Last update: 27 April 2013
Two conjectures are stated in [Ram1998-2, (1.3) and (1.11)] when W is a crystallographic reflection group. The first gives necessary and sufficient conditions for (as defined in (2.16)) to be nonempty when is dominant and the second determines the form of as an interval in the weak Bruhat order when is dominant and integral. Loszoncy [Los1999] proved the second conjecture (Theorem 5.2 below). His theorem implies the nonemptiness conjecture of [Ram1998-2] under the additional assumption that is integral. Here we review Loszoncy’s proof and prove the nonemptiness conjecture in full generality. We give an example (Example 5.4) to show that integrality is necessary in Theorem 5.2. Finally, we provide Example 5.7, which shows that one cannot expect analogous statements to hold when is noncrystallographic.
Let be the root system of a finite real reflection group and fix a set of positive roots in A set of positive roots is closed if it satisfies the condition:
The following theorem characterizes the sets which appear as inversion sets of elements of Recall that denotes the inversion set of see equation (2.4). This result is in [Bjo1984, Proposition 2], but is stated there without proof and we are not aware of a published proof. The following proof was shown to us by J. Stembridge and appears in the thesis of D. Waugh [Wau1995].
Theorem 5.1. Let be a real reflection group. A set of positive roots is equal to for some element if and only if is closed and is closed.
Let and suppose that and is a positive root. Then is a negative root since and are both negative roots. So is closed. Similarly, one shows that is closed.
Assume that is closed and that is closed. We will construct w such that by finding a reduced word for This is done by induction on the size of with the induction step being the combination of the two steps below.
Step 1: contains a simple root.
Let be a root of minimal height in and assume that is not simple. Then
Since is not simple, and so both and are positive roots. Since and both have lower height than then both must be in But then the equation
contradicts the assumption that is closed. So is simple.
Step 2: Let be a simple root in and let
Claim: is closed and is closed.
Let and assume that is a positive root. Then
The second is impossible since is not a positive root. So and is closed.
Let and suppose that is a positive root. Since and are not in So Thus is closed.
An element is dominant (resp. integral) if (resp. for all simple roots The closure of a set of positive roots is the smallest closed set of positive roots containing
Theorem 5.2. Let be a crystallographic reflection group and let be the crystallographic root system of Let be dominant and integral, and set
Let be such that
Then there exist elements such that
where denotes the complement of in and denotes the interval between and in the weak Bruhat order.
By Theorem 5.1, the element will exist if is closed. Assume that where We must show that or Since
We will decompose into two pieces and via the following inductive procedure. Since
By reindexing the we can assume that Thus or and we may assume that Since is a root and is crystallographic, is also a root. If is a negative root, then
gives the desired decomposition. If then
and so we may inductively apply this procedure to decompose
In this way we conclude that, after possible reindexing of the either
where and are positive roots such that In the first case it is immediate that In the second case and so and Thus, since is dominant and integral, one of is in and the other is in If and the condition on implies that Similarly, if then Thus or So is closed. Since is closed and is closed, Theorem 5.1 shows that there is an element such that
The same method can be used to establish the existence of one must show that is closed and this is accomplished by similar arguments.
By the definition of an element is in if
Since the weak Bruhat order is the order determined by inclusions of [Bjo1984, Proposition 3] the result is a consequence of the existence of the elements and
Remark 5.3. An alternative way to establish the existence of in the proof of Theorem 5.2 is to use the conjugation involution
where is the minimal length coset representative of and is the longest element of That this is a well defined involution is proved in [Ram1998-2, (1.7)]. This involution takes for to for In terms of the weak Bruhat order, the structure of the interval is the same as the structure of the interval but with all relations reversed.
Example 5.4. The integrality of is necessary in Theorem 5.2. Let be the dihedral group of order 8 (the Weyl group of type The root system for type is determined by simple roots
where is an orthonormal basis of Let be the parameters for If (see Figure 3), then and is dominant but is not integral. The set satisfies the condition in Theorem 5.2, but is not an inversion set for any since is not closed.
The following method of reducing to the integral root subsystem of a weight is standard in the theory of highest weight modules for finite dimensional complex semisimple Lie algebras; see [Jan1980]. This method turns out to be an efficient tool for reducing the nonemptiness conjecture of [Ram1998-2] to the statement in Theorem 5.2.
Let For any
and so is a root system with Weyl group If then the set of is
Theorem 5.5. Let be a crystallographic reflection group and let be the crystallographic root system of Let such that is dominant and set
Let be such that
Then is nonempty.
Since is dominant and integral for the root system it follows from Theorem 5.2 that there is an element in such that
where Usually is strictly larger than but it is still true that
since all roots of and are in So
When is crystallographic, we can use the method of the proof of Theorem 5.5 in combination with the result of Theorem 5.2 to give a precise description of the set for all central characters By choosing appropriately in its we may assume that is dominant.
Each has a unique expression
In this way the elements of are coset representatives of the cosets in
Since and it follows that
Since and is dominant and integral for the root system Theorem 5.2 has the following corollary.
Corollary 5.6. With notations and assumptions as in Theorem 5.5
where, is as in (5.3), and in are determined by and where the complement is taken in the set of positive roots of
This refined version of Theorem 5.2 is reminiscent of the reduction to real central character given in [BMo1993].
The following example shows that Theorem 5.5 does not naturally extend to noncrystallographic reflection groups. Note that such a generalization necessarily involves modifying the closure condition on to be
Example 5.7. Let be the dihedral group of order even, with root system chosen as in Section 3 (so all roots are the same length). Let be such that and (this is an example of in Table 1). Then the subset satisfies the generalized closure condition above since cannot be written as for any However, since there are no chambers which are on the positive side of both and and on the negative side of both and
This is an excerpt of a paper entitled Representations of graded Hecke algebras, written by Cathy Kriloff (Department of Mathematics, Idaho State University, Pocatello, Idaho 83209-8085) and Arun Ram.
Research of the first author supported in part by an NSF-AWM Mentoring Travel Grant. Research of the second author supported in part by National Security Agency grant MDA904-01-1-0032 and EPSRC Grant GR K99015.