Essays on Coxeter groups Geometric root systems
Bill Casselman
University of British Columbia [email protected]
Contents
1. Hyperplane partitions 2. Geometric root configurations 3. Coxeter groups and Cartan matrices 4. Root configurations and Coxeter groups 5. Consequences
6. Galleries 7. Subsystems
8. The Cartan matrix of a finite system 9. More about Cartan representations
10. Generators and relations for the Weyl group 11. Consequences for integral root systems 12. Remarks about generalizations
13. References
Associated to any finite integral root system (V, Σ, V∨, Σ∨) are two important partitions of V by hyper- planes. The first is that of hyperplanes λ∨= 0 for λ∨in Σ∨. This is the one we are primarily interested in. However, also important is the partition by affine hyperplanes λ∨ = k in V where k is an integer.
This affine configuration occurs in the structure theory of compact groups (in the Stiefel diagram), that of reductive groups over p-adic fields, and that of loop groups. Any of these configurations is stable under Euclidean reflections in the hyperplanes of the partition. Many of the properties of root systems are a direct consequence of the geometry of hyperplane arrangements stable under reflections rather than the algebra of roots, and it is useful to isolate geometrical arguments.
Our principal goal in the first several sections is to show that the connected components of the com- plement of the hyperplanes in either of these situations are open fundamental domains for the group generated by the reflections in the walls in one of these connected components, and to relate geometric properties of this partition to combinatorial properties of this group. But I shall work in a somewhat more general situation, one of relevance as well to the theory of certain infinite-dimensional Lie algebras and groups—instead of partitions of all of Euclidean space, I’ll consider partitions of an open convex cone in a finite-dimensional vector space. This causes very little extra work, and has the virtue of forcing us to be careful.
The principal result is the equivalence of certain geometric partitions and the realizations of Coxeter systems. After that is proved, geometry will be applied to verify properties of Coxeter systems.
Much of the substance of material here, as well as its spirit, can be found in [Vinberg:1971].
Contents
1. Hyperplane partitions
In this preliminary section I shall look more generally at an arbitrary partition of an open convex cone associated to a locally finite family of hyperplanes. This will be a simple exercise in convexity.
Thus suppose for the moment V to be any finite-dimensional real space, C an open convex cone in it (possibly V itself, possibly an open half-space of V ). I recall that a coneis a set invariant under multiplication by positive scalars. An open cone will contain 0 if and only if it is all of V . A collection Hof linear hyperplanes I’ll call apartitionofC if every hyperplane in the collection intersects C and the collection is locally finite inC.
Of course any finite collection of hyperplanes in a real vector space will partition it. For a more interesting example, if we are given a locally finite collection of affine hyperplanes partitioning V itself, it corresponds to a partition of an open half-space by linear hyperplanes in one dimension higher. This is done by embedding V into V ⊕ R, v 7→ (v, 1), and associating to every hyperplane H in V the union of all lines through the origin and (v, 1) with v in H.
A connected component C of the complement of the hyperplanes in H inC will be called achamber. If H is in H then C will be contained in exactly one of the two open half-spaces determined by H, since by definition C cannot intersect H.
Let DH(C) be the intersection of this half space with C. It is convex and open. The following is extremely elementary.
Lemma 1.1.If C is a chamber then
[allh]
C = \
H∈H
DH(C) .
Proof. Of course C is contained in the right hand side. On the other hand, suppose that x lies in C and that y is contained in the right hand side. If H is in H then the closed line segment [x, y] cannot intersect H, since then C and y would lie on opposite sides. So y lies in C also.
Many of the hyperplanes in H will be far away from C, and they can be removed from the intersection without harm. Intuitively, only those hyperplanes that border C closely need be considered, and the next result, only slightly less elementary, makes this precise.
A panel of C is an open face of C of codimension one, a subset of codimension one in the boundary of C. A point lies on a panel of C if it lies on exactly one hyperplane in H as well as in the closure of C. Equivalently, a point x lies on a panel if and only if there exists a linear function f and a neighbourhood U of x such that (a) f (x) = 0; (b) the region {f > 0} ∩ U is contained in C; (c) the region {f < 0} ∩ U is disjoint from C.
The linear support of a panel will be called awall. A panel with support H is a connected component of the complement of the union of the H∗ ∩ H as H∗ runs through the other hyperplanes of H. All faces of a chamber, including panels, are convex.
A chamber might very well have an infinite number of panels, for example if H is the set of tangent lines to the parabola y = x2at points with integral coordinates.
Proposition 1.2.If C is a chamber then
[chambers]
C = \
H∈HC
DH(C)
where now HCis the family of walls of C.
That is to say, there exists a collection of affine functions f such that C is the intersection of the regions f > 0, and each hyperplane f = 0 for f in this collection is a wall of C.
Proof. Suppose y to be in the intersection of all DH(C) as H varies over the walls of C. We want to show that y lies in C itself. Suppose that we can find x in C with the property that the line segment from y to x intersects hyperplanes of H only in panels. If it in fact intersects any hyperplanes at all, then the last one (the one nearest x) will be a panel of C whose wall separates C from y, contradicting the assumption on y. Therefore it intersects no hyperplanes, and y and x are both in C.
It remains to find such a point x. Choose a small closed ball contained in C, and then another one containing y and contained inC. Let B be the convex hull of these two balls. It is relatively compact and contained in the interior ofC. Since the collection of hyperplanes is locally finite in the interior of C, B intersects only a finite subset {Hi} of them. Let k be the collection of intersections Hi∩ Hj of pairs of these. The intersection of B with these intersections will be relatively closed in B, of codimension two (possibly empty). Let S be the union of all lines through y and a point in k, whose intersection with B will be of codimension one. Its complement in B will be open and dense. If x is a point of C in this complement then the intersection points of the line segment from x to y will each be on exactly one hyperplane in H, hence on a panel.
Lemma 1.3.The number of H in H separating two chambers is finite.
[separating-finite]
Proof. A closed line segment connecting them is compact as well as contained in the convex setC. It can therefore meet only a finite number of the H since they are locally finite inC.
2. Geometric root configurations
Let V be a finite-dimensional real vector space, V∨its linear dual. Ageometric root configurationCin V∨consists of
• an open convex cone C in V∨;
• a hyperplane partition H of C;
• a chamber C of H;
• for each wall H of C a reflection sHin H such that
• C is a polyhedral cone whose panels are contained in C;
• both C and the family of hyperplanes in H are stable under each sH.
