Back to main page

Throughout this page, we will let $\Phi$ be a root system of rank $\ell$ in a euclidean space $E$, with Weyl group $W$. Unlike in Humphreys, we will follow the convention that $\innerprod{-}{-}$ is the inner product of $E$.

10.1 Bases and Weyl chambers

Definition (Base)

[definition] A subset $\Delta$ of $\Phi$ is called a base if:

1. $\Delta$ is a basis of $E$,
2. each root $\beta \in \Phi$ can be written as a $\ZZ$-linear combination of elements of $\Delta$ where the coefficients are all nonnegative or all nonpositive.

The elements of $\Delta$ are called simple roots. The height of a root $\beta = \sum_{\alpha \in \Delta} k_\alpha \alpha$ (relative to $\Delta$) is the sum

If all $k_\alpha \ge 0$ (resp. all $k_\alpha \le 0$), then we say $\beta$ is positive (resp. negative) and write $\beta \succ 0$ (resp. $\beta \prec 0$). We define $\Phi^+$ to be the set of positive rotos and $\Phi^{-}$ to be the set of all negative roots. [/definition]

Angle between simples is obtuse

[lemma] If $\Delta$ is a base of $\Phi$, then $\langle \alpha, \beta\rangle \le 0$ for $\alpha \neq \beta$ in $\Delta$, and $\alpha - \beta$ is not a root. [/lemma]

[proof] The second claim is obvious by the second axiom for a base of a root system. Now we show the first claim. Suppose that $\innerprod{\alpha}{\beta} > 0$. Since $\alpha \neq \beta$, and we also have that $\alpha \neq -\beta$ (otherwise $\Delta$ wouldn’t be a base), we claim that $\alpha - \beta$ is, in fact, a root. By the classification of angles between roots, it must be the case that $\innerprod{\alpha}{\beta^\vee} = 1$ or $\innerprod{\alpha^\vee}{\beta} = 1$. Without loss of generality, we will assume the former. Then

[/proof]

Definition of Weyl chambers

[definition] The hyperplanes

for $\alpha \in \Phi$ partition $E$ into finitely many regions. We call the connected components of $E - \bigcup_{\alpha\in\Phi} P_\alpha$ the (open) Weyl chambers of $E$. [/definition]

Definition of regular and singular vectors

[definition] For every vector $\gamma \in E$, define

We say that $\gamma \in E$ is regular if it belongs to one of the Weyl chambers of $E$ (which we will denote by $\mathfrak{C}(\gamma)$). Otherwise, $\gamma$ is singular. [/definition]

Notice that if $\gamma$ is regular, then

Indeed, note that if $\gamma$ is singular, then it is orthogonal too one of the roots. So, at best, we’d only have $\supseteq$.

Definition of (in)decomposable

[definition] Let $\gamma \in E$ be regular. We say that $\alpha \in \Phi^+$ is decomposable if

for some $\beta_i \in \Phi^+(\gamma)$, indecomposable otherwise. [/definition]

Every root system admits a base

[theorem] Let $\gamma \in E$ be regular. Then the set $\Delta(\gamma)$ of all indecomposable roots in $\Phi^+$ is a base of $\Phi$, and every base is obtainable in this manner. [/theorem]

Weyl chambers are in bijection with bases

Notice that $\gamma, \gamma’$ with the property that $\mathfrak{\gamma} = \mathfrak{\gamma}$ means that $\gamma,\gamma’$ lie on the same side of each hyperplane. So $\Phi^+(\gamma) = \Phi^+(\gamma’)$ and so $\Delta(\gamma) = \Delta(\gamma’)$.

For a fixed base $\Delta = \Delta(\gamma)$, we say that $\mathfrak{C}{\gamma}$ is the fundamental Weyl chamber relative to $\Delta$ and we denote it as $\mathfrak{C}(\Delta)$.

Weyl group action on bases and chambers

The Weyl group sends one Weyl chamber onto another one (because it is generated by the simple reflections and simple reflections are reflectoins across a hyperplane). Since the Weyl group consists of orthogonal isomorphisms, $W$ sends bases to bases.

10.2 Lemmas on simple roots

Lemma A

[lemma] If $\alpha$ is positive but not simple, then $\alpha - \beta$ is a root (necessarily positive) for some $\beta \in \Delta$. [/lemma]

Corollary

[corollary] Each $\beta \in \Phi^+$ can be written in the form $\alpha_1 + \cdots + \alpha_k$ (where each $\alpha_i$ is not necessarily distinct) in such a way that each partial sum $\alpha_1 + \cdots + \alpha_i$ is a root. [/corollary]

Lemma B

[lemma] Let $\alpha$ be simple. Then $s_\alpha$ permutes the positive roots other than $\alpha$. [/lemma]

Corollary

[corollary] Set

Then $s_\alpha(\delta) = \delta - \alpha$. [/corollary]

Notice this is obvious from Lemma B.

Lemma C

[lemma] Let $\alpha_1, \ldots, \alpha_t \in \Delta$ (not necessarily distinct). Write $s_i = s_{\alpha_i}$. If

is negative, then for some index $1\le s < t$,

[/lemma]

Corollary

[corollary] If $s = s_1 \cdots s_t$ is an expression for $s \in W$ is a reduced word, then $s(\alpha_t)$ is a negative root. [/corollary]

10.3 The Weyl group

Properties of the Weyl group

[theorem] Let $\Delta$ be a base of $\Phi$.

1. If $\gamma in E$ is regular, there exists $s\in W$ such that $\innerprod{s(\gamma)}{\alpha} > 0$ for all $\alpha \in \Delta$ (i.e. $W$ acts transitively on Weyl chambers).
2. If $\Delta’$ is another base of $\Phi$, then $s(\Delta’) = \Delta$ for some $s \in W$ (i.e. $W$ acts transitively on bases).
3. If $\alpha$ is any root, there exists $s \in W$ such that $s(\alpha) \in \Delta$.
4. $W$ is generated by the $s_\alpha$ for $\alpha \in \Delta$.
5. If $s(\Delta) = \Delta$ with $s \in W$, then $s$ is the identity. [/theorem]

Definition (reduced and length)

[definition] Let $w \in W$ and write

so that each $\alpha_i \in \Delta$ and $t$ is minimal. Then we say that $w$ is of reduced expression and write $\ell(w) = t$: the length of $w$ relative to $\Delta$. [/definition]

Lemma A (alternate characterization of length)

[lemma] Let $w \in W$ and define $n(w)$ to be the number of positive roots $\alpha$ for which $w\alpha \prec 0$. Then, $\ell(w) = n(w)$. [/lemma]

Lemma B (closure of fundamental Weyl chamber is a fundamental domain of $W$)

[lemma] Let $\lambda, \mu \in \overline{\mathfrak{C}{\Delta}}$. If $w\lambda = \mu$ for some $w \in W$, then $w$ is a product of simple reflections which fix $\lambda$. In particular, $\lambda = \mu$. [/lemma]