4.4 Weyl chambers

By studying the structure of a reduced root system, we may choose a base/fundamental system $\Delta$ to get a Cartan matrix

$$(n({\alpha }_i,{\alpha }_j)),\text{ where }n({\alpha }_i,{\alpha }_j)=2\frac{({\alpha }_i,{\alpha }_j)}{({\alpha }_j,{\alpha }_j)}.$$

two equivalent definitions of root system

Definition.1. A root system is a pair $(V,R)$ where $V$ is a finite dimensional Euclidean space and $R\subseteq V\backslash \{0\}$ is a finite set spanning $V$ and satisfying the conditions:

• for each $\alpha \in R$, $S_{\alpha }(R)=R$;
• for $\alpha ,\beta \in R$, $S_{\alpha }(\beta )-\beta$ is an integer multiple of $\alpha$.

Definition.1.’ A root system is a pair $(V,R)$ where $V$ is a finite dimensional real vector space and $R\subseteq V\backslash \{0\}$ is a finite set spanning $V$ and satisfying

• for each $\alpha \in R$, there is a symmetry with vector $\alpha$ s.t. $S_{\alpha }(R)=R$, where a symmetry with vector $\alpha$ is a linear isomorphism $S:V\to V$ s.t. $S(\alpha )=-\alpha$ and $S$ fixes a hyperplane;
• for $\alpha ,\beta \in R$, $S_{\alpha }(\beta )-\beta$ is an integer multiple of $\alpha$.

Remark. Definition 1 and Definition 1’ are equivalent. For Definition 1’, we can put an inner product on $V$ by

$$(v_1,v_2{)}_V=\sum _{w\in W}(w{v}_1,w{v}_2)$$

where $W=⟨S_{\alpha }:\alpha \in R⟩$ is a permutation group of $R$. Then each $S_{\alpha }$ preserves $(\cdot ,\cdot )$. For any $v$ in the fixed point of $S_{\alpha }$, since

$$(-\alpha ,v{)}_V=(S_{\alpha }(\alpha ),S_{\alpha }(v){)}_V=(\alpha ,v{)}_V$$

we have $(v,\alpha {)}_V=0$. So $(V,R)$ is a root system according to Definition 1, where $V$ is a Euclidean space with $(\cdot ,\cdot {)}_V$.

In fact, the inner product defined above is unique up to scalar. See exercise Q1.

Observation. For a root system $(V,R)$ with $\alpha ,\beta \in R$, if the angle $\varphi$ between $\alpha$ and $\beta$ are acute, then $\alpha -\beta ,\beta -\alpha \in R$.

\begin{proof} Recall local picture, for any $\alpha ,\beta \in R$ with the angle between them acute, there is

$$n(\alpha ,\beta )n(\beta ,\alpha )=4{\cos }^2\varphi \in \{1,2,3\}.$$

Then one of $n(\alpha ,\beta )=1$ and so one of $\alpha -\beta \in R$. Whence $\alpha -\beta ,\beta -\alpha \in R$. \end{proof}

the global picture

Let $(V,R)$ be a root system. Define $V\backslash {\cup }_{\alpha \in R}{H}_{\alpha }={\bigsqcup }_i{C}_i$ where $C_i$ are connected components and they are called the Weyl chambers. Let $C$ be a Weyl chamber. Then for each $\alpha \in R$, $(\alpha ,\cdot )$ does not change the sign on $C$. Define $R_{+}=\{\alpha \in R:(\alpha ,t)>0,t\in C\}$ and $R_{-}=\{\alpha \in R:(\alpha ,t)<0,t\in C\}$. Define $\Delta =\{\text{ indecomposable positive roots }\}$. Then:

• $\Delta \ne \varnothing$
• $\Delta$ is linearly independent
• each $\beta \in R^{+}$ is a linear combination of elements in $\Delta$ with non-negative coefficients

\begin{proof} i) Choose $\alpha \in R,t\in C$ s.t. $(\alpha ,t)=\min \{(\beta ,t):\beta \in R_{+}\}$. Then $\alpha$ is indecomposable.

ii) Claim that $({\alpha }_1,{\alpha }_2)\le 0$ for any ${\alpha }_1,{\alpha }_2\in \Delta$. Otherwise, ${\alpha }_1-{\alpha }_2,{\alpha }_2-{\alpha }_1\in R$. One of them would be positive, which contradicts the assumption that ${\alpha }_1,{\alpha }_2$ are indecomposable.