I’ll call the configurationsimplicialif C is a simplicial cone, and in this case refer to asimplicial root decomposition. Each s = sHis determined by a pair (αs, α∨s) such that
s: v∨7−→ v − hαs, v∨iα∨s α(C) > 0 .
Since s is a reflection, hαs, α∨si = 2. This pair is unique up to simultaneous scalar multiplication by c > 0, 1/c. Let ∆ = ∆Cbe the set of all α in V such that H = ker(α) bounds C and α > 0 on C, and for each α in ∆ let α∨be the unique vector in V∨determining sHα. By Proposition 1.2
♣[chambers]
C = {v∨| hα, v∨i > 0 for all α ∈ ∆} ,
so specifying C and the reflections in its walls is equivalent to specifying ∆ and the map
∆ → V∨: α 7→ α∨.
A reflection s together with one side of its mirror H is called anoriented reflection. Specifying one of these is equivalent to specifying the set of all α with Hα = ker(α) together with the map α 7→ α∨ determining s. In other words, the configuration is given a set of oriented reflections in the walls of a cone.
In an integral root system the linear function α defining a root hyperplane is determined up to a factor of two by integrality conditions, but in a geometric root system there is no such determination—no good notion of analgebraic root. I define a(geometric) rootof the configuration to be one of the half-spaces defined by one of its hyperplanes, or equivalently the set of all α with ker(α) = H and α≥ 0 on the half space. A root α ispositiveif the half-space contains C, or equivalently ifhα, Ci > 0.
The group W generated by the reflections sHis called theWeyl groupof the configuration.
The case most commonly considered is that where the hyperplanes in H are the root hyperplanes of a finite root system. In this case there exists a W -invariant Euclidean norm, and the reflection sH is the unique orthogonal one fixing the elements of H.
Another important case is the partition of a half-space in dimension n + 1 by a partition of a space of dimension n by affine hyperplanes. The initial structure is an affine hyperplane partition of a Euclidean space V , with each sH orthogonal, and the partition stable under all sH. This translates to a root configuration in V ⊕ R. The formula for an affine reflection is
v 7−→ v − (hα, vi + k) α∨
withhα, α∨i = 2, and the corresponding linear reflection in dimension one greater is v 7−→ v − hF, vif
where
F, (v, c) = hα, vi + kc, f = (α, 0) . The function F in the first term is also (α, kδ) wherehδ, (v, c)i = c.
We have seen two-dimensional examples already, in the discussion of reflections. Here are a few more of higher rank.
First pack the plane z = 1 by equilateral triangles. Then pack the half-plane z > 0 by the tetrahedral cones slicing through them. The figure at right shows the set of lines λ = k for roots λ of SL3. Other rank two root systems give other packings of the plane. The irreducible ones give rise to packings by triangles, but the reducible ones give rise to packings by rectangles.
The planes of symmetry of any regular polyhedron in Eu- clidean space form a Euclidean root configuration. Here are shown the planes of symmetry of a dodecahedron and the corresponding triangulation.
A hexagonal root configuration in non-Euclidean geometry.
This is the realization in the Poincar´e disk of a linear root configuration in the cone Q(a, b, c) =−a2−bc > 0, b−c > 0.
This quadratic form may be identified with the determinant of matrices
a b c −a
of trace 0 on which the group PGL2(R) acts by conjugation.
The full orthogonal group is generated by this together with the reflection (a, b, c) 7→ (−a, b, c). The spin group of this form is SL2(R).
A triangular non-Euclidean configuration, again in the the Poincar´e disk. The union of the transforms of C is different from its interiorC
Suppose given a geometric root decomposition (H,C, C, S). If F is a face of C, let VFbe its linear support.
Let HFbe the collection of hyperplanes containing VF,C + VFthe translation ofC by VF, C + VF that of C, SF the subset of reflections in the walls of C + VF. These are the same as the walls of C containing F . They are also the s in S fixing elements of F .
Proposition 2.1.These data form a geometric root configuration.
[grc-quotient]
Proof. It is immediate thatC + VF is open and convex, and that reflections in SFare stable on HF. Since C ⊆ C + VF, every H in HFcertainly intersectsC + VF. Since HF ⊆ H and H is locally finite, so is HF. This configuration determines, and is determined by, a configuration on V /VF, the quotient configuration.
3. Coxeter groups and Cartan matrices
Suppose S to be a finite set and (ms,t) a symmetric matrix of integers ms,t(possibly∞) indexed by S, subject to the conditions ms,s= 1 and ms,t≥ 2 for s 6= t. The Coxeter system associated to this matrix is the pair (W, S) where W is the group defined by generators in S and relations
s2= 1 for all s ∈ S
st . . . = ts . . . (ms,tterms on each side, for s6= t)
(The second relation is called thebraid relation.) That is to say, the group W is defined to be the set of all words w = s1. . . snmodulo the equivalence relation according to which w1≡ w2if and only if one can be obtained from the other by (a) deletion of a pair ss, (b) insertion of a pair ss, or (c) replacement of one braid st . . . by its ‘twin’ ts . . . The two relations above imply that
(st)ms,t = 1 in W . If ms,t= 3, for example,
(st)3= sts tst ≡ sts sts ≡ st ts ≡ s s ≡ 1
A Coxeter group can also be defined as that with generators in S and relations (st)ms,t = 1. For example, starting with (st)3= 1, we get
sts ≡ sts · tt ≡ sts · tsst ≡ sts · tsttst ≡ (st)3tst ≡ tst .
As we’ll see shortly, a Coxeter group W can always be realized by a linear representation in which the generators in S are represented by reflections, and in which the geometry of the representation is closely related to the combinatorics of W . Why and how this works depends essentially on what happens for groups of two generators s, t in two dimensions. Suppose s = sα,α∨ and tβ,β∨to be distinct reflections in R2. Let C be the wedge αs > 0, αt > 0. Changing αsor αtto its negative changes C, so varying αsand αtgives you a choice of four possible C. Choosing C thus amounts to an orientation of the pair of reflections. The numbers cs,t = hαs, α∨ti, ct,s= hαt, α∨si are not invariant under conjugation of the triple (s, t, C), but their product ns,tis, and so are their signs.
LetC be the interior of the union of all the wC. I call the oriented pair of reflections aVinberg pairif C is a fundamental domain of W for its action onC. This depends only on the conjugacy class of the pair.
