The Chamber Graph

Finite Reflection Groups

1.4 Cell Decomposition

1.4.9 The Chamber Graph

Definition 1.50.Two chambers C andC are adjacent if they are distinct and have a common panelA.

Note thatCandCare then the two chambers havingAas a face, and their sign sequences differ in exactly one position. Thus the hyperplaneH := suppA is the unique element ofHseparatingCfromC; in particular,d_H(C, C) = 1, where d_H is our metric on the setC :=C(H) of chambers (Definition 1.39).

Moreover, A = C∩C. [One can prove this last assertion by a dimension argument, or by looking at sign sequences, or simply by checking the definition of C∩C above.] We will often say, in this situation, that “C and C are adjacent along the wallH.”

Example 1.51.IfHis the braid arrangement (Section 1.4.7), then chambers are labeled by permutations, viewed as lists of numbers. Two chambers are adjacent if and only if the lists differ by the interchange of two consecutive elements. See Figure 1.7.

Definition 1.52.The chamber graph associated with His the graph whose vertex set is the set C of chambers, with an edge joining two chambersC, C if and only if they are adjacent.

We can visualize the chamber graph by putting a dot in each chamber and an edge cutting across each panel, as in Figure 1.8. We will sometimes draw the schematic diagram

C C

H

(1.8)

1.4 Cell Decomposition 31

Fig. 1.8.The chamber graph is a hexagon.

to indicate that C and C are adjacent along H. The horizontal line is in- tended to suggest an edge in the chamber graph, and the dashed vertical line represents the wall that is crossed in going fromC toC.

There is a canonical metric on the vertices of any graph, where the dis- tance between two vertices is the minimal length of a path joining them. The usual convention is that the distance is∞ if the two vertices are in different connected components. But we will see below that the chamber graph is in fact connected and that moreover, the graph metric on the set of chambers coincides with the metricd_H of Definition 1.39. Before proceeding to this, we introduce some terminology that we will be using throughout the book.

Definition 1.53.A path in the chamber graph is called a gallery. Thus a gallery is a sequence of chambers Γ = (C0, C1, . . . , Cl) such that consecutive chambers C_i−1 and C_i (i= 1, . . . , l) are adjacent. The integerl is called the length ofΓ. We will write

Γ:C₀, . . . , C_l

and say that Γ is a gallery from C0 to Cl, or that Γ connects C0 and Cl. The minimal lengthl of a gallery connecting two chambersC, Dis called the gallery distance betweenC and D and is denoted d(C, D). Finally, a gallery C=C₀, . . . , C_l=Dof minimal lengthl=d(C, D) is called aminimal gallery from C toD. This is the same as what is commonly called a geodesic in the chamber graph.

Once we have proven thatd =d_H, we will no longer need the notation d_H, nor will we need to refer to the distance as “gallery distance,” though we may still do so occasionally for emphasis.

We sometimes represent a gallery schematically by means of a diagram Γ: C0 C1 C2 · · · Cl ,

which may be further decorated with hyperplanes as in the diagram (1.8).

Warning. In some of the literature, including the precursor [53] of the present book, galleries are defined more generally to be sequences as above in which consecutive chambers are either equal or adjacent. Such sequences do come up naturally, as we will see, and we will call them pregalleries. A pregallery can be converted to a gallery by deleting repeated chambers.

We noted above that the metricd_H of Definition 1.39 has the property thatd_H(C, C) = 1 ifC andC are adjacent, i.e., if they are connected by an edge in the chamber graph. This motivates the following:

Proposition 1.54.The chamber graph is connected, and the gallery dis- tance d(C, D)is equal tod_H(C, D) for any two chambersC, D.

The crux of the proof is the following result:

Lemma 1.55.For any two chambersC=D, there is a chamberC adjacent toC such that d_H(C, D) =d_H(C, D)−1.

Proof. Since C is defined by its set of walls (Proposition 1.32), there must be a wall of C that separatesC from D. [Otherwise, we would haveD⊆C, contradicting the fact that distinct chambers are disjoint.] LetA be the cor- responding panel ofC, and letC be the projectionAD(Section 1.4.6). Then C is adjacent toC, andd_H(C, D) =d_H(C, D)−1.

Proof of the proposition. Given two chambersC, D, we may apply the lemma finitely many times to obtain a gallery of length d_H(C, D) fromC to D. In particular, the chamber graph is connected andd≤d_H. To prove the opposite inequality, consider a gallery

C=C0, C1, . . . , Cl=D

of minimal lengthl=d(C, D). Thend_H(Ci−1, Ci) = 1 fori= 1, . . . , l, whence

d_H(C, D)≤l.