Note that a finite set $\Delta$ of vectors in $\{v\in V:(v,t)>0\}$ with pairwise angles $\in [\pi /2,\pi )$ must be linear independent. Otherwise there are $c_i$ such that ${\sum }_i{c}_i{\alpha }_i=0$. Then ${\sum }_{c_i>0}{c}_i{\alpha }_i=-{\sum }_{c_j<0}{c}_j{\alpha }_j:=v$ and

$$0\le (v,v)=\sum _{i,j}{c}_i(-c_j)({\alpha }_i,{\alpha }_j)\le 0.$$

Whence $v=0$. However, $0=(v,t)={\sum }_i{c}_i({\alpha }_i,t)=-{\sum }_j{c}_j({\alpha }_j,t)$ and $({\alpha }_i,t),({\alpha }_j,t)\ge 0$ yields $c_i=c_j=0$.

iii) Otherwise, we call $\beta \in R^{+}$ a bad root if it cannot expressed as a linear combination of elements in $\Delta$ with non-negative coefficients. Let ${\beta }_0$ be such a root with $({\beta }_0,t)=\min \{(\beta ,t):\beta \text{ is bad }\}$. Then ${\beta }_0$ is indecomposable and so ${\beta }_0\in \Delta$, contradiction. \end{proof}

Definition

Let $(V,R)$ be a root system. A base, or a fundamental system of this root system is a subset $\Delta \subseteq R$ such that

• $\Delta$ is a basis of $V$;
• each $\beta \in R$ is a linear combination of elements in $\Delta$ with coefficients either all in ${\mathbb{Z}}_{\ge 0}$ or all in ${\mathbb{Z}}_{\le 0}$.

And roots in a bases are called simple roots w.r.t. this base.

Proposition

Let $(V,R)$ be a root system. Then there is a one-to-one correspondence

$$\{\text{ Weyl chambers }\}⟷\{\text{ bases }\}.$$

\begin{proof} One direction is done before. Now for any given base ${\Delta }_t=\{{\alpha }_1,\cdots ,{\alpha }_n\}$, choose $t$ such that $({\alpha }_i,t)\ge 0$. Let $C$ be a Weyl chamber containing $t$. Then $R_{+}$ according $C$ equals

$$\{\beta \in R:\beta \in \sum {\mathbb{Z}}_{\ge 0}{\alpha }_i\}.$$

It is easy to show that $\Delta =\{\text{ indecomposible roots w.r.t. }C\}$ and hence is the base associated to $C$. \end{proof}

Remark. A base $\Delta$ together with the number $\{n(\alpha ,\beta ):\alpha ,\beta \in \Delta \}$ would determine the root system, that is why we define Cartan matrix.

Explanation of the remark. The set of numbers $\{n(\alpha ,\beta ):\alpha ,\beta \in \Delta \}$ first determines the simple reflections $\{S_{\beta }:\beta \in \Delta \}$, since the action $S_{\beta }$ on a basis vector $\alpha$ is given by $s_{\beta }(\alpha )=\alpha -n(\alpha ,\beta )\beta$. The inner product $(\cdot ,\cdot )$ is therefore determined up to a scalar factor. Furthermore, one can show that the Weyl Group $W=⟨S_{\beta }:\beta \in \Delta ⟩$ and hence $W$ is determined. One can further show that for any root $\gamma \in \Phi$, there exists $w\in W$ and $\alpha \in \Delta$ such that $\gamma$ $=w(\alpha )$. Thus, the entire root system $\Phi$ is determined.

Theorem

Let $(V,R)$ be a reduced root system and $W:=⟨S_{\beta }:\beta \in R⟩$ be the Weyl group. Choose a Weyl chamber $C$ and an according base $\Delta$. Then:

• for any $t\in V$ there is a $w\in W$ such that $wt\in \overline{C}$;
• if $C^{\prime }$ is a Weyl chamber, then there exists a $w\in W$ such that $w{C}^{\prime }=C$;
• for any $\alpha \in R$, there is a $w\in W$ such that $w\alpha \in \Delta$;
• $W$ is generated by $\{S_{\alpha }:\alpha \in \Delta \}$.

\begin{proof} Firstly, we have the following observations.

