Reading on Root Systems

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Basic Definitions

Roughly speaking, a root system is a collection of vectors having the properties that are satisfied by a semisimple Lie algebra [Hall].

[definition=%counter% (Etingof Definition 21.1)] Let $E$ be a finite-dimensional $\RR$ vector space with inner product $\langle-, -\rangle$. An abstract root system is a finite set $R \subseteq E\setminus 0$ such that

  • R1: $R$ spans $E$;
  • R2: For all $\alpha, \beta \in R$, the number $n_{\alpha\beta} \coloneqq \frac{2\langle\alpha, \beta\rangle}{\langle\alpha,\alpha\rangle} \in \ZZ$;
  • R3: If $\alpha, \beta \in R$, then $s_\alpha(\beta) \coloneqq \beta - n_{\alpha\beta}\alpha \in R$.

The elements of $R$ are called roots and $\dim_\RR(E)$ is the rank of $R$. [/definition]

[definition=%counter% (Etingof Definition 21.2)] A root system $R$ is reduced if for $\alpha, c\alpha \in R$, we have $c = \pm 1$. [/definition]

[remark=%counter%] The $s_\alpha$’s are geometrically interpreted as orthogonal reflections that pointwise fix the hyperplane

\[\langle \alpha, x \rangle = 0.\]

All these claims can be verified easily by just direct computation. [/remark]

[example=%counter% (Etingof Example 21.4)] The root system of $\lie{sl}_n$ is called $A_{n-1}$ where the roots are given by $e_i - e_j$ for $i\neq j$ (where $e_i$ is the ith standard basis vector of $\RR^n$). We then set $V$ to be the span of such vectors and we claim that

\[V = \{x \in \RR^n \mid \text{components of } x \text{ sum to zero}\}.\]

Note that $\subseteq$ is apparent and $\supseteq$ takes a little bit of work.

So then notice that $s_{e_i - e_j}$ acting on $v = \sum_{k=1}^n c_k e_k \in V$ gives

\[\begin{align*} v - \frac{2}{\left< e_i-e_j, e_i-e_j\right>}\left<e_i - e_j ,\; v\right>(e_i-e_j) \end{align*}\]

This simplifies as

\[\begin{align*} v - (c_i - c_j)(e_i - e_j) = v + (-c_i + c_j)e_i + (-c_j + c_i)e_j. \end{align*}\]

Thus, $s_{e_i - e_j}$ effectively permutes the $i$th and $j$th coordinates and can be realized as the transposition $(ij)$ in $S_n$. [/example]

Rank $1$ root systems

If $R$ is a reduced root system of rank one, then $R$ is of the form $\{\pm \alpha\}$.

Rank $2$ root systems

Let $\alpha$ and $\beta$ be L.I. roots in $R$, set

\[E' = \RR\alpha \oplus \RR\beta \subseteq E.\]

Then $R’ = R \cap E’$ is a root system in $E’$ of rank $2$. A first step in classifying higher rank root systems is to classify the reduced rank $2$ root systems.

[theorem=%counter% (Etingof Theorem 21.10)] Let $R$ be a reduced root system and $\alpha, \beta$ be two linearly independent roots with $|\alpha| \ge |\beta|$. Let

\[\phi = \arccos\frac{\left<\alpha, \beta\right>}{2|\alpha||\beta|}\]

be the angle between $\alpha$ and $\beta$. Then we have one of the following possibilities:

  • (1) $\phi = \pi/2$, $n_{\alpha\beta} = n_{\beta\alpha} = 0$;
  • (2a) $\phi = 2\pi/3$, $|\alpha|^2 = |\beta|^2$, $n_{\alpha\beta} = n_{\beta\alpha} = -1$;
  • (2b) $\phi = \pi/3$, $|\alpha^2| = |\beta^2|$, $n_{\alpha\beta} = n_{\beta\alpha} = -1$;
  • (3a) $\phi = 3\pi/4$, $|\alpha|^2 = 2|\beta|^2$, $n_{\alpha\beta} = -1$, $n_{\beta\alpha} = -2$;
  • (3b) $\phi = \pi/4$, $|\alpha|^2 = 3|\beta|^2$, $n_{\alpha\beta} = 1, n_{\beta\alpha} = 2$;
  • (4a) $\phi = 5\pi/6$, $|\alpha|^2 = 3|\beta|^2$, $n_{\alpha\beta} = -1$, $n_{\beta\alpha} = -3$;
  • (4b) $\phi = \pi/6$, $|\alpha|^2 = 3|\beta|^2$, $n_{\alpha\beta} = 1$, $n_{\beta\alpha} = 3$. [/theorem]

[proof=Proof Idea] Recall that

\[\phi = \arccos\left(\frac{\left<\alpha, \beta\right>}{2|\alpha||\beta|}\right).\]

So we obtain that

\[n_{\alpha\beta} = 2\frac{|\beta|}{|\alpha|} \cos\phi.\]

Thus,

\[n_{\alpha\beta}n_{\beta\alpha} = 4\cos^2\phi.\]

Since the LHS is an integer, it follows that the RHS can only take on the values $0, 1, 2, 3$ (after recalling that $\alpha$ and $\beta$ are L.I. so $\phi \neq 0$ nor $\phi \neq \pi$). [/proof]

All of these possibilities are realized:

  • $\phi = \pi/2$, we get $A_1 \times A_1$ ($\lie{sl}_2(\CC) \oplus \lie{sl}_2(\CC)$ root system);
  • $\phi = \pi/3$ or $2\pi/3$ we get $A_2$ ($\lie{sl}_3$ root system).
  • $\phi = \pi/4$ or $3\pi/4$ we get $B_2$ ($\lie{sp}4 \cong \lie{so}_5$ root system).
  • $\phi = \pi/6$ or $5\pi/6$ we get $G_2$.

The type $G_2$ root system is generated by $\alpha, \beta$ with

\[\left<\alpha, \alpha\right> = 6, \quad \left<\beta, \beta\right> = 2, \quad \left<\alpha, \beta\right> -3\]

and

\[R = \{\pm \alpha, \pm \beta, \pm(\alpha + \beta), \pm(\alpha + 2\beta), \pm(\alpha + 3\beta), \pm(2\alpha + 3\beta)\}.\]

See Wikipedia for the pictures of the rank $2$ root systems: https://en.wikipedia.org/wiki/Root_system. Note that $A_2 \cong D_2$

[theorem=%counter% (Etingof Theorem 21.10)] Any reduced rank $2$ root system $R$ is of the form $A_1 \times A_1$, $A_2$, $B_2$ or $G_2$. [/theorem]

Positive and Simple Roots

[definition=%counter% (Etingof)] Let $R$ be a reduced root system and $t \in E^*$ be such that $t(\alpha) \neq 0$ for any $\alpha \in R$. We say that $\alpha$ is positive with respect to $t$ if $t(\alpha) > 0$ and negative if $t(\alpha) < 0$.

We denote the set of positive roots by $R_+$ and the negative ones by $R_-$. By linearity, we have

\[R_+ = -R_-\]

and $R = R_+ \sqcup R_-$. This decomposition of $R$ is called a polarization of $R$ since it depends on the choice of $t$. [/definition]

[example=%counter% (Etingof Example 21.12)] Let $R$ be type $A_{n-1}$. Then if we let

\[t = \sum_{i = 1}^n t_i e_i^*.\]

Then $t(\alpha) \neq 0$ for all $\alpha$ iff $t_i \neq t_j$ for $i \neq j$.

