## Combinatorics of Local Regions

Last update: 27 April 2013

## Combinatorics of Local Regions

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 ${ℱ}^{\left(\gamma ,J\right)}$ (as defined in (2.16)) to be nonempty when $\gamma$ is dominant and the second determines the form of ${ℱ}^{\left(\gamma ,J\right)}$ as an interval in the weak Bruhat order when $\gamma$ 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 $\gamma$ 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 $W$ is noncrystallographic.

Let $R$ be the root system of a finite real reflection group $W$ and fix a set of positive roots ${R}^{+}=\left\{\alpha >0\right\}$ in $R\text{.}$ A set of positive roots $S$ is closed if it satisfies the condition:

$If α,β∈S and a,b>0 are such that aα+ bβ∈R+, then aα+ bβ∈S.$

The following theorem characterizes the sets which appear as inversion sets of elements of $W\text{.}$ Recall that $R\left(w\right)$ denotes the inversion set of $w\text{;}$ 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 $W$ be a real reflection group. A set of positive roots $S$ is equal to $R\left(w\right)$ for some element $w\in W$ if and only if $S$ is closed and ${S}^{c}={R}^{+}\S$ is closed.

 Proof. $⟹:$ Let $w\in W$ and suppose that $\alpha ,\beta \in R\left(w\right)$ and $a\alpha +b\beta$ is a positive root. Then $w\left(a\alpha +b\beta \right)=a\left(w\alpha \right)+b\left(w\beta \right)$ is a negative root since $w\alpha$ and $w\beta$ are both negative roots. So $R\left(w\right)$ is closed. Similarly, one shows that $R{\left(w\right)}^{c}$ is closed. $⟸:$ Assume that $S$ is closed and that ${S}^{c}$ is closed. We will construct w such that $R\left(w\right)=S$ by finding a reduced word $w={s}_{{i}_{1}}\dots {s}_{{i}_{k}}$ for $w\text{.}$ This is done by induction on the size of $S,$ with the induction step being the combination of the two steps below. Step 1: $S$ contains a simple root. Let $\alpha$ be a root of minimal height in $S$ and assume that $\alpha ={\sum }_{i}{c}_{{\alpha }_{i}}{\alpha }_{i},$ ${c}_{{\alpha }_{i}}\in {ℝ}_{\ge 0},$ is not simple. Then $⟨α,αi⟩>0 for some i,since 0<⟨α,α⟩ =∑i=1ncαi ⟨a,αi⟩.$ Since $\alpha$ is not simple, $\alpha \ne {\alpha }_{i},$ and so both ${s}_{{\alpha }_{i}}\alpha$ and ${\alpha }_{i}$ are positive roots. Since ${s}_{{\alpha }_{i}}{\alpha }_{i}=\alpha -⟨\alpha ,{\alpha }_{i}^{\vee }⟩{\alpha }_{i}$ and ${\alpha }_{i}$ both have lower height than $\alpha ,$ then both must be in ${S}^{c}\text{.}$ But then the equation $α=sαiα+ ⟨α,αi∨⟩ αi$ contradicts the assumption that ${S}^{c}$ is closed. So $\alpha$ is simple. Step 2: Let ${\alpha }_{{i}_{1}}$ be a simple root in $S$ and let ${S}_{1}={s}_{{i}_{1}}\left(S\\left\{{\alpha }_{{i}_{1}}\right\}\right)\text{.}$ Claim: ${S}_{1}$ is closed and ${S}_{1}^{c}$ is closed. Let $\alpha ,\beta \in {S}_{1}$ and assume that $a\alpha +b\beta$ is a positive root. Then $si1 (aα+bβ)=a si1α+b si1β∈Sand aα+bβ∈S1,$ or $asi1α+b si1β=αi1 andaα+bβ=- αi1.$ The second is impossible since ${s}_{{i}_{1}}{\alpha }_{{i}_{1}}$ is not a positive root. So $a\alpha +b\beta \in {S}_{1}$ and ${S}_{1}$ is closed. Let $\alpha ,\beta \in {S}_{1}^{c}$ and suppose that $a\alpha +b\beta$ is a positive root. Since ${s}_{{i}_{1}}\alpha$ and ${s}_{{i}_{1}}\beta$ are not in $S,$ ${s}_{{i}_{1}}\left(a\alpha +b\beta \right)\notin S\text{.}$ So $a\alpha +b\beta \notin {S}_{1}\text{.}$ Thus ${S}_{1}^{c}$ is closed. $\square$