[Vinberg:1971] shows that there are precisely three situations in which this happens:
(a) ns,t= 4 cos2(π/m) for some integer m = ms,t≥ 2 (so 0 ≤ ns,t< 4);
(b) ns,t= 4 and cs,t< 0;
(c) ns,t> 4 and cs,t< 0.
In case (a) the group W is the dihedral group of size 2m; in case (b) it is an infinite group with a subgroup of index 2 generated by a single shear; and in case (c) it is a discrete group of two-dimensional hyperbolic transformations. In effect, in case (c) we have ns,t= 4 cosh2(θ) for some θ. Each case actually occurs:
(a): cs,t= ct,s= −2 cos(π/ms,t) (b): cs,t= ct,s= −2
(c): cs,t= ct,s= −√ns,t= −2 cosh(θ) .
Given ns,t, different choices of cs,tare possible, and significantly different. The choice just laid out is called thestandardone.
ACartan matrixindexed by S and associated to (ms,t) is a matrix Γ = (cs,t) of real numbers for which each cs,s= 2 and each distinct pair satisfies Vinberg’s conditions.
Every Cartan matrix Γ gives rise to a representation ρΓof the Coxeter group with matrix (ms,t). Let V = R|S|, with basis αsfor V . Define α∨t in V∨by the condition
hαs, α∨ti = cs,t
for all s. The representation ρΓ takes s to the reflection sαs,α∨s. The braid relations follow from the remark made above about pairs of reflections, so ρ is a representation of the Coxeter group W .
Suppose Θ to be a subset of ∆, T = TΘthe subset of t∈ S with αtin Θ, and VΘ= ∩α∈Θker α .
Then WT acts trivially on VΘand its representation on V /VΘis that determined by the submatrix of Γ indexed by T . Let CΘbe the region where α > 0 for α in Θ.
Given ρΓ, let ∆ be the set of αs, C the region where αs> 0 for all s, C the interior of the union of the wC, Hthe collection of W -transforms w{α = 0} for α in ∆. We are also given a map α 7→ α∨from ∆ to V∨. Theorem 3.1.These data form a simplicial root configuration.
[cartan-configuration]
The proof will take a while, but will give us along the way a crucial relation between geometry and combinatorics. For each w in W , let ℓS(w) be the length of the smallest expression for w as a product of elements of S. For w in WT, this is to be distinguished from ℓT(w) (although we shall see eventually that they agree). We have ℓS(w) ≤ ℓT(w). We also have ℓS(xy) ≤ ℓS(x) + ℓS(y), and ℓS(sw) = ℓS(w) ± 1 for s in S. I write sw <Sw if ℓS(sw) < ℓS(w).
The crucial relation is this:
Proposition 3.2.For w in W , sw >S w if and only if αsis positive on wC.
[geom-comb]
Equivalently, sw <S w if and only if C and wC lie on different sides of α = 0. One important consequence is that α = 0 does not intersect wC.
Proof. We begin with:
Lemma 3.3. Suppose T ⊆ S. Every w in W may be expressed as w = ux such that (a) u ∈ WT, (b)
[wxy]
tx >Sx for all t in T , and (c) ℓS(w) = ℓS(u) + ℓS(x).
This result will be made more precise later on, where we discuss the cosets WT\W in more detail.
Proof of the Lemma. The following algorithm computes u and x:
u := 1 x := w
while tx < x for some t in T:
u := ut x := tx
By definition ℓS(tx) decreases in every iteration of the loop. Hence the algorithm certainly stops. When it does so, tx >S x for all t in T , and u is in WT. Since ux = ut· tx for every t in T , the product ux remains invariant in each loop, hence w = ux at the end.
It must now be verified that conditions (b) and (c) hold whenever the loop is entered. They certainly hold at the first test, so it remains to see that they are not destroyed in the loop.
Since ℓS(w) = ℓS(u) + ℓS(x) at the start, it remains to show that ℓS(u) + ℓS(x) remains invariant in every iteration. Suppose ℓS(w) = ℓS(u) + ℓS(x) at the start of a loop. Since w = ux = ut · tx, we have then ℓS(u) + ℓS(x) ≤ ℓS(ut) + ℓS(tx), or
ℓS(ut) ≥ ℓS(u) + ℓS(x) − ℓS(tx) .
By assumption ℓS(x) − ℓS(tx) = 1, so ℓS(ut) > ℓS(u), and ℓS(ut) = ℓS(u) + 1. Therefore ℓS(ut) + ℓS(tx) = ℓS(u) + 1 + ℓS(x) − 1 = ℓS(u) + ℓS(x) .
Back to the proof of the Proposition. It is true when|S| = 2, because of the observation of Vinberg referred to in the previous section.
We now prove Proposition 3.2 by induction on ℓ(w). If w = 1 there is no problem.
♣[geom-comb]
Suppose ℓ(w) > 1. If x = sw <S w it must be shown that αsis negative on wC. But then wC = sxC;
by induction αsis positive on xC, hence sxC lies in the region where αs< 0.
Now suppose sw >S w. It must be shown that αs > 0 on wC. Choose t such that tw <S w. Find u in Ws,tand x in W satisfying the conditions of the Lemma. Since tw <S w, ℓS(x) <S ℓS(w). Since sx >S x and tx >S x, induction lets us see that xC is contained in the region Cs,twhere αs> 0 and αt> 0. Since ℓS(w) = ℓS(u) + ℓS(x), ℓS(su) = ℓS(u) + 1. Since ℓS ≤ ℓT, this is still valid for ℓT. But since the Proposition is true for|S| = 2, αs> 0 on the region uCs,t, hence on wC = uxC as well.
The W -transforms of the half-spaces bounding C are called therootsof the system. A positive root is one containing C. The transforms of the hyperplanes themselves are called root hyperplanes. The positive roots are those containing C, the negative ones those disjoint from C. One consequence of Proposition
♣[geom-comb]
3.2:
Corollary 3.4.If wC = C then w = 1.
[fixC]
Hence the representation ρ is faithful.
Proof. By Proposition 3.2, because C is the intersection of the half-spaces determined by its walls.
♣[geom-comb]
Corollary 3.5.Every root is either positive or negative.
[psoneg]
If x is any point of V∨that does not lie on any root hyperplanes, it is clear what it means for a root hyperplane to separate x from C, since no root hyperplanes intersect C. If x is an arbitrary point of V∨and H a root hyperplane, choose λ so H = ker(λ) andhλ, Ci > 0. I say H separates x from C if hλ, xi ≤ 0.