Suppose in particular that

\[t_1 > t_2 > \cdots > t_n.\]

Then $e_i - e_j \in R_+$ iff $i < j$. [/example]

[proof=Proof of first claim] ($\implies$) Write $\alpha = e_i - e_j$ for some $i \neq j$. Then

\[t(\alpha) = t_i - t_j\]

and so the claim follows.

($\impliedby$) Identical argument. [/proof]

The second claim is obvious via the first claim so we omit the proof.

[definition=%counter% (Etingof Definition 21.13)] A root $\alpha \in R_+$ is simple if it is not a sum of two other positive roots. [/definition]

[lemma=%counter% (Etingof Lemma 21.14)] Every positive root is a sum of simple roots. [/lemma]

[proof] Let $\alpha \in R_+$. If $\alpha$ is simple, there is nothing to prove. So suppose $\alpha$ is not simple. Then

\[\alpha = \beta + \gamma\]

where $\beta, \gamma \in R_+$. By linearity,

\[t(\alpha) = t(\beta) + t(\gamma)\]

so then

\[t(\beta), t(\gamma) < t(\alpha).\]

If $\beta$ or $\gamma$ is not simple, we continue recursively. This process terminates since $t$ has finitely many values on $R$ as $R$ itself is finite. [/proof]

[lemma=%counter% (Etingof Lemma 21.15)] If $\alpha, \beta \in R_+$ are simple roots, then

\[\left<\alpha, \beta\right> \le 0.\]

[/lemma]

[proof] Suppose $\left<\alpha, \beta\right> > 0$. Then

\[\left<-\alpha, \beta\right> \le 0.\]

So since $-\alpha, \beta$ are independent roots, $\gamma = \beta - \alpha$ is a root via the classification of rank $2$ root systems. If $\gamma$ is positive, then $\beta = \alpha + \gamma$ is not simple. If $\gamma$ is negative, then $-\gamma$ is positive so $\alpha = \beta + (-\gamma)$ is not simple. [/proof]

[theorem=%counter%] The set $\Pi\subseteq R_+$ of simple roots is a basis of $E$. [/theorem]

[remark=%counter%] Since $E \cong \RR^r$ as an $\RR$-vector space, it follows that

\[\Pi = (\alpha_1, \ldots, \alpha_r)\]

where each $\alpha_i \in R_+$ is a simple root. [/remark]

In the following, the polarization is given by

\[t = \sum_{i=1}^n t_i e_i^*\]

where

\[t_1 > t_2 > \cdots > t_n.\]

[example=%counter% (Etingof Example 21.18 Part 1)] If $R$ is type $A_{n-1}$ (i.e. $\lie{sl}_n$), then the simple roots are

\[\alpha_i = e_{i} - e_{i+1}\]

for $1 \le i \le n - 1$. [/example]

[example=%counter% (Etingof Example 21.18 Part 3)] If $R$ is type $B_n$ (i.e. $\lie{so}_{2n+1}$), then the simple roots are

\[\alpha_1 = e_1 - e_2, \quad \ldots, \quad a_{n-1} = e_{n-1} - e_n, \quad \alpha_n = e_n.\]

[/example]

[example=%counter% (Etingof Example 21.18 Part 2)] If $R$ is type $C_n$ (i.e. $\lie{sp}_{2n}$), then the simple roots are

\[\alpha_1 = e_1 - e_2, \quad \ldots, \quad a_{n-1} = e_{n-1} - e_n, \quad \alpha_n = 2e_n.\]

[/example]

[example=%counter% (Etingof Example 21.18 Part 2)] If $R$ is type $D_n$ (i.e. $\lie{so}_{2n}$), then the simple roots are

\[\alpha_1 = e_1 - e_2, \quad \ldots, \quad a_{n-1} = e_{n-1} - e_n, \quad \alpha_n = e_{n-1} + e_n.\]

[/example]

