Coxeter Complexes
3.3 Construction of Chamber Maps
where the asterisk denotes the join operation. [Recall that thejoin ∆ of two simplicial complexes ∆ and ∆ with vertex sets V and V has vertex set equal to the disjoint union V V and has one simplex A ∪A for every A ∈ ∆ and A ∈ ∆. From the poset point of view, then, ∆ is simply the Cartesian product of∆ and∆. But its geometric realization|∆|isnot the Cartesian product |∆| × |∆|; in fact, ∆ does not even have the right dimension for this to be true.]
Returning to the question whether a reducible Coxeter group can yield a Coxeter complex that is a manifold, the essential point is that the join of two manifolds that are not spheres is generally not a manifold. [But the join of two spheres is a sphere.]
This discussion suggests that the Coxeter complexΣ(W, S) is not always the “best” geometric model for a Coxeter group W. For example, it would seem more reasonable to use a product of Euclidean spaces rather than a join of Euclidean spaces in the case of a reducible Euclidean reflection group.
The result is a cell complex whose cells are products of simplices rather than simplices. We will return to this circle of ideas in Chapter 12.
3.3 Construction of Chamber Maps 125 The compatibility condition, then, is that φ takes s-equivalent chambers to φ(s)-equivalent chambers.
Here, now, is the precise result:
Lemma 3.31.Endomorphisms φ of Σ are in 1–1 correspondence with pairs (φ, φ), where φ is a function W → W, φ is a permutation of S, and φ(ws) =φ(w)orφ(w)φ(s)for allw∈W ands∈S.
Proof. Let φ be an endomorphism of Σ. Then the restriction of φ to the chambers yields a function φ: W →W. (Recall that the chambers are the singleton standard cosets and are identified with the elements ofW.) We also have a type-change map φ∗ (Proposition A.14), which is a bijection φ :=
φ∗:S→S. Thenφtakess-adjacent chambers toφ(s)-equivalent chambers, i.e.,φ(ws) =φ(w) orφ(w)φ(s).
Note thatφis completely determined by the pair (φ, φ). For ifA:=wWJ
is an arbitrary simplex ofΣ, thenAis the face of cotypeJ of the chamberw;
soφ(A) must be the face ofφ(w) of cotypeφ(J); in other words,φ(wWJ) = φ(w)Wφ(J).
Finally, we must show that every pair (φ, φ) as in the statement of the lemma arises from an endomorphismφ. To this end we simply defineφ, as we must, byφ(wWJ) =φ(w)Wφ(J). It is easy to check thatφis a well-defined chamber map that inducesφ on the chambers andφ on the types.
The next two subsections illustrate the lemma.
3.3.2 Automorphisms
Recall that theW-action onΣis simply transitive on the chambers; in partic- ular, this action is faithful, in the sense that the corresponding homomorphism W →AutΣis injective. Here AutΣdenotes the group of simplicial automor- phisms ofΣ.
Proposition 3.32.The image ofW →AutΣis the normal subgroupAut0Σ consisting of the type-preserving automorphisms ofΣ.
(This shows, for the second time, that W is determined up to isomorphism by its Coxeter complexΣ.)
Proof. We already know thatW acts as a group of type-preserving automor- phisms ofΣ. Conversely, supposeφis an arbitrary type-preserving automor- phism, and letφandφbe its “components” as in Lemma 3.31. Thenφis the identity, soφ(ws) =φ(w)sfor allwands. [The possibilityφ(ws) =φ(w) is excluded becauseφis an automorphism.] It follows easily thatφ(w) =φ(1)w for allw, soφis left multiplication byw1:=φ(1), and henceφis given by the action of w1. This proves everything except the normality of Aut0Σ, which
is left as an exercise.
There is a second obvious source of automorphisms ofΣ. Namely, there is a homomorphism Aut(W, S) → AutΣ, where Aut(W, S) is the group of automorphisms ofW stabilizingS; for such an automorphism takes standard cosets to standard cosets and hence induces an automorphism ofΣ.
Proposition 3.33.The homomorphism Aut(W, S)→ AutΣ just defined is injective, and its image is the group Aut(Σ, C) consisting of the automor- phisms ofΣ that stabilize the fundamental chamber C= 1.
Proof. Givenf ∈Aut(W, S), its imageφ∈AutΣhas componentsφ =f and φ :=f|S. This shows that the homomorphism is injective. Andφstabilizes C because f(1) = 1. Conversely, suppose we are given φ∈ Aut(Σ, C), and let φ, φ be its components. Thenφ is a bijection satisfying φ(1) = 1 and φ(ws) = φ(w)φ(s). It follows that φ(s1· · ·sd) = φ(s1)· · ·φ(sd) for all s1, . . . , sd ∈ S. This implies that φ is a homomorphism, hence an automor- phism, and thatφ(s) =φ(s) for alls∈S. Thusφ is in Aut(W, S) andφis
its image in Aut(Σ, C).