If x is a point of V∨, let Λx be the λ > 0 such thathλ, xi ≤ 0. These are in bijection with the root hyperplanes separating x from C.
Lemma 3.6.Suppose x to be a point of V∨not lying on any root hyperplane, withhα, xi > 0. Then
[separating-lemma]
Λsx = {αs} ⊔ sΛx.
Proof. The hyperplanes separating sx from C are the transforms by s of those separating x from sC. If λ > 0 on sC but < 0 on C, then C and sC are adjacent with common face α = 0, so λ = α.
Proposition 3.7.The length ℓS(w) is equal to the number of root hyperplanes separating C from wC.
[lengthw]
Equivalently, it is the number of roots λ > 0 such that w−1C < 0. In particular, this number is finite.
Proof. By induction on ℓS(w). The case ℓ(w) = 0 is trivial. We can check one more case easily—if w = sα, then on the one hand there is exactly one hyperplane separating C from wC, and on the other ℓS(w) = 1. In general, the Proposition follows by induction from Lemma 3.6.
♣[separating-lemma]
Lemma 3.8.Suppose x not on any root hyperplane. If Λxis finite, then x∈ wC for some w in W .
[w-transitive]
Proof. By induction. If α = 0 separates x from C, with α ∈ ∆, then there is one less hyperplane separating sx from C than x from C.
Proposition 3.9. The points of C are those with a finite number of root hyperplanes separating them
[characterizationC]
from C.
Proof. Both halves require proof, but the proof of one half will prove the other as well.
Suppose that the number of root hyperplanes separating x from C is finite. One may adjust the segment so it crosses into C inside a panel α = 0. The line segment from x to some point inside C is thus partitioned into a finite number of segments by root hyperplanes. Let y be a point of the open end segment containing x in its closure. By the previous Lemma there exists w with wy∈ C. So we may as well assume y in C, and x in C.
Therefore the original x lies in W C. It remains to be shown that th current x now lies in its interiorC.
Suppose x to lie in CΘ. I claim that WΘis finite. If x belongs to a panel, this is clear. If not, let H be a wall of C containing x. Since the subgroup of W fixing H has order two, WΘ{H} is an infinite set of root hyperplanes containing x, contradicting the initial assumption, which implied that x lay on only a finite number of root hyperplanes.
So WΘ is finite. Therefore the projection of C in V/VΘ is all of V /VΘ. Therefore the projection of x has a neighbourhood entirely contained in WΘ applied to the projection of C, which implies that x is contained in the interior of WΘC.
THe same argument will prove the other half.
Corollary 3.10.The setC is convex.
[convexityV]
Proof. Suppose x, y inC, z = (1 − t)x + ty on the line segment connecting them. For any positive root λ hλ, zi = (1 − t)hlambda, xi + thλ, yi
so that ifhlambda, zi ≤ 0 then at least one of hlambda, xi or hλ, yi must be ≤ 0.
Corollary 3.11.The partition ofC by root hyperplanes is locally finite.
[locallyfiniteV]
This concludes the proof of Theorem 3.1.
♣[cartan-configuration]
The following was proved in the course of the proof of Proposition 3.9.
♣[characterizationC]
Proposition 3.12.The group W is finite if and only ifC = V .
[wfinite]
Proposition 3.13. A subgroup of W is finite if and only if if it is conjugate to a subgroup of some finite
[wtfinite]
WT.
Proof. If G ⊆ W is finite, the average of any orbit of a point of calC will give us a point x of C fixed by G. Conjugation will bring x to a point of C, and in some WΘ. But the previous result implies that if there exists a point of CΘinC then WΘis finite.
4. Root configurations and Coxeter groups
Suppose given a simplicial root configuration H,C, C, S. Choose positive roots and coroots {α, α∨} defining S. These define a matrix M = (hα, β∨i). Changing the α by positive scalars dαwill change M to dM d−1, where d is the diagonal matrix with entries dα.
Proposition 4.1.The matrix M associated to a simplicial root configuration is a Cartan matrix.
[configuration-cartan]
Condition (a) in the specification of a Cartan matrix is a matter of definition. As for the others, they depend only on the system Hα,β. This follows from Vinberg’s result about pairs of reflections.
Theorem 4.2.Any simplicial root configuration is the one determined by its Cartan matrix.
[rootcofig-coxeter]
Proof. What must be shown is that the group W acts transitively on chambers. Suppose C∗ to be a chamber. According to Lemma 1.3, the number of hyperplanes in the partition separating C∗from C is
♣[separating-finite]
finite. The proof goes by induction on this number. But if α = 0 separates C∗from C, with α∈ ∆, then sαC∗is separated from C by one less root hyperplane from C.
5. Consequences
We know now that simplicial root configurations and representations of a Coxeter group through Cartan matrices are equivalent. We shall now use geometry to prove properties of Coxeter systems.
From Corollary 3.4 follows:
♣[fixC]
Theorem 5.1.The group W acts faithfully onC with fundamental domain C.
[Fundamental-domain]
Proposition 5.2. Suppose w in WS. If v and wv both lie in C, then wv = v and w belongs to the group
[stabilizers]
generated by the reflections in S fixing v.
Proof. By induction on ℓ(w). If ℓ(w) = 0 then w = 1 and the result is trivial.
If ℓ(w) > 1 then let x = sw with ℓ(x) = ℓ(w)−1. Then C and wC are on opposite sides of the hyperplane αs= 0, by Proposition 3.2. Since v and wv both belong to C, the intersection C∩ wC is contained in the
♣[geom-comb]
hyperplane as= 0 and wv must be fixed by s. Therefore wv = xv. Apply the induction hypothesis.
In particular, the only elements of WSfixing elements of a wall H of C are 1 and sH.
The regionC called theTits cone(although he and Vinberg seem to have discovered it independently).
An old result in new language:
Proposition 5.3.The following are equivalent:
[fd-cone]
(a) The group W is finite;
(a) The set of roots is finite;
(a) The Tits cone is all of V .
Proposition 5.4.The face CT lies in the interior of <C > if and only if WT is finite.
[face]
I call a geometric root configurationrigidif every reflection in one of its hyperplanes that stabilizes the entire configuration is in the group W generated by the reflections given by the system.
The configuration arising from a finite root system is rigid, since sHis the unique orthogonal reflection in H. But not all root configurations arise from orthogonal reflections. There is one simple geometric condition that guarantees rigidity.