Observation. Let $(V,R)$ be a reduced system and $\Delta$ be a base. Then:

i) for $\alpha \in \Delta$, $S_{\alpha }$ leaves $R_{+}\backslash \{\alpha \}$ invariant. Proof: For $\beta \in R_{+}\backslash \{\alpha \}$, $\beta =\sum n_{{\alpha }^{\prime }}{\alpha }^{\prime }+n(\alpha )\alpha$ with not all $n_{\alpha }^{\prime }=0$. Then $S_{\alpha }(\beta )=\sum n_{{\alpha }^{\prime }}\alpha +(n(\alpha )-n(\beta ))\alpha$. Since some $n_{{\alpha }^{\prime }}$ is positive, $S_{\alpha }(\beta )$ is still in $R_{+}\backslash \{\alpha \}$.

ii) Put $\rho =1/2{\sum }_{\beta \in R^{+}}\beta$. Then for $\alpha \in \Delta$, $S_{\alpha }(\rho )=\rho -\alpha$. It is easy to prove by i).

Now we can prove the theorem.

i) The Weyl chamber according to $\Delta$ is $C=\{v\in V:(v,\alpha )\ge 0,\alpha \in \Delta \}$. So we need to find $w\in W$ such that $(wt,\alpha )\ge 0$. Note that

$$(S_{\alpha }v,\rho )=(v,S_{\alpha }\rho )=(v,\rho -\alpha )=(v,\rho )-(v,\alpha ).$$

Let $w$ be an element such that $(wt,\rho )={\max }_{w^{\prime }\in W}(w^{\prime }t,\rho )$. Then for any $\alpha \in \Delta$,

$$(wt,\rho )\ge (S_{\alpha }wt,\rho )=(wt,\rho )-(wt,\alpha ).$$

Thus $(wt,\alpha )\ge 0$ and $w$ would send $t$ to $\overline{C}$.

ii) By i), choose $t^{\prime }\in C^{\prime }$ and $w{t}^{\prime }\in \overline{C}$. Then $w{C}^{\prime }=C$.

iii) WLOG we may assume $\alpha \in R_{+}$. (Otherwise that $\alpha =S_{\alpha }\alpha$.) Write $\alpha ={\sum }_{i=1}^n{n}_i{\alpha }_i$, where $\Delta =\{{\alpha }_1,\cdots ,{\alpha }_n\}$. We may do induction on

$$\mathrm{ht}(\alpha )=\sum _{i=1}^n{n}_i.$$

When $\mathrm{ht}(\alpha )=1$, we have done. When $\mathrm{ht}(\alpha )\ge 2$, there are at least two $n_i\in {\mathbb{Z}}_{\ge 0}$. The idea is applying some $S_{{\alpha }_i}$ to $\alpha$ such that $\mathrm{ht}(\alpha )$ decreases.

Note that there exists some $i$ such that $(\alpha ,{\alpha }_i)\ge 0$, because for any ${\alpha }_i,{\alpha }_j\in \Delta$, the angle between them is obtuse. Then $S_{{\alpha }_i}(\alpha )$ is of smaller height and still belongs to $R_{+}$. Apply the induction hypothesis, $w^{\prime }{S}_{{\alpha }_i}(\alpha )\in \Delta$ for some $w^{\prime }\in W$. So $w=w^{\prime }{S}_{{\alpha }_i}$ meets the purpose. In fact, we have actually proved that for any $\alpha \in R_{+}$, there is $w\in ⟨S_{\gamma }:\gamma \in \Delta ⟩$ such that $w\alpha \in \Delta$.

iv) It suffices to show $S_{\beta }\in W^{\prime }:=⟨S_{\alpha }:\alpha \in \Delta ⟩$ for each $\beta \in R_{+}$. By previous observation, there is $w\in W^{\prime }$ such that $\beta =w\alpha$ for some $\alpha \in \Delta$. Then $S_{\beta }=w{S}_{\alpha }{w}^{-1}$. \end{proof}

Remark. Textbook gives us a more natural proof. See page 61.

Theorem

Let $(V,R)$ be a reduced root system and $\Delta$ a base. Then:

• $W\cong$ the group generated by $\{S_{\alpha }:\alpha \in \Delta \}$ subject to the relations $(S_{\alpha }{S}_{\beta }{)}^{m_{\alpha ,\beta }}=1$ where $m_{\alpha ,\beta }$ is the smallest positive integer $m$ satisfying $(S_{\alpha }{S}_{\beta }{)}^m=1$;

• for $w\in W$, let $l(w)$ be the smallest number $l$ such that $w=S_{{\alpha }_{i1}}\cdots S_{{\alpha }_{il}}$. Then

l(w)=#{\beta\in R_+:w\beta\in R_{-}}.

Remark. The proof of this theorem can be found in the textbook (pages 63 and 66).