Remark 3.34.The group Aut(W, S) is quite easy to understand, in view of the Coxeter presentation of W: An element of this group is determined by giving a permutationπofSthat is compatible with the Coxeter matrix, in the sense that m(π(s), π(t)) =m(s, t) for all s, t∈S. More concisely, Aut(W, S) is simply the group of automorphisms of the Coxeter diagram of (W, S).
Exercises
3.35.Show that the full automorphism group ofΣis the semidirect product Aut0ΣAut(Σ, C). Hence AutΣ∼=W Aut(W, S).
3.36.SupposeW is an irreducible finite reflection group. By looking at the list in Section 1.5.6 of possible Coxeter diagrams, show that with one excep- tion, Aut(W, S) is either trivial or of order 2. [The exception is the group of type D4.] So, with one exception,W is either the full automorphism group of Σ or a subgroup of index 2.
3.37.Specialize now to the case that W is the group of symmetries of a regular solid X, and note (again by looking at the list) that Aut(W, S) is of order 2 if and only ifX is self-dual. Explain this geometrically. More precisely, explain why an isomorphism from X to its dual induces a “type-reversing”
automorphism ofΣ.
3.3.3 Construction of Foldings
As a final illustration of Lemma 3.31, we will construct maps that, intuitively,
“fold Σonto a half-space along a wall.” The significance of this will become clear in the next section.
3.3 Construction of Chamber Maps 127 Proposition 3.38.Let C1 and C2 be adjacent chambers of Σ = Σ(W, S).
Then there is an endomorphismφof Σ with the following properties:
(1)φis a retraction onto its imageα.
(2)Every chamber inαis the image of exactly one chamber not in α.
(3)φ(C2) =C1.
To constructφ, we may assume that C1 is the fundamental chamberC, in which caseC2is necessarilysCfor somes∈S. Before beginning the proof based on Lemma 3.31, we remark that there is a very short proof that uses the Tits cone instead of the proposition. Namely, identify Σ with the set of cells in the latter, and letα+(resp.α−) be the set of cells in the closed half- spaceU+(s) (resp.U−(s)), where the notation is that of Sections 2.5 and 2.6.
Then we can takeφto be the map given by the reflectionsonα− and by the identity onα+. It is well defined becausesis the identity onα+∩α−, which consists of the cells inHs.
But we will give a purely combinatorial proof using Lemma 3.31. The crux of the proof is the next lemma, which constructs the φ component of the desiredφ(still assuming that C1 =C andC2=sC). Recall, for motivation, that there are two possibilities for an elementw∈W: eitherl(sw) =l(w)−1 or l(sw) = l(w) + 1. In the first case, w admits a reduced decomposition starting with s, so there is a minimal gallery of the formC, sC, . . . , wC. We therefore expect that there is a “wall” that separatesCfromsC, and this wall should also separate C from wC. Thus we should havewC /∈αin this case.
In the second case, there is a minimal gallery of the formC, sC, . . . , swC. So we expect thatswCis not inαbut that its “mirror image”wCis inα. These considerations motivate the following lemma and its proof:
Lemma 3.39.Fix s ∈ S. Then there is a function φs:W → W with the following properties:
(1)φis a retraction onto its imageαs, which consists of the elementsw∈W such that l(sw) =l(w) + 1.
(2)Each element of αs is the image under φs of exactly one element of the complementαs.
(3)The left-translation action ofs onW interchanges the setsαsandαs. (4)For eacht ∈S, φs takes t-adjacent elements ofW to elements that are
either equal or t-adjacent.
Proof. It is clear how we should defineφs: φs(w) :=
w ifl(sw) =l(w) + 1, sw ifl(sw) =l(w)−1.
And it is immediate from this definition that (1)–(3) hold. It remains to verify (4). We will prove a more precise result, which should be plausible in view of the “folding” interpretation: Consider two t-adjacent elements w andwt for somew∈W. Then we claim:
(a) Ifwandwt are both inαsor both inαs, thenφs(wt) =φs(w)t.
(b) Ifwis in αs andwtis in αs, thenφs(w) =w=φs(wt).
Assertion (a) is immediate from the definition of φs. To prove (b), note that the assumptions imply that l(sw) =l(w) + 1 and l(swt) = l(wt)−1. This implies, first, that l(wt) =l(w) + 1. For we have
l(wt) =l(swt) + 1≥l(sw) =l(w) + 1.
We can now apply the folding condition (F) of Section 2.3.1 to conclude that swt=w; henceφs(wt) =swt=w, as claimed.
Remark 3.40.The proof explains why we called condition (F) the folding condition.
Proof of Proposition 3.38. Assuming still thatC1 =C andC2 =sC, we can set φ = φs and φ = idS. Everything should be clear now, except perhaps for (1), which can be expressed by saying thatφisidempotent, i.e., thatφ2= φ. But φ2 and φare type-preserving chamber maps that agree on chambers;
hence they agree on all simplices.