It’s nice that Weyl groups are Coxeter groups and all, but the definition of a Coxeter group as a group with a particular kind of representation doesn’t immediately tell us why this is the appropriate level of generalization (although the faithfulness of the geometric representation is a good sign). It turns out there is a structural property, the strong exchange condition, which completely characterizes Coxeter groups among groups generated by involutions. Today we will prove this property.
More about roots
Recall that we showed that the roots of a Coxeter system can be divided into the positive roots and the negative roots .
Proposition: a) Any simple reflection sends to its negative and permutes the remaining positive roots. b) equals the number of positive roots sent by to negative roots.
Proof. a) If is a positive root not equal to , then it has non-negative coefficients and some . Applying doesn’t change the coefficient , so the root , which cannot be equal to , must be positive.
b) Let denote the set of positive roots sent by to negative roots. If , then , so . Similarly, if , then , so . It follows by induction that for all .
Corollary: If is infinite, then the length function takes arbitrarily large values, hence is infinite. If is finite, then has a unique element of maximal length, the longest element, hence is finite.
Proof. Since has finitely many generators, there are only finitely many words of a given length, so the length function (and hence the number of positive roots which can be sent to negative roots) is unbounded if and only if is infinite. If are two elements of maximal length with finite, then for all , so each sends all positive roots to negative roots. (In particular, there are at most positive roots.) It follows that sends every positive root to a positive root, hence has length zero and must be the identity.
Example. In the symmetric group , the longest element is the permutation in one-line notation. It has length , as can be deduced from an enumeration of the positive roots. The geometric representation can be placed in the hyperplane , on which acts by permutation. The simple roots are the vectors of the form , and the positive roots are their images under permutation in which the occurs before the .
Associated to any positive root there is a reflection conjugate to a simple reflection which negates . A short computation shows that
hence that the behavior of doesn’t depend on a choice of , but only on the root ; we will therefore denote this reflection by . More generally, if where are two roots, then . It follows that we may freely identify the set
of conjugates of simple reflections with the set of positive roots.
Proposition: Let be a positive root. If , then .
Proof. Again we proceed by induction. If the result is clear. If , then pick such that . Then
By the inductive hypothesis, . If , then since sends exactly one root to its negative we must have , hence . This implies that , hence that . But we know that ; contradiction. Hence .
The strong exchange condition
The following theorem completely characterizes Coxeter groups among groups generated by involutions.
Theorem (strong exchange): Let where (not necessarily a reduced expression). Suppose satisfies . Then there exists an index such that (where the hat indicates that a factor has been omitted). If , then is unique.
Proof. Let . Since , we know that . Since , there exists some index such that but . Since the only positive root that negates is , it follows that , hence that
hence that as desired. A short computation now shows that if , then , which is impossible when .
Corollary (deletion): Suppose with . Then there exist indices such that .
Proof. If is not a reduced expression, then there exists such that . The exchange condition then implies that .
On the other hand, Humphreys shows in Chapter 1 that any group generated by involutions satisfying the deletion condition must be a Coxeter group. So the strong exchange and deletion conditions must, in principle, be enough to answer any question one could ask of an arbitrary Coxeter group.