Proposition 5.5.Suppose that for each s in S there exists t such that Ws,tms,t< ∞. Then the system is
[rigidity]
rigid.
Proof. Suppose σ and τ to be reflections fixing the same hyperplane H, and suppose also that they stabilizeC. Conjugating by an element of W . we may assume H is some Hs. Choose t such that
ms,t< ∞, and choose x in Hs∩ Ht. The product στ is a transvection along Hs. The hyperplanes στ Ht
all lie in H and contain x, but are not a locally finite family.
6. Galleries
Suppose given a simplicial geometric root configuration.
Agallerybetween two chambers C and C∗is a chain of chambers C = C0, C1, . . . , Cn = C∗in which each two successive chambers are distinct and share a common face of codimension 1. The integer n is called thelengthof the gallery. The gallery is calledminimalif there exist no shorter galleries between C0
and Cn. Thecombinatorial distancebetween two chambers is the length of a minimal gallery between them.
Fix a chamber C, assumed simplicial, and let S be as usual the set of reflections in its walls. Expres- sions w = s1s2. . . sn with each siin S can be interpreted in terms of galleries—each such expression determines a gallery linking C to w C. This can be seen inductively. The trivial expression for 1 in terms of the empty string just comes from the gallery of length 0 containing just C0 = C. A single element w = s1 of S corresponds to the gallery C0 = C, C1 = s1C. If we have constructed the gallery for w = s1. . . sn−1, we can construct the one for s1. . . sn in this fashion: the chambers C and snC share the wall H where sn = sH, and therefore the chambers wC and wsnC share the wall wH. The pair Cn−1 = wC, Cn = wsnC continue the gallery from C to wsnC. This associates to every expression s1. . . sna gallery. The converse is also straightforward, as can be seen by an induction argument.
There is another way to see how galleries relate to expressions in S. We know that every chamber C∗is wC for a unique w in WS. If F is a face of C, the only elements of W that take F to itself are in the group WF generated by the reflections sHin the wall containing F . If F∗is any face panel of C∗there exists a unique face F of C with some w(F ) = F∗. Therefore, we can label the faces of an arbitrary chamber by faces of the fixed chamber C. In particular, each panel is associated to a unique generator s in S. A gallery C = C0, C1, . . . therefore gives rise to a sequence (si) according to the rule that the face between Ci−1and Ciis labeled by si. Galleries of minimal length leading from C to wC correspond to reduced expressions for w.
C
wC
w = s 1s
2s 3s
2s 1s
2s 3s
2s 1s
3s 2 s1
s2 s3
s2 s1
s2
s3 s2 s1 s3 s
2
roots-images/gallery1.eps
Let
Rw= {λ > 0 | wλ, 0}
Lw= {λ > 0 | w−1λ, 0} . Each of these is a finite set because of Lemma 1.3.
♣[separating-finite]
Proposition 6.1.The following are equivalent:
[lwrw]
(a) ℓ(xy) = ℓ(x) + ℓ(y);
(b) Rxy= y−1Rx⊔ Ry; (c) Lxy= xLy⊔ Lx.
Of courseLw= Rw−1. There is a very simple interpretation ofLw. We have λ > 0 if and only if λ > 0 on C, and w−1λ < 0 if and only if λ < 0 on wC. Therefore:
Proposition 6.2.The positive root λ lies inLwif and only if λ = 0 separates C from wC .
[roots-separate]
Proposition 6.3.A minimal gallery never crosses and recrosses a root hyperplane.
[crosses-recrosses]
Proof. If a gallery first crosses to λ < 0 at sk+1and recrosses back to λ < 0 at sm+1, then s1. . . skHk+1= Hλ
s1. . . sksk+1. . . smHm+1= Hλ
sk+1. . . smHm+1= Hk+1. This means that sk+1is conjugate to sm+1:
sk+1. . . smsm+1sm. . . sk+1= sk+1
sk+1. . . smsm+1= sk+2. . . sm
s1. . . sksk+1sk+2. . . smsm+1= s1. . . sksksk+1. . . sm
= s1. . . sk−1sk+1. . . sm
which is shorter than the original.
Proposition 6.4. For any w,|Lw| is exactly the number of hyperplane crossings of any minimal gallery
[crosses-recrosses-cor]
from C to wC.
Corollary 6.5. For x and y in W , ℓ(xy) = ℓ(x) + ℓ(y) if and only if Lxyis the disjoint union of Lxand
[rw]
xLy.
Corollary 6.6.If Lx= Lythen x = y.
[rw-cor]
Finally, suppose that the configuration we are considering hasC equal to all of U. There are only a finite number of hyperplanes in the configuration, because of local finiteness at the origin. Since−C is then also a chamber:
Proposition 6.7. When C = U, there exists in WS a unique element wℓ,S of maximal length, with
[longest]
wℓC = −C. Thus swℓ,S < wℓ,Sfor all s in S. Conversely:
Ifwlies inWSandsw < wfor allsthenWSis finite andw = wℓ,S. 6.8.
[longest-cor]
Proof. Follows by induction fromProposition 3.2.
♣[geom-comb]
Corollary 6.9.For T ⊆ S, ℓT = ℓS.
[ltls]
Proposition 6.10.If α lies in ∆, thenLsα = Rsα = {α}.
[lara]
In other words, sαpermutes the complement of{α} in the positive roots.
7. Subsystems
Let CΘbe the open face of C where α = 0 for α in Θ, α > 0 for α in ∆ but not in Θ. If F is a face of any chamber, the Proposition tells u it will be W -equivalent to a unique Θ⊆ ∆. The faces of chambers are therefore canonically labeled by subsets of ∆.
If Θ is a subset of ∆, let ΣΘbe the roots which are integral linear combinations of elements of Θ. These, along with V , V∨and their image in Σ∨form a root system. Its Weyl group is the subset WΘ. Recall that to each subset Θ⊆ ∆ corresponds the face CΘof C where λ = 0 for λ∈ Θ, λ > 0 for λ ∈ ∆ − Θ.
According to Proposition 5.2, an element of W fixes a point in CΘif and only if it lies in WΘ.
♣[stabilizers]
Proposition 7.1. In each coset WΘ\W there exists a unique representative x of least length. This element
[cosets]
is the unique one in its coset such that x−1Θ > 0. For any y in WΘwe have ℓ(yx) = ℓ(y) + ℓ(x).