[corollary=%counter% (Etingof Corollary 21.19)] Any root $\alpha$ can be uniquely written as

\[\alpha = \sum_{i=1}^r n_i \alpha_i\]

where $n_i \in \ZZ$. If $\alpha$ is positive, then $n_i \ge 0$ for all $i$ and if $\alpha$ is negative then $n_i \le 0$ for all $i$. [/corollary]

[proof]For integrality, note that

\[\alpha - \sum_{\substack{i=1 \\ i \neq j}}^r n_i \alpha_i = n_j\alpha_j.\]

The uniqueness part holds via the fact that $\Pi$ is a basis for $E$. For integrality, we note that $R = R_+ \sqcup R_-$ with $R_+ = -R_-$ so it suffices to prove the claim for $\alpha \in R_+$. But notice that since every positive root is just a sum of simple roots, the claim immediately follows. In fact, this argument even shows the second claim. [/proof]

Duality

[definition=%counter%] Let $R$ be a root system. If $\alpha \in R$, we call the linear functional

\[\alpha^\vee \coloneqq 2\frac{\left<\alpha, -\right>}{\left<\alpha, \alpha\right>}\]

a coroot. The set of coroots

\[R^\vee \coloneqq \{\alpha^\vee \mid \alpha \in R\} \subseteq E^*\]

is called the dual root system to $R$. [/definition]

[remark=%counter%] The inner product on $E$ induces an inner product on the dual space. In particular, there is an isomorphism via the Riesz representation theorem

\[\begin{align*} \nu:E &\to E^* \\ f \mapsto \left<f, -\right> \end{align*}\]

the induced inner product on the dual is

\[\left<f, g \right>_{E^*} = \left<\nu^{-1}(f), \nu^{-1}(g)\right>_E.\]

By definition, it follows that

\[\alpha^\vee = \frac{2\nu(\alpha)}{\left<\alpha, \alpha\right>}.\]

So then

\[\begin{align*} \left<\alpha^\vee, \alpha^\vee\right>_{E^*} &= \frac{4}{\left<\alpha, \alpha\right>^2} \left<\nu(\alpha), \nu(\alpha)\right>_{E^*}\\ &= \frac{4}{\left<\alpha, \alpha\right>^2}\left<\alpha, \alpha\right> = \frac{4}{\left<\alpha, \alpha\right>}. \end{align*}\]

And so

\[\begin{align*} (\alpha^\vee)^\vee (f) &= \frac{2\left<\alpha^\vee, f\right>}{\left<\alpha^\vee, \alpha^\vee\right>} \\ &= \left<\nu(\alpha), f\right>_{E^*} \\ &= \left<\alpha, \nu^{-1}(f)\right>. \end{align*}\]

But since $f = \left<R_f, -\right>$ where $R_f$ is the Riesz representor of $f$, we have that

\[\left<\alpha, \nu^{-1}(f)\right> = \left<R_f, \alpha\right> = f(\alpha).\]

Thus,

\[(\alpha^\vee)^\vee(f) = f(\alpha)\]

for all $f \in E^*$. So it follows that $(\alpha^\vee)^\vee = \alpha$.

So more generally it actually follows that $(R^\vee)^\vee = R$. [/remark]

[remark=%counter%] Polarizations of $R$ inducce polarizations of $R^\vee$ induce a polarization on $R^\vee$ via the isomorphism $E \to E^*$ so that the corresponding system $\Pi^\vee$ of simple roots consists of $\alpha_i^\vee$ for $\alpha_i\in \Pi$. [/remark]

[definition=%counter%] Let $R_1 \subseteq E_1$ and $R_2 \subseteq E_2$ be root systems. An isomorphism of root systems

\[\phi:R_1 \to R_2\]

is an isomorphism $\phi:E_1 \to E_2$ which maps $R_1 \to R_2$ and preserves $n_{\alpha\beta} = \frac{2\left<\alpha, \beta\right>}{\left<\alpha, \alpha\right>}$. [/definition]