An element $\gamma \in {𝔥}_{ℂ}$ is dominant (resp. integral) if $\gamma \left({\alpha }_{i}\right)\in {ℝ}_{\ge 0}$ (resp. $\gamma \left({\alpha }_{i}\right)\in ℤ\text{)}$ for all simple roots ${\alpha }_{i}\text{.}$ The closure $\stackrel{‾}{S}$ of a set of positive roots $S$ is the smallest closed set of positive roots containing $S\text{.}$

Theorem 5.2. Let $W$ be a crystallographic reflection group and let $R$ be the crystallographic root system of $W\text{.}$ Let $\gamma \in {𝔥}_{ℂ}$ be dominant and integral, and set

$Z(γ)= {α>0 | ⟨γ,α⟩=0} andP(γ)= {α>0 | ⟨γ,α⟩=1} .$

Let $J\subseteq P\left(\gamma \right)$ be such that

$if β∈J,α∈ Z(γ)and β-α∈R+, then β-α∈J,$

and set

$ℱ(γ,J)= { w∈W | R(w) ∩Z(γ)=∅, R (w)∩P(γ)=J } .$

Then there exist elements ${w}_{\text{min}},{w}_{\text{max}}\in W$ such that

$R(wmin)= J‾,R (wmax)= (P(γ)\J)∪Z(γ)‾c ,and ℱ(γ,J)= [wmin,wmax],$

where ${K}^{c}$ denotes the complement of $K$ in ${R}^{+}$ and $\left[{w}_{\text{min}},{w}_{\text{max}}\right]$ denotes the interval between ${w}_{\text{min}}$ and ${w}_{\text{max}}$ in the weak Bruhat order.