Given a minimal galleryC =C0, . . . , Cl =D, let H1, . . . , Hl∈ H be the hyperplanes such that C_i−1 and Ci are adjacent along Hi. [Warning: This notation has nothing to do with our original indexing of the elements ofHas {H_i}_i∈I; we will have no further need for that indexing.] We will refer to the H_i as the “walls crossed” by the gallery. Since exactly one component of the sign sequence changes as we move from one chamber to the next, and since exactly l =d(C, D) signs must change altogether, it is clear that H₁, . . . , H_l are distinct and are precisely the elements of H that separate C from D.

Conversely, suppose we have a gallery fromC to D that does not cross any wall more than once. If k is the length of the gallery, then exactly k signs change, sok=l and the gallery is minimal. This proves the following:

Proposition 1.56.A gallery from C to D is minimal if and only if it does not cross any wall more than once. In this case the walls that it crosses are

precisely those that separateC from D.

1.4 Cell Decomposition 33 Since the setC=C(H) of chambers is a metric space, it has a well-defined diameter, which we will also refer to as the diameter of Σ; by definition, it is the maximum distanced(C, D) between two chambersC, D. The following result is immediate from the interpretation of the metric onC asd_H:

Proposition 1.57.The diameter of C is m := |H|. For any chamber C, there is a unique chamberDwithd(C, D) =m, namely, the opposite chamber

D=−C.

Observe that for any chambersCandD,

d(C, D) +d(D,−C) =m . (1.9) Indeed, every hyperplane in H separates D from either C or −C, but not both. Thus if we concatenate a minimal gallery fromC toD with a minimal gallery fromDto−C, we get a minimal gallery fromCto−C. Consequently:

Corollary 1.58.For any chambersC, D, there is a minimal gallery from C

to−C passing through D.

We have confined ourselves so far to distances and galleries between cham- bers. But it is also possible to consider distances and galleries involving cells other than chambers. The basic facts about these are easily deduced from the chamber case via the theory of projections (Section 1.4.6); see Exercises 1.61 and 1.62 below.

Exercises

1.59.LetC be a chamber.

(a) IfAis a cell that is not a chamber, show that AC is not opposite toC.

(b) Conversely, ifD is any chamber not oppositeC, then D=AC for some panelAofD.

1.60.Arguing as in the proof of Proposition 1.54, prove the following criterion for recognizing the distance function on a graph. LetGbe a graph with vertex setV, and letδ:V ×V →Z+be a function, whereZ+is the set of nonnegative integers. Call two verticesincident if they are connected by an edge. Assume:

(1)δ(v, v) = 0 for all verticesv.

(2) Ifv andv are incident, then|δ(v, w)−δ(v, w)| ≤1 for all verticesw.

(3) Given verticesv=w, there is a vertexvincident tovsuch thatδ(v, w)<

δ(v, w).

ThenGis connected, andδis the graph metric.

1.61.GivenA, C∈Σ withCa chamber, consider galleries C₀, . . . , C_l=C

with A ≤ C₀. Such a gallery will be said to connect A to C. Show that a gallery fromAtoCof minimal length must start withC₀=AC. Deduce that the minimal length d(A, C) of such a gallery is |S(A, C)|, where S(A, C) is the set of hyperplanes in Hthat strictly separate A from C. [A hyperplane is said to strictly separate two subsets if they are contained in opposite open half-spaces.]

1.62.More generally, given any two cells A, B ∈ Σ, consider galleries Γ of the form

C₀, . . . , C_l

with A≤C0 andB ≤Cl. In other words,Γ is a path in the chamber graph starting in C_≥A and ending in C_≥B. Show that the minimal length d(A, B) of such a gallery is|S(A, B)|, whereS(A, B) has the same meaning as in the previous exercise. More concisely,

d(C_≥A,C_≥B) =|S(A, B)|,

where the left side denotes the usual distance between subsets of a metric space. Moreover, the chambers C0 that can start a minimal gallery are pre- cisely those havingABas a face.

A glance at Dress–Scharlau [97] is illuminating in connection with the previous exercise.

1.63.Generalize Corollary 1.58 as follows: For any cell A and chamber D, there is a minimal gallery fromAto−Apassing throughD.

1.64.Proposition 1.57 can be viewed as giving a characterization of the cham- ber−Copposite a given chamberCin terms of the metric onC. In this exercise we extend that characterization to arbitrary cells.

(a) Fix a cellA∈Σ, and consider the maximum value ofd(A, B), asBvaries over all cells. Show thatd(A, B) achieves this maximum value if and only ifB≥ −A.

(b) Deduce that for anyB∈Σ, we haveB=−Aif and only if dimB≤dimA andd(A, B) = max{d(A, B)|B ∈Σ}.

1.65.LetD be a nonempty set of chambers. Show that the following condi- tions are equivalent:

(i) For anyD, D ∈ D, every minimal gallery fromDtoDis contained inD. (ii)D is the set of chambers in an intersection of half-spaces bounded by

hyperplanes inH.