[example=%counter%] Type $A_{n-1}$ is self-dual. Indeed, recall the roots are

\[\alpha_{ij} = e_i - e_{j}\]

for $i \neq j$. So then the roots are all uniform length and we can renormalize the inner product so that $\left<\alpha, \alpha\right> = 2$ for all $i$. So then

\[\alpha_{ij}^\vee = \left<e_i - e_{j}, -\right>.\]

Now we define $\phi:R \to R^\vee$ to be the isomorphism $E \to E^*$ via the inner product restricted to $R$ (which, by the above remark, restricts to $R^\vee$). So it follows that

\[\phi(\alpha_{ij}) = \alpha_{ij}^\vee.\]

So then if $\alpha, \beta$ are in $R$, we have that

\[n_{\alpha^\vee\beta^\vee} = \left<\alpha^\vee, \beta^\vee\right>_{E^*} = \left<\alpha, \beta\right>_{E} = n_{\alpha\beta}.\]

by definition. [/example]

[example=%counter%] By similar argument, $D_n$ and $G_2$ are also self-dual, though, $G_2$ requires more care. On the other hand, $B_n$ is dual to $C_n$. [/example]

Weight Lattices

[definition=%counter%] A lattice in a real vector space $E$ is a subgroup $Q \subseteq E$ generated by a basis of $E$. Every lattice is conjugate to $\ZZ^n \subseteq \RR^n$ by an element of $\GL_n(\RR)$.

Let $Q$ be a lattice. The dual lattice $Q^{*}$ is the set of $f \in E^{*}$ so that $f(v) \in \ZZ$ for all $v \in Q$. So, in particular, if $Q$ is generated by a basis $e_i$ of $E$, then $Q^*$ is generated by the dual basis $e_i^{*}$. [/definition]

[definition=%counter%] Let $R$ be a root system. The root lattice $Q\subseteq E$ of $R$ is the lattice generated by the simple roots with respect to some polarization of $R$.

The coroot lattice $Q^\vee \subseteq E^*$ generated by $\alpha^\vee$ is the root lattice of $R^\vee$.

The weight lattice $P\subseteq E$ is the dual lattice to $Q^\vee$ and the coweight lattice $P^\vee \subseteq E^*$ is the dual lattice to $Q$. Thus,

\[P = \{\lambda \in E \mid \left<\lambda, \alpha^\vee\right> \in \ZZ, \forall \alpha \in R\}\]

and

\[P^\vee = \{\lambda \in E^* \mid \left<\lambda, \alpha\right> \in \ZZ, \forall \alpha \in R\}.\]

Here, we are using the usual shorthand so that $\left<\alpha^\vee, \beta\right> = n_{\alpha\beta} \in \ZZ$. So $Q\subseteq P$ and $Q^\vee \subseteq P^\vee$. [/definition]

[definition=%counter%] Given a system of simple roots $\Pi = \{\alpha_1, \ldots, \alpha_r\}$, we define fundamental coweights $\omega_i^\vee$ to be the dual basis to $\alpha_i$ and fundamental weights $\omega_i$ to be the dual basis to $\alpha_i^\vee$. So then $P$ is generated by $\omega_i$ and $P^\vee$ by $\omega_i^\vee$. [/definition]

[example=%counter% (Etingof Example 21.20)] Let $R$ be type $A_1$. Then $\left<\alpha, \alpha^\vee\right> = 2$ for the unique positive root (which is also a simple root). So then the fundamental weight is dual to $\alpha^\vee = 2\left<\alpha, -\right>/\left<\alpha, \alpha\right>$. So, in particular, $\omega = \frac{1}{2}\alpha$. [/example]

Dynkin Diagrams

[definition=%counter%] A root system $R$ is reducible if there exists a (nontrivial) orthogonal decomposition $E = E_1 \oplus E_2$ so that every element of $R$ is either in $E_1$ or $E_2$. Otherwise $R$ is said to be irreducible. [/definition]

[remark=%counter%] Every root system is uniquely realized as a union of irreducible root systems. This decomposition is obtained via a maximal decomposition of $\Pi$ into mutually orthogonal systems of simple roots. Therefore, it suffices to classify the irreducible root systems. [/remark]

[definition=%counter%] The Cartan matrix $A$ of $R$ is defined by

\[a_{ij} = n_{\alpha_j\alpha_i} = \left<\alpha_i^\vee, \alpha_j\right>.\]

[/definition]

By previous results, the next result is an obvious corollary.