 Proof. By Theorem 5.1, the element ${w}_{\text{min}}\in W$ will exist if ${\stackrel{‾}{J}}^{c}$ is closed. Assume that $\beta ={\beta }_{1}+{\beta }_{2}$ where $\beta \in \stackrel{‾}{J},$ ${\beta }_{1},{\beta }_{2}\in {R}^{+}\text{.}$ We must show that ${\beta }_{1}\in \stackrel{‾}{J}$ or ${\beta }_{2}\in \stackrel{‾}{J}\text{.}$ Since $\beta \in \stackrel{‾}{J},$ $β=δ1+…δm with δi∈J.$ We will decompose $\beta ={\delta }_{1}+\dots +{\delta }_{m}$ into two pieces ${\beta }_{1}={\delta }_{1}+\dots +{\delta }_{k}+{\eta }_{1}$ and ${\beta }_{2}={\eta }_{2}+{\delta }_{k+2}+\dots +{\delta }_{m},$ via the following inductive procedure. Since $0< ⟨ β1+β2, β1+β2 ⟩ =∑i ⟨β1+β2,δi⟩ ,then ⟨β1+β2,δj⟩ >0for some j.$ By reindexing the ${\delta }_{i}$ we can assume that $j=1\text{.}$ Thus $⟨{\beta }_{1},{\delta }_{1}⟩>0$ or $⟨{\beta }_{2},{\delta }_{1}⟩>0$ and we may assume that $⟨{\beta }_{1},{\delta }_{1}⟩>0\text{.}$ Since ${s}_{{\delta }_{1}}{\beta }_{1}={\beta }_{1}-⟨{\beta }_{1},{\delta }_{1}^{\vee }⟩{\delta }_{1}$ is a root and $R$ is crystallographic, ${\beta }_{1}-{\delta }_{1}$ is also a root. If ${\beta }_{1}-{\delta }_{1}$ is a negative root, then $β1=β1andβ =(δ1-β1)+ δ2+…+δm,$ gives the desired decomposition. If ${\beta }_{1}-{\delta }_{1}\in {R}^{+},$ then $β1+β2= δ1+ ((β1-δ1)+β2) and (β1-δ1)+ β2=δ2+…+ δm,$ and so we may inductively apply this procedure to decompose $\beta \prime =\left({\beta }_{1}-{\delta }_{1}\right)+{\beta }_{2}={\delta }_{2}+\dots +{\delta }_{m}\text{.}$ In this way we conclude that, after possible reindexing of the ${\delta }_{i},$ either $β1=δ1+…+δk andβ2=δk+1 +…+δm,$ or $β1=δ1+…+δk +η1andβ2= η2+δk+2+…+ δm,$ where ${\eta }_{1}$ and ${\eta }_{2}$ are positive roots such that ${\eta }_{1}+{\eta }_{2}={\delta }_{k+1}\text{.}$ In the first case it is immediate that ${\beta }_{1},{\beta }_{2}\in \stackrel{‾}{J}\text{.}$ In the second case $⟨\gamma ,{\delta }_{k+1}⟩=⟨\gamma ,{\eta }_{1}+{\eta }_{2}⟩=1,$ and so $⟨\gamma ,{\eta }_{1}⟩\le 1$ and $⟨\gamma ,{\eta }_{2}⟩\le 1\text{.}$ Thus, since $\gamma$ is dominant and integral, one of ${\eta }_{1},{\eta }_{2}$ is in $Z\left(\gamma \right)$ and the other is in $P\left(\gamma \right)\text{.}$ If ${\eta }_{1}\in Z\left(\gamma \right),$ ${\eta }_{2}={\delta }_{k+1}-{\eta }_{1}$ and the condition on $J$ implies that ${\eta }_{2}\in J\text{.}$ Similarly, if ${\eta }_{2}\in Z\left(\gamma \right),$ then ${\eta }_{1}\in J\text{.}$ Thus ${\beta }_{1}\in \stackrel{‾}{J}$ or ${\beta }_{2}\in \stackrel{‾}{J}\text{.}$ So ${\stackrel{‾}{J}}^{c}$ is closed. Since $\stackrel{‾}{J}$ is closed and ${\stackrel{‾}{J}}^{c}$ is closed, Theorem 5.1 shows that there is an element ${w}_{\text{min}}\in W$ such that $R\left({w}_{\text{min}}\right)=\stackrel{‾}{J}\text{.}$ The same method can be used to establish the existence of ${w}_{\text{max}}:$ one must show that ${\stackrel{‾}{\left(P\left(\gamma \right)\J\right)\cup Z\left(\gamma \right)}}^{c}$ is closed and this is accomplished by similar arguments. By the definition of ${ℱ}^{\left(\gamma ,J\right)}$ an element $w\in W$ is in ${ℱ}^{\left(\gamma ,J\right)}$ if $J‾⊆R(w)⊆ {\stackrel{‾}{\left(P\left(\gamma \right)\J\right)\cup Z\left(\gamma \right)}}^{c} .$ Since the weak Bruhat order is the order determined by inclusions of $R\left(w\right)$ [Bjo1984, Proposition 3] the result is a consequence of the existence of the elements ${w}_{\text{min}}$ and ${w}_{\text{max}}\text{.}$ $\square$

Remark 5.3. An alternative way to establish the existence of ${w}_{\text{max}}$ in the proof of Theorem 5.2 is to use the conjugation involution

$ℱ(γ,J) ⟷1-1 ℱ(γ,J)′ w ⟷ wu-1 where (γ,J)′= (-uγ,-u(P(γ)\J)), (5.1)$

where $u$ is the minimal length coset representative of ${w}_{0}{W}_{\gamma }$ and ${w}_{0}$ is the longest element of $W\text{.}$ That this is a well defined involution is proved in [Ram1998-2, (1.7)]. This involution takes ${w}_{\text{max}}$ for ${ℱ}^{\left(\gamma ,J\right)}$ to ${w}_{\text{min}}$ for ${ℱ}^{\left(\gamma ,J\right)}\prime \text{.}$ In terms of the weak Bruhat order, the structure of the interval ${ℱ}^{\left(\gamma ,J\right)}\prime$ is the same as the structure of the interval ${ℱ}^{\left(\gamma ,J\right)}$ but with all relations reversed.

