Finite Reflection Groups
1.4 Cell Decomposition
1.4.6 Semigroup Structure
We return now to the general setup. Thus H = {Hi}i∈I is not necessarily essential, and V is not assumed to be equipped with an inner product. We saw in Section 1.4.2 that the setΣ:=Σ(H) of cells is a poset under the face relation. What is less obvious, and perhaps surprising, is that there is a natural way to multiply cells, so that Σ becomes a semigroup. This product was introduced by Bland in the early 1970s in connection with linear programming, and it eventually led to one approach to the theory of oriented matroids;
see [37]. Tits [247] discovered the product independently (in the setting of Coxeter complexes and buildings), although he phrased his version of the theory in terms of “projection operators” rather than products.
We proceed now to the definition of the product. Given two cellsA, B∈Σ, choose x∈Aand y∈B, and consider a typical pointpt:= (1−t)x+ty on the line segment [x, y] (0 ≤t ≤1). For eachi ∈ I and all sufficiently small t >0, the sign of fi(pt) is the same as the sign offi(x) unlessfi(x) = 0, in which case the sign offi(pt) is the same as the sign offi(y). Hence there is a cellCthat containspt for all sufficiently smallt >0, and its sign sequence is given byσi(C) =σi(A) unlessσi(A) = 0, in which caseσi(C) =σi(B). Note that the sign sequence of C, and hence C itself, depends only on A and B, not on the choice ofx∈Aandy ∈B. We will callC theproductofAandB:
Definition 1.38.Given two cellsA, B ∈Σ, theirproductis the cellABwith sign sequence
σi(AB) =
σi(A) ifσi(A)= 0,
σi(B) ifσi(A) = 0. (1.4) The cell AB is characterized by the property that if we choose x ∈ A and y∈B, then (1−t)x+tyis in ABfor all sufficiently smallt >0.
See Figure 1.5 for a simple example, where A and B are half-lines and AB turns out to be a chamber. For a second example, letA be the half-line oppositeAin the same figure; thenAA=A. One can easily check from (1.4)
AB
B
A
Fig. 1.5.The product of two half-lines.
that the associative law holds:
A(BC) = (AB)C (1.5)
for allA, B, C∈Σ. In fact, the triple product, with either way of associating, can be characterized by the property thatσi(ABC) isσi(A) unlessσi(A) = 0, in which case it is σi(B) unless σi(B) = 0, in which case it is σi(C). So Σ is indeed a semigroup. It has an identity, consisting of the cell
i∈IHi with sign sequence (0,0, . . . ,0).
Following Tits [247], we will often callAB theprojection of B on Aand write
AB= projAB .
This may serve as a reminder of the geometric meaning of the product. We will see, however, that the product notation is quite useful, especially to facil- itate application of the associative law. Note that the associative law, in the language of projections, takes the complicated form
projA(projBC) = projproj
ABC . (1.6)
Equation (1.6) appears (in a slightly different context from ours) in Tits’s appendix [249] to Solomon’s paper [221] on the descent algebra, and the ob- servation that (1.6) is actually an associative law can be used to give a much simpler treatment; see [55].
1.4 Cell Decomposition 27 The geometry of projections is especially clear when the second factor is a chamber. In order to state the result, we introduce a metric on the set C:=C(H) of chambers. We will temporarily denote this metric bydH(−,−);
later, after showing thatdH coincides with another naturally defined metric, we will drop the subscriptH.
Definition 1.39.The distance dH(C, D) between two chambers C, D is the number of hyperplanes in Hseparating C and D. Equivalently, dH(C, D) is the number of positions at which the sign sequences of CandD differ.
The following result justifies the term “projection.”
Proposition 1.40.Given a cellA and a chamberC, the productAC (or the projection ofC onA) is a chamber having A as a face; among the chambers having Aas a face, it is the unique one at minimal distance fromC.
Proof. To minimize the distance toCof a chamberD≥A, we must maximize the number of indices i such thatσi(D) =σi(C). We have no choice about σi(D) whenever σi(A) = 0, so the best we can do is make σi(D) = σi(C) whenever σi(A) = 0. This is precisely what the definition of AC in (1.4)
achieves.
Finally, sinceΣis now both a poset and a semigroup, it is natural to ask how these structures interact. We record a few simple results in the following proposition, whose proof is routine and is left to the reader.
Proposition 1.41.Let AandB be arbitrary cells.
(1)A≤AB, with equality if and only ifsuppB ≤suppA.
(2)A≤B if and only ifAB=B.
(3) suppA= suppB if and only if AB=A andBA=B.
(4)AB and BAhave the same support, which is the intersection of the hy-
perplanes inHcontaining both A andB.
Exercises
1.42.Prove the following more precise version of Proposition 1.40: For any chamber D≥A,
dH(C, D) =dH(C, AC) +dH(AC, D). (1.7) In the language of Dress–Scharlau [97], this says that the setC≥Aof chambers D ≥ A is a gated subset of the metric space of chambers. Here AC is the
“gate” through which one entersC≥A to get fromC to an arbitrary chamber D≥A. See Figure 1.6 for a schematic illustration.
1.43.We say that cells A, B, . . . are joinable if they have an upper bound in the poset Σ. Show that this holds if and only if they commute with one another in the semigroup Σ, in which case their product is their least upper bound.
C D
AC C≥A
Fig. 1.6.The gate property.
1.44.IfA andB have the same support, show that left multiplication byA gives a bijection Σ≥B → Σ≥A, with inverse given by multiplication by B.
This holds, for example, ifAandB areopposite, i.e., A=−B.
1.45.GivenA∈Σ, show that the posetΣ≥Ais isomorphic to the set of cells of a hyperplane arrangement.