# Root systems

In this post, I will record some properties of reduced root systems, following Bourbaki. The reference is their “Lie groups and algebras, Ch. 4-6”. There will be no proofs, just a summary of the main properties. The post consists mainly of definitions and lists their properties.

Throughout, all root systems are assumed reduced.

Definition 1 A reduced root system is the data of a real vector space ${V}$ and a subset ${R\subset V}$ of roots such that

1. (i) ${R}$ is finite and generates ${V}$
2. (ii) ${\forall \alpha\in R}$ there exists ${\alpha^{\vee} \in V^{\vee}}$ such that
• ${\langle\alpha,\alpha^{\vee}\rangle=2}$
• ${s_\alpha:x\mapsto x-\langle\alpha^{\vee}\rangle\alpha}$ takes ${R}$ to ${R}$

Given ${(i)}$, note that ${\alpha^{\vee}}$ is unique.

3. (iii) ${\forall \alpha\in R}$ we have ${\alpha^{\vee}(R)\subset {\mathbb Z}}$
4. (iv) If ${\alpha\in R}$ then ${2\alpha\notin R}$

Remark 2

1. (i) The map ${s_\alpha}$ fixes ${L_\alpha:=\ker(\alpha^{\vee})}$ and takes ${\alpha\mapsto -\alpha}$. Therefore, ${\alpha\in R}$ gives ${-\alpha\in R}$.
2. (ii) The group generated by the ${s_\alpha}$ is called the Weyl group of ${R}$, denoted ${W(R)}$.
3. (iii) The coroots ${\alpha^{\vee}}$ form ${R^{\vee}\subset V^{\vee}}$ the dual root system, and ${W(R)\cong W(R^{\vee})}$. Warning: The bijection ${\alpha \leftrightarrow \alpha^{\vee}}$ is not linear.
4. (iv) The root system ${R}$ is irreducible if it cannot be written non-trivially as ${\oplus R_i \subset \oplus V_i}$.

— 1. The scalar product —

On an irreducible root system, up to scaling their is a unique scalar product on ${V}$ which is invariant under ${W(R)}$. Denote it by ${(x|y)}$. We could take it to be, for example

$\displaystyle (x|y) := \sum_{\alpha\in R} \alpha^{\vee}(x)\alpha^{\vee}(y)$

If we identify ${V}$ with ${V^{\vee}}$ using this scalar product, then we have

$\displaystyle \alpha^{\vee} = \frac{2\alpha}{(\alpha|\alpha)}$

Note that ${s_\alpha}$ become orthogonal reflections in ${L_\alpha:=\ker(\alpha^{\vee})}$. Then ${W(R)}$ acts transitively on the roots of the same length.

— 2. Possible angles —

Let ${n(\alpha,\beta):=\langle\alpha\beta^{\vee}\rangle\in {\mathbb Z}}$. Then we have

$\displaystyle \begin{array}{rcl} n(\alpha,\alpha) =& 2\\ s_\beta(\alpha) =& \alpha - n(\alpha,\beta)\beta\\ n(\alpha,\beta) = &\frac{2(\alpha|\beta)}{(\alpha|\alpha)} \end{array}$

We have the following cases:

• If ${|n(\alpha,\beta)|=|n(\beta,\alpha)|=2}$ then ${\alpha=\pm\beta}$.
• ${n(\alpha,\beta)=0}$
Then ${\measuredangle(\alpha,\beta)=\pi/2}$ and ${\|\alpha\|=\|\beta\|}$.

• ${n(\alpha,\beta)=n(\beta,\alpha)=1}$
Then ${\measuredangle(\alpha,\beta)=\pi/3}$ and ${\|\alpha\|=\|\beta\|}$.

• ${n(\alpha,\beta)=n(\beta,\alpha)=-1}$
Then ${\measuredangle(\alpha,\beta)=2\pi/3}$ and ${\|\alpha\|=\|\beta\|}$.

• ${n(\alpha,\beta)=1}$ but ${n(\beta,\alpha)=2}$
Then ${\measuredangle(\alpha,\beta)=\pi/4}$ and ${\|\beta=\sqrt{2}\|\alpha\|}$.

• ${n(\alpha,\beta)=-1}$ but ${n(\beta,\alpha)=-2}$
Then ${\measuredangle(\alpha,\beta)=3\pi/4}$ and ${\|{\beta}=\sqrt{2}\|\alpha\|}$.

The next cases arise only for ${G_2}$

• ${n(\alpha,\beta)=1}$ but ${n(\beta,\alpha)=3}$
Then ${\measuredangle(\alpha,\beta)=\pi/6}$ and ${\|\beta\|=\sqrt{3}\|\alpha\|}$.

• ${n(\alpha,\beta)=-1}$ but ${n(\beta,\alpha)=-3}$
Then ${\measuredangle(\alpha,\beta)=5\pi/6}$ and ${\|\beta\|=\sqrt{3}\|\alpha\|}$.