Example 5.4. The integrality of $\gamma$ is necessary in Theorem 5.2. Let $W={I}_{2}\left(4\right)$ be the dihedral group of order 8 (the Weyl group of type ${C}_{2}\text{).}$ The root system for type ${C}_{2}$ is determined by simple roots

$α1=2ε1and α2=ε2-ε1$

where $\left\{{\epsilon }_{1},{\epsilon }_{2}\right\}$ is an orthonormal basis of ${𝔥}_{ℝ}^{*}={ℝ}^{2}\text{.}$ Let ${c}_{1}={c}_{2}=1$ be the parameters for $ℍ\text{.}$ If $\gamma =\left(1/2\right){\epsilon }_{2}$ (see Figure 3), then $Z\left(\gamma \right)=\left\{{\alpha }_{1}\right\},$ $P\left(\gamma \right)=\left\{{\alpha }_{1}+2{\alpha }_{2}\right\},$ and $\gamma$ is dominant but $\gamma \left({\alpha }_{2}\right)$ is not integral. The set $J=P\left(\gamma \right)$ satisfies the condition in Theorem 5.2, but $\stackrel{‾}{J}=J$ is not an inversion set for any $w\in W$ since ${\stackrel{‾}{J}}^{c}$ 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 ${R}_{\left[\gamma \right]}=\left\{\alpha \in R | ⟨\gamma ,{\alpha }^{\vee }⟩\in ℤ\right\}\text{.}$ For any $\alpha ,\beta \in {R}_{\left[\gamma \right]},$

$⟨γ,(sαβ)∨⟩ =⟨sαγ,β∨⟩= ⟨γ,β∨⟩- ⟨γ,α∨⟩ ⟨α,β∨⟩∈ℤ,$

and so ${R}_{\left[\gamma \right]}$ is a root system with Weyl group ${W}_{\left[\gamma \right]}=⟨{s}_{\alpha } | \alpha \in {R}_{\left[\gamma \right]}⟩\subseteq W\text{.}$ If $\tau \in {W}_{\left[\gamma \right]},$ then the ${R}_{\left[\gamma \right]}\text{-inversion}$ set of $\tau$ is

$R[γ](τ)= { α>0 | τα< 0,α∈R[γ] } =R(τ)∩R[γ] .$ $Hα1 Hα2 Hα1+2α2 Hα1+α2+δ Hα1+α2 Hα1+2α2+δ Hα1+2α2-δ Hα2+δ Hα2-δ Hα1-δ Hα1+δ Hα1+α2-δ γ$

Theorem 5.5. Let $W$ be a crystallographic reflection group and let $R$ be the crystallographic root system of $W\text{.}$ Let $\gamma \in {𝔥}_{ℂ}$ such that $\text{Re}\left(\gamma \right)$ is dominant and set

$Z(γ)= {α>0 | ⟨γ,α⟩=0} andP(γ)= {α>0 | ⟨γ,α⟩=1}.$

Let $J\subseteq P\left(\gamma \right)$ be such that

$if β∈J, α∈ Z(γ) and β -α∈R+,then β-α∈J.$

Then ${ℱ}^{\left(\gamma ,J\right)}=\left\{w\in W | R\left(w\right)\cap Z\left(\gamma \right), R\left(w\right)\cap P\left(\gamma \right)=J\right\}$ is nonempty.

 Proof. Since $\gamma$ is dominant and integral for the root system ${R}_{\left[\gamma \right]},$ it follows from Theorem 5.2 that there is an element $w$ in ${W}_{\left[\gamma \right]}$ such that $R[γ](w)∩ Z(γ)=∅and R[γ](w)∩ P(γ)=J,$ where ${R}_{\left[\gamma \right]}\left(w\right)=\left\{\alpha \in {R}_{\left[\gamma \right]} | \alpha >0,w\alpha <0\right\}\text{.}$ Usually $R\left(w\right)$ is strictly larger than ${R}_{\left[\gamma \right]}\left(w\right)$ but it is still true that $R(w)∩Z(γ)=∅ andR(w)∩ P(γ)=J,$ since all roots of $P\left(\gamma \right)$ and $Z\left(\gamma \right)$ are in ${R}_{\left[\gamma \right]}\text{.}$ So $w\in {ℱ}^{\left(\gamma ,J\right)}\text{.}$ $\square$

When $W$ 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 ${ℱ}^{\left(\gamma ,J\right)}$ for all central characters $\gamma \in {𝔥}_{ℂ}\text{.}$ By choosing $\gamma$ appropriately in its $W\text{-orbit}$ we may assume that $\text{Re}\left(\gamma \right)$ is dominant.