Proof. The region in V∨where α > 0 for α in Θ is a fundamental domain for WΘ. For any w in W there exists y in WΘsuch that xC = y−1wC is contained in this region. But then x−1α > 0 for all α in Θ. In fact, x will be the unique element in WΘw with this property.
The element x can be found explicitly. Let [WΘ\W ] be the set of these distinguished representatives, [W/WΘ] those for right cosets. These distinguished representatives can be found easily. Start with x = w, t = 1, and as long as there exists s in S = SΘsuch that sx < x replace x by sx, t by ts. At every moment we have w = tx with t in WΘand ℓ(w) = ℓ(t) + ℓ(x). At the end we have sx > x for all s in S.
I’ll set for future reference the notation I used here:
[WΘ\W ] = {w ∈ W | w; −1]((Θ) > 0}
[W/WΘ] = {w ∈ W | w(Θ) > 0} .
Here is another way to phrase this:
Corollary 7.2.Suppose Θ⊆ ∆. Every w in W can be expressed uniquely as w = xy with x ∈ WΘand
[coset-factorization]
y ∈ [W/WΘ]. In this situation
ℓ(w) = ℓ(x) + ℓ(y) .
Applying this to w−1, we get also factorizations w = yx with y∈ [WΘ\W ], x ∈ WΘ.
Next I want to do something similar for double cosets WΘ\W/WΦ. The best approach to this apparently first appeared in [Bergeron et al/:1992]. We’ll follow their argument, but we’ll need the following fact:
Lemma 7.3.Suppose Θ⊆ ∆, If wΘ > 0 then ∆ ∩ wΣ+Θ= ∆ ∩ w(Θ).
[panels]
Proof. Let DΘbe the region where α > 0 for all α∈ Θ. Then Σ+Θis set of all hyperplanes containing DΘ, and the panels of DΘare those defined by Θ.
By assumption wΣ+Θ contains a panel D of C. This means that we have a linear function f on V such that f ≥ 0 on C, f = 0 contains a panel of C, and f ≥ 0 on wDΘ. But since wΘ > 0, wDΘcontains C, so f = 0 contains a panel of wDΘ, which means that w−1D is a panel of DΘ.
Proposition 7.4.Suppose Θ, Ω⊆ ∆, x in [W/WΩ]. If we write
[doublecosets]
x = uv (u ∈ WΘ, v ∈ [WΘ\W ] , then
u ∈ WΘ∩ [W/WΘ∩v(Ω), v ∈ [WΘ\W ] ∩ [W/WΩ] . Conversely, if u and v satisfy these conditions then uv∈ [W/WΩ.
Proof. Factor x = uv as in the Lemma, with u ∈ WΘ, v−1(Θ) > 0. Since ℓ(uv) = ℓ(u) + ℓ(v), Rx= Ruv= Rv∪ v−1Ru. This implies (a) that Rv⊆ Rx⊆ Σ+− Σ+Ω, hence v(Ω) > 0 and
v ∈ [WΘ\W ] ∩ [W/WΩ] ;
(b) that v−1Ru⊆ Σ+− Σ+Ω, which implies that u v(Ω) > 0, hence in particular u Θ ∩ v(Ω) > 0 .
Now assume conversely that
u ∈ WΘ∩ [W/WΘ∩v(Ω)], v ∈ [WΘ\W ] ∩ [W/WΩ] We want to deduce that if x = uv then
xΩ > 0 . The assumptions on u mean that
u Σ+− Σ+Θ > 0, u Σ+Θ∩vΩ > 0 . We have
vΩ ⊆ Σ+= (Σ+− Σ+Θ) ∪ Σ+Θ
Ω ⊆ v−1 Σ+− Σ+Θ) ∪ v−1 Σ+Θ . But
Ω ∪ v−1Σ+Θ = Ω ∩ v−1Θ . so
vΩ ⊆ Σ+= (Σ+− Σ+Θ) ∪ Σ+Θ∩vΩ
so
uvΩ ⊆ u Σ+− Σ+Θ ∪ u Σ+Θ∩vΩ ⊆ Σ+. Let [WΘ\W/WΩ] = [WΘ\W ] ∩ [W/WΩ].
Corollary 7.5.The map taking v to WΘvWΩis a bijection of [WΘ\W/WΩ] with WΘ\W/WΩ. If we factor
[doublecosetrepresentatives]
x = uvw in WΘvWΩas in the Proposition then ℓ(x) = ℓ(u) + ℓ(v) + ℓ(w).
Corollary 7.6.If x is in [WΘ\W/WΩ] then WΘ∩ xWΩx−1= WΘ∩xΩ.
[coxeterassociates]
Maximal elements if W finite. Involutions? Associates?
For x in WΘ\W ], maximal in WΘx is wℓ,Θx. Maximal in [WΘ\W ] is wℓ,Θwℓ. Call it wℓΘ. For x in [WΘ\W/WΩ], maximal in WΘxWΩis wΘ∩xΩℓ,Θ xwℓ,Ω.
[Hohlweg-Skandera:2005]:
Proposition 7.7.If x and y are maximal, the closure of WΘyWΩcontains WΘxWΩif and only if x≤ y.
[hohlweg]
8. The Cartan matrix of a finite system
Now assume thatC = V . The family H itself is finite, and so is W . We can find a Euclidean norm with respect to which the elements of W are orthogonal. If we fix a chamber C then by•W = WS where S is
♥[rigidity-b]
♣[rigidity-b]
the set of orthogonal reflections in the panels of C.
For each wall H of C pick α such that α = 0 on H and α > 0 on C. Let ∆ be the set of all such α. It is unique up to scalar c > 0. Let α∨be the corresponding eigenvector.
The matrixhα, β∨i for α, β in ∆ is called theCartan matrixof the system (and the choice of elements of
∆). Since hα, α∨i = 2 for all roots α, its diagonal entries are all 2. According to the discussion of rank two systems, its off-diagonal entries
hα, β∨i = 2 α•β β•β
= α•β•
are all non-positive. Furthermore, one of these off-diagonal entries is 0 if and only if its transpose entry is. This equation means that if D is the diagonal matrix with entries 2/α•α and M is the matrix (α•β) then
A = M D . Proposition 8.1.The set ∆ is finite.
[walls-finite]
Proof. If V has dimension n, the unit sphere in V is covered by the 2n hemispheres xi > 0, xi< 0. By •,
♥[non-pos]
♣[non-pos]
each one contains at most one of theGRAD(α) in ∆.
The next few results of this section all depend on understanding the Gauss elimination process applied to M .
