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

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