Define

$W[γ]= { σ∈W | R (σ)∩R[γ] =∅ } .$

Each $w\in W$ has a unique expression

$w=στwithσ∈ W[γ],τ∈ W[γ],$

and

$R(w)∩R[γ]= R(τ)∩R[γ]= R[γ](τ).$

In this way the elements of ${W}^{\left[\gamma \right]}$ are coset representatives of the cosets in $W/{W}_{\left[\gamma \right]}\text{.}$

Since $P\left(\gamma \right)\subseteq {R}_{\left[\gamma \right]}$ and $Z\left(\gamma \right)\subseteq {R}_{\left[\gamma \right]}$ it follows that

$ℱ(γ,J)= { στ∈W | σ∈ W[γ],τ∈ ℱ[γ](γ,J) } , (5.2)$

where

$ℱ[γ](γ,J)= { τ∈W[γ] | Rγ(τ)∩P(γ) =J,R(w)∩ Z(γ)=∅ } . (5.3)$

Since ${ℱ}^{\left(\gamma ,J\right)}={ℱ}^{\left(\text{Re}\left(\gamma \right),J\right)}$ and $\gamma$ is dominant and integral for the root system ${R}_{\left[\gamma \right]},$ Theorem 5.2 has the following corollary.

Corollary 5.6. With notations and assumptions as in Theorem 5.5

$ℱ(γ,J)= ℱ[γ](γ,J)= W[γ]· [τmax,τmin],$

where, ${ℱ}_{\left[\gamma \right]}^{\left(\gamma ,J\right)}$ is as in (5.3), ${\tau }_{\text{max}}$ and ${\tau }_{\text{min}}$ in ${W}_{\left[\gamma \right]}$ are determined by ${R}_{\left[\gamma \right]}\left({\tau }_{\text{max}}\right)=\stackrel{‾}{J}$ and ${R}_{\left[\gamma \right]}\left({\tau }_{\text{min}}\right)={\stackrel{‾}{\left(P\left(\gamma \right)\J\right)\cup Z\left(\gamma \right)}}^{c},$ where the complement is taken in the set of positive roots of ${R}_{\left[\gamma \right]}\text{.}$

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 $J$ to be

$if β∈J,α∈Z(γ) ,a∈ℝ>0, and β-a α∈R+,then β-aα∈J.$

Example 5.7. Let $W={I}_{2}\left(n\right)$ be the dihedral group of order $2n,$ $n$ even, with root system chosen as in Section 3 (so all roots are the same length). Let $\gamma$ be such that $Z\left(\gamma \right)=\left\{{\beta }_{0}\right\}$ and $P\left(\gamma \right)=\left\{{\beta }_{n/4},{\beta }_{n/2},{\beta }_{3n/4}\right\}$ (this $\gamma$ is an example of ${\gamma }_{q}$ in Table 1). Then the subset $J=\left\{{\beta }_{n/4},{\beta }_{3n/4}\right\}\subseteq P\left(\gamma \right)$ satisfies the generalized closure condition above since ${\beta }_{n/2}$ cannot be written as ${\beta }_{n/4}-a{\beta }_{0}$ for any $a\in {ℝ}_{>0}\text{.}$ However, ${ℱ}^{\left(\gamma ,J\right)}=\varnothing$ since there are no chambers which are on the positive side of both ${H}_{{\beta }_{0}}$ and ${H}_{{\beta }_{n/2}}$ and on the negative side of both ${H}_{{\beta }_{n/4}}$ and ${H}_{{\beta }_{3n/4}}\text{.}$

## Notes and References

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.