In fact, suppose for the moment that M is an arbitrary symmetric n× n matrix with m1,1 > 0. It will be the matrix ei •ejfor some inner product (not necessarily positive definite) on Rn. Here (ei) is the standard basis. Let’s look at just one step of elimination, reducing all but one entry in the first row and column to 0. Since m1,1 = e1 •e1 > 0, it replaces each vector ei with i > 1 by its projection onto the space perpendicular to e1: ei7→ e⊥i for i > 1, where
v⊥ = v − v•e1
e1 •e1
e1.
We have
v = v•e1
e1 •e1
e1+ v⊥ so that v⊥ = 0 if and only if v is a scalar multiple of e1.
If by convention I set e⊥1 = e1, the new matrix M⊥has entries e⊥i •e⊥j, and we have the matrix equation LMtL = M⊥
with L a unipotent lower triangular matrix
L = 1 ℓ
tℓ I
ℓi= −e1 •ei
e1 •e1 ≥ 0 . If M is invertible, then
M−1 =tL (M⊥)−1L .
This argument and induction have a number of consequences.
Proposition 8.2. If M is any symmetric matrix, it is positive definite if and only if Gauss elimination
[pd-criterion]
leads to an equation
LMtL = D
with L lower triangular unipotent and D a positive diagonal matrix.
Proposition 8.3.If ∆ is a subset of non-zero vectors in Rnwith αi •αj≤ 0 for all i 6= j then the vectors
[delta-ind]
in ∆ are linearly independent.
Proof. Let M be the matrix with rows and columns indexed by ∆ and mi,j = αi •αj. The argument above shows that one step of elimination leads to a matrix that is the orthogonal sum of a scalar matrix α1 •α1and complement M⊥with entries
α⊥i •α⊥j
for i, j > 1. Since α1 •αi≤ 0 for i > 1 we cannot have any α⊥i = 0, and for the same reason for i 6= j
α⊥i •α⊥j = αi •αj−(αi •α1)(αj •α1) α1 •α1 ≤ 0 so we can apply induction to see that M is non-singular.
Corollary 8.4. Suppose A = (ai,j) to be a matrix such that ai,i > 0, ai,j ≤ 0 for i 6= j. Assume D−1A
[lummox]
to be a positive definite matrix for some diagonal matrix D with positive entries. Then A−1 has only non-negative entries.
Lemma 8.5.Suppose ∆ to be a set of vectors in a Euclidean space V such that α•β ≤ 0 for α 6= β in ∆.
[titsA]
If there exists v such that α•v > 0 for all α in ∆ then the vectors in ∆ are linearly independent.
Proof. By induction on the size of ∆. The case|∆| = 1 is trivial. But the argument just before this handles the induction step, since if v•α > 0 then so is v•α⊥.
As an immediate consequence:
Proposition 8.6.The set ∆ is a basis of V (Σ).
[tits]
That is to say, a Weyl chamber for a finite geometric root system is a simplicial cone. Its extremal edges are spanned by the columns ̟α in the inverse of the Cartan matrix, which therefore have positive coordinates with respect to ∆. Hence:
Proposition 8.7.Suppose ∆ to be a set of linearly independent vectors such that α•β ≤ 0 for all α 6= β
[roots-inverse]
in ∆. If D is the cone spanned by ∆ then the cone dual to D is contained in D.
Proof. Let
̟ =X
cαα be in the cone dual to D. Then for each β in ∆
̟•β =X
cα(α•β) .
If A is the matrix (α•β), then it satisfies the hypothesis of the Lemma. If u is the vector (cα) and v is the vector (̟•α), then by assumption the second has non-negative entries and
u = A−1v so that u also must have non-negative entries.
9. More about Cartan representations
Every Coxeter group possesses at least one realization, as we shall see in a moment.
Cartan matrices with integral matrices determine Kac-Moody Lie algebras. In this case the representation of its Weyl group on the lattice of roots is the one associated to this Cartan matrix. Coxeter groups which occur as the Weyl groups of Kac-Moody algebras are calledcrystallographic. and are distinguished by the property that for them the numbers ms,tare either 2, 3, 4, 6 or infinite.
<p> Two Cartan matrices C1and C2will give rise to isomorphic representations of a Coxeter group if and only if there exists a positive diagonal matrix D with C2 = DC1D−1. In particular, those Cartan matrices giving rise to realizations equivalent to the standard one aresymmetrizable.
Proposition 9.1. If each ms,tis finite, then the isomorphism classes of Cartan realizations are parame-
[isomorphisms]
trized by H1(Γ, R) (where Γ is the Coxeter diagram).
Proof. If all ms,tare finite, then all Cartan matrices are of the form cs,tds,twhere cs,t= −2 cos2(π/ms,t) and ds,t is an arbitrary matrix of positive real numbers with dt,s = 1/ds,t. Two of these will give isomorphic represntations of W when the entries differ by factors ds/dt, with all ds > 0. But the assignment of ds,tto (s, t) defines a cocycle on the Coxeter graph with values in the multiplicative group of positive real numbers, and assignments ds/dtare coboundaries. Conclude by applying the logarithm.
Corollary 9.2.If W is finite, then the Coxeter diagram has no circuits.
[xxx]
Proof. Because if the diagram were not a tree, the group would possess a continuous family of non- isomorphic representations of dimension r.
Distinct classes can give rise to realizations with very different geometric properties. We have seen this already in the case of the infinite dihedral group, and here are the pictures for two different realizations of the Coxeter group whose Coxeter diagram is
images/diagram343.eps 4
images/kld.eps
The Klein model of the standard realization
images/kvd.eps
The Kac-Vinberg realization
The first of these is associated to the standard Cartan matrix, and the second to the integral matrix
2 −1 −1
−1 2 −1
−2 −1 2
which is that of a certain hyperbolic Kac-Moody Lie algebra. It is the second, therefore, which is likely to have intrinsic significance.
10. Generators and relations for the Weyl group
Suppose given a simplicial root decomposition. Let C be a chamber of such a root configuration; let S be the set of reflections in the walls of C. which we know (by rigidity) to generate W . For each s6= t in S let ms,tbe the order of st.
We have a surjection onto W from the abstract group W∗with generators s in S and relations s2= 1, (st)ms,t= 1 .
Proposition 10.1.This surjection from W∗to W is an isomorphism.