[proposition=%counter% (Etingof Proposition 23.4)] Let $A$ be the Cartan matrix of $R$.

  1. $a_{ii} = 2$.
  2. $a_{ij} \in \ZZ_{\le 0}$.
  3. $a_{ij}a_{ji} = 4\cos^2\phi \in {0, 1, 2, 3}$ where $\phi$ is the angle between $\alpha_i$ and $\alpha_j$.
  4. If $d_i = |\alpha_i|^2$, then the matrix $d_ia_{ij}$ is symmetric (obvious) and positive definition (less obvious). [/proposition]

[definition=%counter%] Let $A$ be the Cartan matrix of $R$. The Dynkin diagram of $R$ is given by:

  • Vertices are indices $i$.
  • Vertices $i$ and $j$ are connected by $a_{ij}a_{ji}$ lines.
  • If $a_{ij} \neq a_{ji}$, then $|\alpha_i|^2 \neq |\alpha_j|^2$ and so we draw an arrow from the long root to the short root. [/definition]

[proposition=%counter% (Etingof Proposition 23.6)] The Cartan matrix determines the root system uniquely. [/proposition]

[theorem=%counter% (Etingof Theorem 23.7)] Connected Dynkin diagrams are classified by the diagrams $A_n, B_n, C_n, D_n, E_6, E_7, E_8, F_4, G_2$. [/theorem]

See Wikipedia for a picture.

[definition=%counter%] We say that a Dynkin diagram is simply laced if all the edges are simple. In other words, the Cartan matrix is symmetric and the roots all have the same length. [/definition]

Weyl and Coxeter Group Stuff

[definition=%counter% (Etingof Definition 21.6)] The Weyl group of a root system $R$ is the group of automorphisms of $E$ generated by $s_\alpha$. [/definition]

[proposition=%counter% (Etingof Proposition 21.7)] $W$ is a finite subgroup of $O(E)$ that preserves $R$. [/proposition]

[definition=%counter% (Humphreys)] A Coxeter system is a pair $(W, S)$ where $W$ is a group generated by $S \subseteq W$ subject only to the relations

\[(ss')^{m(s,s')} = 1\]

where

  • $m(s, s) = 1$;
  • $m(s, s’) = m(s’, s) \ge 2$ for $s \neq s’$;
  • $m(s, s’) = \infty$ (no relations).

We call $|S|$ the rank of $(W, S)$ and typically refer to $W$ as a Coxeter group. [/definition]

[remark=%counter%] Every Weyl group is a Coxeter system with $S$ given by $s_\alpha$. [/remark]

[definition=%counter%] Given a Coxeter system $(W, S)$ (with $S$ finite), there is a canonical matrix $A$, called the Coxeter matrix, indexed by $S$ with entries in $\ZZ\cup \infty$ given by $m(s, s’)$. [/definition]

[definition=%counter%] Given a Coxeter system $(W, S)$ (with $S$ finite), the associated Coxeter graph is the undirected graph $\Gamma$ with $S$ as a vertex set joining vertices $s$ and $s’$ by an edge labeled $m(s, s’)$ whenever this number is at least $3$ (including $\infty$). If distinct vertices $s$ and $s’$ are not joined, then $m(s, s’) = 2$. [/definition]

[remark=%counter%] The Coxeter graph of a Weyl coincides with the Dynkin graph. [/remark]

References