and 0 otherwise.

PROPOSITION 22 [5].LetU be a set of sign types of regions ofS such that thereexists someY ∈ U which can be obtained from anyZ ∈ U by replacing somenonzero signs by zero signs. In this case there exists an elementy in the regionwith sign typeY which is minimal in the weak order among all elements of allregions with sign types inU .

PROPOSITION 23. If x0 and y0 are the minimal elements of two regions ofS,thenx0 and y0 have an upper bound if and only if they have an upper boundz0

which is the minimal element of a region ofS.Proof.Let z be an upper bound forx0 andy0. LetX,Y , andZ denote the regions

of S containingx0, y0 andz, respectively, and letz0 denote the minimal elementof Z. Sincez ≥ y0, the sign type ofy0 can be obtained from the sign type ofz by replacing some nonzero signs by zero signs, that is the sign type ofY canbe obtained from the sign type ofZ by such a replacement, and therefore settingU = {Y,Z} in Proposition 22, we have thaty0 is minimal inY ∪Z and in particulary0 ≤ z0, as desired. By symmetry, the same argument applies tox0.

UPPER BOUNDS IN AFFINE WEYL GROUPS UNDER THE WEAK ORDER 87

Figure 2. Finite subposets of minimal elements ofS for (A2,<L) and(B2,<L).

The converse is trivial. 2THEOREM 24. Two elementsx, y ∈ W have an upper bound inW if and only ifthe minimal elements of the regions ofS containingx andy have an upper boundin the finite subposet ofW formed by the set of minimal elements of regions ofS.

Proof.Apply Propositions 19 and 23. 2The Hasse diagrams for the finite subposets of minimal elements ofS in the casesof A2 andB2 appear in Figure 2.

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