[coxeter]
The case when S has two generators is elementary. When S has more, this is a very special case of a very general result about groups acting on simply connected spaces. If G acts on a simply connected space X with a polyhedral fundamental domain D, it is generated by those g permuting the panels of D. Let them make up a set S. If s and t are two panels that intersect in a face of codimension two, the cycle around that face will give some relation involving only s and t, and G will be isomorphic to the group with generators from S satisfying those relations. This is explained thoroughly—perhaps a bit too thoroughly—in [Macbeath:1964]. In our case, the space X is the quotient of the Tits coneC by the positive scalars. If W is finite, for example, of rank n, this will be the sphere Sn−1. It would be interesting to see this argument made into an effective algorithm.
The classic example is SL2(Z) acting by fractional linear transformations s: z 7−→ −1/z, t: z 7−→ z + 1
acting on the upper half plane. The fundamental domain D is the region|z| ≤ 1/2, |z| ≥ 1, assigned a triangulation by adding the vertical line from i to∞. The transformations s and t are generators and the relations are s2= 1 since s(i) = i and (st)3= 1 since st(ω) = ω where ω = −1/2 +√
−3/2.
But now I give a purely combinatorial proof, one found in the classic notes [Steinberg:1968]. It suggests an effective algorithm.
Proof. The group W∗is the monoid ofwordsw = s1•s2•. . .•snmodulo certain equivalences, I’ll recall in a moment. For each such word w, let w = s1s2. . . snbe its image in W . Define the inverse of a word to be its reverse.
What is the equivalence of words defining W∗? If x and y are words, I say that x ∼ y if one can be obtained from the other by a sequence of these elementary operations: (a) insertions of s•s; (b) deletions of s•s; (c) deletions of (s•t)ms,t. Let x−1 be the reverse of x. Thus x•x−1 ∼ 1, x ∼ 1 if and and only if x−1 ∼ 1, and x ∼ y if and only if xy−1 ∼ 1. I say x can becollapsedif x∼ 1. If w = x•y and x and y can both be collapsed, so can w.
It is easy to check that this is an equivalence relation, with concatenation x•y inducing the group multiplication. Keep in mind also that w can be collapsed if and only if any of its cyclic shifts sk•. . . sn•s1•. . . •sk−1can be collapsed, since if s2•. . . •sn•s1∼ 1 then
s1•s2•. . . •sn∼ s1•s2•. . . •sn•s1•s1∼ s1•s1∼ 1 .
Proposition 10.2.If w = 1 then w∼ 1.
[steinberg]
Proof. Steinberg’s proof will even suggest an explicit algorithm for finding a chain of elementary operations that exhibit the reduction of w to the empty word.
Suppose that w = s1. . . sn= 1. Since the determinant of each siis−1, n must be even, say n = 2r.
The case when w has length 0 is trivial, and when it has length 2 it can be collapsed by a single deletion.
Let wi = s1. . . sifor each i.
We proceed by induction on the length of the expression. For r > 1 the corresponding gallery from C back to itself starts off by crossing the hyperplane α1 = 0 and must recross it later on, say in the step from wkto wk+1= wksk+1. This means that
wkHk+1 = s1. . . skHk+1 = H1
for some k, and this in turn means that
s1= wksk+1w−1k = s1. . . sksk+1sk. . . s1
or
s1. . . sk= s2. . . sk+1,
a convenient cyclic shift. Suppose at first that k6= r. Replacing w by one of its cyclic shifts if necessary, we may assume that k < r. Let x = s1•. . . •sk, y = sk+1•. . . •sn, so that w = x•y. Let z = s2•. . . •sk•sk+1. Then w∼ x•z−1•z•y. By assumption, x = z and x•z−1has length 2k < 2r, so by induction x•z−1can be collapsed. But
z•y = s2. . . sksk+1sk+1sk+2. . . sn= s2. . . sksk+2. . . sn
which has length n− 2. Again by induction zy ∼ 1, hence w ∼ 1 also.
Applying the same reasoning to the shifts of w, we may now assume that si. . . sr+iHr+1+i= Hi
for all i (subscripts taken modulo 2r).
So now we have
s1. . . srHr+1= H1
s2. . . srHr+2= H2
s1. . . sr= s2. . . sr+1
s2. . . sr+1= s3. . . sr+2
and proceed
1 = s1s2s3. . . sr+1sr+2. . . s2r
= s1s2s3. . . sr+1sr+2sr+3. . . s2r
= s1s2s2. . . srsr+1sr+3. . . s2r
= s1s3. . . srsr+1sr+3. . . s2r.
This last expression has length n− 2, hence by induction it is collapsible. The original expression will thus be collapsible if
s3•s2•s3. . . •sr∼ s4•. . . •sr+2•sr+1
or, equivalently,
s3•s2•s3•. . . •sr•sr+1•sr+2. . . •s4
is collapsible.
If s36= s1, we are back in the first case since H36= H1= s2s3. . . srHr+1, so this last expression is indeed collapsible, but from this collapsibility follows that of our original expression.
So now we may assume that s3 = s1. Applying the same argument to cyclic shifts, we may in fact assume that s1= s3 = . . ., s2= s4 = . . .. But in this case we are looking at a power of st, and collapse is immediate.
This argument is very close to that used by Fokko du Cloux in justifying his clever algorithm for finding normal forms of elements in a Coxeter group (which we’ll see later on). I have always found the crucial step in Fokko’s argument very mysterious, but until recently I had not seen that of Steinberg. Whether Fokko himself got a hint from it, I do not know.
The standard argument for proving this theorem is apparently due to Matsumoto, and to be found in [Bourbaki:1972]. It is a lot less direct.
We know now that the group W is a Coxeter group, and for any expression of an element of W representing 1 we can find a sequence of insertions and deletions that collapses the expression. In [Tits:1968] it is shown that in fact only deletions are necessary.
For every word w, let D(w) (D for ‘descendants’) be the set of words obtained from w by a sequence of reflection deletions or braid relations (not insertions).
Proposition 10.3. Two words x and y represent the same element of W if and only if D(x) and D(y)
[Tits-Rw]
overlap.
In particular, if x = 1 then the empty string can be obtained from x by deletions and briod replacements.
Proof. I write∼ to mean equivalence through a sequence of braid relations, ≡ to mean ‘maps to the same group element’.
It need only be shown that if x and y are equivalent then D(x) and D(y) overlap, because the converse is trivial.
The proof is by a kind of double induction. Asume x and y equivalent. We may assume ℓ(y)≤ ℓ(x), and we order all such pairs (x, y) according to the following diagram:
