Characterization of Coxeter Complexes

Coxeter Complexes

3.4 Roots

3.4.2 Characterization of Coxeter Complexes

to have gained wide acceptance, probably because there is too much existing literature referring to the Bruhat order. See Humphreys [133, Sections 5.9–

5.11] for more information about the Bruhat order and for further references.

3.61.Let±αbe a pair of opposite roots, letC, C be adjacent chambers with C ∈αand C ∈ −α, and letv be the vertex of C not in the common panel C∩C. Show thatv /∈ −α.

3.62.LetΣ be the Coxeter complexΣ(W, S). For any simplexA∈Σ, show that the stabilizerWA ofAinW is generated by the reflectionssH, whereH ranges over the walls containingA.

3.63.You have now seen the standard uniqueness argument applied several times. Try to write down a lemma that includes all of these applications.

[Warning: Unless you have incredible foresight, you can expect to have to modify your lemma one or more times as you see further applications of the argument. In fact, this might even happen in the next few pages.]

3.4 Roots 139 Proof of Theorem 3.65 (start). We have already proven the “only if” part.

For the converse, assume that every pair of adjacent chambers is separated by a wall. Choose an arbitrary chamberC, called thefundamental chamber, and letSbe the set of reflections determined by the panels ofC. LetW ≤AutΣ be the subgroup generated byS. We will prove that (W, S) is a Coxeter system and that Σ∼=Σ(W, S).

We could simply repeat, essentially verbatim, the arguments that led to the analogous results for finite reflection groups in Chapter 1. For the sake of variety, however, we will use a different method. This is actually a little longer, but it adds some geometric insight that we would not get by repeating the previous arguments. In particular, it gives a simple geometric explanation of the deletion condition.

We now proceed with a sequence of lemmas, after which we can complete the proof.

Lemma 3.66.W acts transitively on the chambers ofΣ.

Proof. This is identical to the proof given in Chapter 1 for finite reflection

groups (Theorem 1.69).

Lemma 3.67.Σ is colorable.

Proof. LetC be the subcomplexΣ_≤C. It suffices to show thatC is a retract of Σ. The idea for showing this is to construct a retractionρ by folding and folding and folding. . . , until the whole complexΣhas been folded up ontoC.

To make this precise, letC1, . . . , Cn be the chambers adjacent toC, and letφ₁, . . . , φ_n be the foldings such thatφ_i(C_i) =C. Letψ be the composite φ_n◦ · · · ◦φ₁. We claim thatd(C, ψ(D))< d(C, D) for any chamberD=C. To prove this, letΓ: C, C, . . . , Dbe a minimal gallery fromCtoD; we will show thatψ(Γ) has a repetition. Ifφ₁(Γ) has a repetition, we are done. Otherwise, the standard uniqueness argument shows that φ₁ fixes all the chambers ofΓ pointwise. In this case, repeat the argument with φ₂, and so on. Eventually we will be ready to apply the folding φi that takes C to C. If the previous foldings did not already produce a repetition in Γ , then they have fixedΓ pointwise, and the application ofφiyields a pregallery with a repetition. This proves the claim.

It follows that for any chamberD, ψ^k(D) =Cforksufficiently large. Since ψfixesCpointwise, this implies that the “infinite iterate”ρ:= limk→∞ψ^k is

a well-defined chamber map that retractsΣ ontoC.

It will be convenient to choose a fixed type functionτ with S as the set of types, analogous to the canonical type function that we used earlier in the chapter. To this end we assign types to the vertices of the fundamental chamber C by declaring that the panel fixed by the reflection s ∈ S is an s-panel. We then extend this to all of Σ by means of a retraction ρ of Σ ontoC. Note that this type functionτ has a property that by now should be very familiar: For any s∈S, the chambersC andsCares-adjacent.

Lemma 3.68.Foldings and reflections are type-preserving; hence all elements of W are type-preserving. Consequently, wC andwsC ares-adjacent for any w∈W ands∈S.

Proof. A folding φ fixes at least one chamber pointwise, so the type-change map φ_∗ is the identity (see Proposition A.14). This proves that foldings are type-preserving, and everything else follows from this.

IfΓ:C0, . . . , Cd is a gallery andHi is the wall separatingC_i−1 from Ci, then, as usual, we will say thatH1, . . . , Hd are thewalls crossed byΓ. Lemma 3.69.If Γ: C0, . . . , Cd is a minimal gallery, then the walls crossed by Γ are distinct and are precisely the walls separating C0 from Cd. Hence the distance between two chambers is equal to the number of walls separating them.

Proof. SupposeHis a wall separatingC0fromCd. Let±αbe the correspond- ing roots, say with C0 ∈ αand Cd ∈ −α. Then there must be some i with 1 ≤ i ≤ d such that C_i−1 ∈ α and Ci ∈ −α. Since α and −α are convex (Lemma 3.44), it follows that we haveC0, . . . , C_i−1∈αandCi, . . . , Cd∈ −α.

In other words,Γ crossesH exactly once. Now supposeH is a wall that does not separate C₀ from C_d. Then C₀ and C_d are both in the same root α, so the convexity ofαimplies thatΓ does not crossH. The crux of the proof of Lemma 3.69, obviously, is the convexity of roots, which in turn was based on the idea of using foldings to shorten galleries.

We can now use this same idea to prove a geometric analogue of the deletion condition. The statement uses the notion oftypeof a gallery (Definition 3.22).

Lemma 3.70.LetΓ be a gallery of types= (s1, . . . , sd). IfΓ is not minimal, then there is a galleryΓ with the same extremities asΓ such thatΓhas type s= (s₁, . . . ,ˆs_i, . . . ,sˆ_j, . . . , s_d)for somei < j.

Proof. SinceΓ is not minimal, Lemma 3.69 implies that the number of walls separating C0 from Cd is less than d. Hence the walls crossed by Γ cannot all be distinct; for if a wall is crossed exactly once by Γ, then it certainly separates C₀ from C_d. We can therefore find a root α and indicesi, j, with 1≤i < j≤d, such thatC_i−1andC_j are inαbutC_k ∈ −αfori≤k < j; see Figure 3.6. Let φbe the folding with imageα. If we modifyΓ by applyingφ to the portionC_i, . . . , C_j−1, we obtain a pregallery with the same extremities that has exactly two repetitions:

C0, . . . , Ci−1, φ(Ci), . . . , φ(Cj−1), Cj, . . . , Cd.

So we can delete C_i−1 and C_j to obtain a gallery Γ of length d−2. The types ofΓ is (s₁, . . . ,sˆ_i, . . . ,sˆ_j, . . . , s_d) becauseφis type-preserving.

Lemma 3.71.The action of W is simply transitive on the chambers of Σ.

3.4 Roots 141

α

φ

−α Ci

Cj C_j−1 C_i−1

Fig. 3.6. A geometric proof of the deletion condition.

Proof. We have already noted that the action is transitive. To prove that the stabilizer ofCis trivial, note that ifwC=CthenwfixesCpointwise, sincew is type-preserving. But thenw= 1 by the standard uniqueness argument.

It follows from Lemma 3.71 that we have a bijection W → C(Σ) given by w → wC. This yields the familiar 1–1 correspondence between galleries starting atCand wordss= (s₁, . . . , s_d), where the gallery (C_i) corresponding to sis given byCi:=s1· · ·siC fori= 0, . . . , d. In view of Lemma 3.68, the type of this gallery is the wordsthat we started with. So a direct translation of Lemma 3.70 into the language of group theory yields the deletion condition for (W, S). Consequently:

Lemma 3.72.(W, S)is a Coxeter system.

Remark 3.73.Another way to prove that (W, S) is a Coxeter system is to verify condition (A) of Chapter 2 by using the action ofW on the set of roots ofΣ. Indeed, Lemma 3.66 implies that every panel ofΣisW-equivalent to a face ofC. Hence every reflection ofΣisW-conjugate to an element ofS. This shows that the “reflections” inW, in the sense of Definition 2.1, are precisely the reflections of Σ obtained from the theory of foldings. We can therefore identify the set T used in Chapter 2 with the set of reflections ofΣ, and we can identifyT×{±1}with the set of roots ofΣ. The action ofW on the roots therefore yields an action ofW onT× {±1}with the properties required for condition (A). Details are left to the interested reader.

For the next lemma, we need a simplicial analogue of the concept of “strict fundamental domain” (Definition 1.103).

Definition 3.74.If a groupGacts on a simplicial complex∆, then we call a set of simplices∆⊆∆asimplicial fundamental domainif∆is a subcomplex of∆ and is a set of representatives for theG-orbits of simplices.

(This yields a strict fundamental domain |∆| for the action of G on the geometric realization|∆|.)

Lemma 3.75.The subcomplexC:=Σ_≤Cis a simplicial fundamental domain for the action ofW onΣ. Moreover, the stabilizer of the face ofC of cotypeJ is the standard subgroupWJ of W.

Proof. The first assertion follows from the transitivity ofW on the chambers, together with the fact that W is type-preserving. To prove the second, letA be a face of C and letτ(A) =SJ. It follows from the definition ofτ that J is the set of elements ofS that fixApointwise. In particular, the subgroup WJ stabilizesA. To prove thatWJ is the full stabilizer, supposewA=A. We will show by induction on l(w) thatw∈WJ. We may assume w= 1, so we can write w=sw withs∈S andl(w)< l(w). Our correspondence between words and galleries now implies that there is a minimal gallery of the form C, sC, . . . , wC. By Lemma 3.69, then, the wallH corresponding tosseparates C fromwC.

Letαbe the root bounded byH that contains C. Then wC ∈ −α=sα, so we havewC∈α. The equationwA=Anow yields

wA=sA∈α∩sα=H ,

hence A ∈ H and wA = A. We therefore have s ∈ J [because s fixes A pointwise] and w∈WJ by induction; thusw=sw∈WJ. We have now done all the work required to complete the proof of the theorem.

Proof of Theorem 3.65 (end). Recall that we have assumed that every pair of adjacent chambers inΣis separated by a wall, and we are trying to prove that Σ is a Coxeter complex. By Lemma 3.72, we have a Coxeter system (W, S), and Lemma 3.75 easily yields an isomorphism Σ ∼= Σ(W, S). Thus Σ is a

Coxeter complex.

Example 3.76.Let Σ be the plane tiled by equilateral triangles. It is geo- metrically evident that we can construct, for any adjacent chambersC, C, a folding takingC toC. SoΣ is indeed a Coxeter complex, as claimed in Ex- ample 3.7. To see that the Coxeter groupW is the one given in that example, one can compute the orders of pairwise products of fundamental reflections, or one can observe that the link of every vertex is a hexagon.

The last assertion of Lemma 3.69 is the analogue of a fact that we used many times in Chapter 1, giving two different ways of computing the distance between two chambers. The final result of this section generalizes this to arbitrary simplices. Recall that one can talk about the gallery distanced(A, B) between arbitrary simplices (Section A.1.3).

3.4 Roots 143 Definition 3.77.We say that a wall H strictly separates two simplices if they are in opposite roots determined by H and neither is in H. We denote byS(A, B) the set of walls that strictly separate two simplicesAandB.

Proposition 3.78.For any two simplicesA, B in a Coxeter complex Σ, we have

d(A, B) =|S(A, B)|,

i.e.,d(A, B)is equal to the number of wallsH that strictly separateAfromB.

More precisely, the walls crossed by any minimal gallery from A to B are distinct and are precisely the walls in S(A, B).

Proof. A proof from the point of view of the Tits cone was sketched in Sec- tion 2.7. Here is a combinatorial proof: LetΓ:C0, . . . , Cdbe a minimal gallery from AtoB. Then it is also a minimal gallery fromC0 toCd, so it crossesd distinct walls, and these are the walls separatingC0fromCd. It is immediate that S(A, B) ⊆ S(C₀, C_d), so Γ crosses all the walls in S(A, B). We must show, conversely, that every wallH crossed byΓ is inS(A, B). Suppose not.

Then there is a root αbounded byH that contains bothA andB. But then we can get a shorter gallery fromAtoB by applying the folding ofΣontoα.

This contradicts the minimality of Γ.

We close this section by making some remarks that will be useful later, concerning links. Given a simplex Ain a Coxeter complex Σ, recall that its link Σ := lk_ΣA is again a Coxeter complex (Proposition 3.16). We wish to explicitly describe its walls and roots. SupposeH is a wall ofΣcontainingA, and let ±α be the corresponding roots. Then one checks immediately from the definitions that H := H ∩Σ is a wall of Σ, with associated roots

±α :=±α∩Σ.

Proposition 3.79.The function H →H :=H∩Σ is a bijection from the set of walls ofΣcontainingAto the set of walls ofΣ. Similarly, the function α →α := α∩Σ is a bijection from the set of roots of Σ whose boundary containsA to the set of roots of Σ.

Proof. It suffices to prove the first assertion. Since a wall of Σ is completely determined by any panel that it contains, we can reformulate the assertion as follows: For any panel P of Σ, there is a unique wall H of Σ with A∈ H and P ∈ H ∩Σ. Equivalently, there is a unique wall of Σ containing the simplex P :=P∪A. [The equivalence follows from the fact that walls are full subcomplexes by Lemma 3.54.] Since P is a panel ofΣ, the proposition

is now immediate.

Remark 3.80.Recall that we may identify Σ with Σ_≥A viaB →B∪A for B ∈ Σ, and B →BA for B ∈Σ_≥A. If we make this identification, then the bijections in the proposition are still given by intersection. In other words, ifH is a wall ofΣ containingA andH :=H∩Σ, then

{B ∈Σ_≥A|(BA)∈H}=H∩Σ_≥A, and similarly for roots.

Exercises

Assume throughout these exercises that Σis a Coxeter complex.

3.81.Let H be a wall with associated roots ±α, and let A be an arbitrary simplex. Show that A∈H if and only if there are chambersC, C≥Awith C∈αandC∈ −α.

3.82.With the notation of the previous exercise, if A ∈ H show that the chambers C, C can be taken to be adjacent. Thus there is a panel P in Σ such thatA≤P ∈H.

3.83.Assume thatΣis infinite.

(a) Show thatΣhas infinitely many walls.

(b) Assume that Σ is irreducible (i.e., its Coxeter diagram is connected).

For every vertex x of Σ, show that there are infinitely many walls not containingx. [See Lemma 2.92 for the same result expressed in terms of the Tits cone.]