— 3. Chains of roots —

First, we have that

• If ${(\alpha|\beta)>0}$ then ${\alpha-\beta}$ is a root (unless ${\alpha=\beta}$)
• If ${(\alpha|\beta)<0}$ then ${\alpha+\beta}$ is a root (unless ${\alpha=-\beta}$)

Definition 3 Given non proportional roots ${\alpha,\beta\in R}$ the set

$\displaystyle S:=\{\beta + {\mathbb Z}\alpha \}\cap R$

is the ${\alpha}$-chain defined by ${\beta}$. The corresponding integer interval is ${[-q,p]}$. Call ${\beta-q\alpha}$ the origin and ${\beta+p\alpha}$ the end of the chain. The length is ${p+q}$. If ${\gamma}$ is the origin of an ${\alpha}$-chain, then the length of the chain is ${-n(\gamma,\alpha)}$.

— 4. Chambers, bases, Cartan matrix —

Definition 4 The chamber ${C}$ of a root system ${(R,V)}$ is a connected component of ${V\setminus \cup_{\alpha\in R} L_\alpha}$, where ${L_\alpha:=\ker \alpha^{\vee}}$ is a hyperplane.

Here are the properties:

• If ${x,y\in C}$ then ${(x|y)>0}$
• ${W(R)}$ acts transitively on the chambers
• We have dual chambers ${C^{\vee}\subset V^{\vee}}$ under the linear isomorphism given by the scalar product

Definition 5 Given a chamber ${C}$, a basis ${B(C)}$ of ${R}$ is the set of roots ${\{\alpha_1,\ldots,\alpha_l\}}$ such that ${\{L_{\alpha_i}\}}$ form the walls of ${C}$, and ${\alpha_i^{\vee}(x)>0}$ for all ${x\in C}$.

The properties are:

• ${B(C)}$ is a basis of ${V}$ in the usual sense,
• Any root is an integral linear combination of elements in ${B(C)}$, having either all positive, or all negative coefficients.
• For all ${i\neq j }$ we have ${(\alpha_i|\alpha_j)\leq 0}$.
• Any root can be transformed to an element of ${B(C)}$ using ${W(R)}$.
• The relationship with the duals is ${B(C^{\vee})=\{\alpha_1^{\vee}, \ldots, \alpha_n^{\vee}\}}$.

Definition 6 Set ${n_{ij}=n(\alpha_i,\alpha_j)}$ with the ${\alpha_i}$ forming a basis. The matrix ${(n_{ij})_{1\leq i \leq j \leq l}}$ is called the Cartan matrix of the root system.

The properties are:

• The Cartan matrix uniquely determines ${(R,V)}$.
• We have ${n_{ii}=2}$ and ${n_{ij}\in \{0,-1,-2,-3\}}$ for ${i\neq j}$.

— 5. Positivity and order —

Fix a chamber ${C}$ and thus a basis ${B(C)=\{\alpha_1,\ldots,\alpha_n\}}$.

Definition 7 An element ${x\in V}$ is positive if ${x}$ is a linear combination of the ${\alpha_i}$ with positive coefficients. An equivalent definition is that ${x}$ takes positive values on ${C^{\vee}}$.

The basis ${\{\alpha_i\}}$ can be characterized as those positive roots which are not the sum of two positive roots.

The properties are:

• Every root is either positive or negative. Let ${R_+(C)}$ denote the positive roots.
• ${s_{\alpha_i}}$ takes ${\alpha_i\mapsto -\alpha_i}$ and permutes the other elements of ${R_+(C)}$.
• The set of positive elements in ${V}$ contains ${C}$, but is usually bigger.
• The notion of positivity leads to a partial order relation on elements of ${V}$.
• We have that ${x\in C}$ if and only if ${x>w\cdot x}$ for all ${w\in W(R)\setminus id}$.
• Let ${\beta_1+\cdots+\beta_n}$ be a root with each ${\beta_i}$ positive. Then there exists a permutation ${\pi}$ such that for all ${i=1..n}$ we have that ${\beta_{\pi(1)}+\cdots+\beta_{\pi(i)}}$ is also a root.
• There exists a largest root

$\displaystyle \tilde{\alpha}=n_1 \alpha_1+\cdots + n_l \alpha_l$

such that for any root ${\alpha=p_1 \alpha_1 +\cdots p_l \alpha_l}$ we have that ${n_i\geq p_i}$ and ${\|\tilde{\alpha}\|\geq \|\alpha\|}$.

— 6. Weights (“poids” in French) —

Definition 8 The group ${P(R)\subset V}$ which takes integer values for all elements of ${R^{\vee}}$ is called the weight lattice. Let ${\varpi_1,\ldots,\varpi_l}$ be the basis dual to ${\{\alpha_1^{\vee},\ldots,\alpha_l^{\vee}\}=B(C^{\vee})}$. These are the fundamental weights. Positive integer combinations of the ${\varpi_i}$ are called dominant weights, denoted ${P_{++}(R)}$.

We have

$\displaystyle \rho = \frac 12 \sum_{\alpha\in R_+(C)}\alpha = \varpi_1 + \cdots + \varpi_l$