Root SL2 Explorer

Root SL2 Explorer

Choose a construction and inspect the rank-one subgroup attached to a root.

ConstructionGeck minusculeNaturalSteinberg classical

Dynkin TypeABCDE

Rank

Minuscule OrbitΨ₁ = Wϖ₁ (4 weights)Ψ₂ = Wϖ₂ (6 weights)Ψ₃ = Wϖ₃ (4 weights)

Selected Root

Construction	Geck minuscule construction
Root	$\alpha=\alpha_{1}$ (1, 0, 0)
Coroot	$\alpha^\vee=\alpha^\vee_{1}$ (1, 0, 0)
Minuscule orbit	$$\Psi_1=W\varpi_{1}$$ (4 basis vectors)
Root subgroup	$X_\alpha=\{x_\alpha(t)=1+t\bar e_\alpha\}$

Embedded Matrix

The embedding $\varphi_\alpha : \mathrm{SL}_2 \to G$ sends $\begin{pmatrix}a&b\\c&d\end{pmatrix}$ to the following matrix on the displayed basis.

$a$

$b$

$c$

$d$

$ad-bc=1$

$\varphi_{\alpha_{1}}\!\left(\begin{smallmatrix}a&b\\c&d\end{smallmatrix}\right)$ with basis labels shown on rows and columns:

Use fixed w0-guided basis order

The basis is the selected minuscule basis. For each alpha-string$\{\mu,\mu+\alpha\}$ with$(\mu,\alpha^\vee)=-1$, the ordered pair$(z_{\mu+\alpha},z_\mu)$ carries the standard two-dimensional representation; orthogonal weights are fixed.

	$z_{-\varpi_{1} + \varpi_{2}}$	$z_{-\varpi_{2} + \varpi_{3}}$	$z_{-\varpi_{3}}$	$z_{\varpi_{1}}$
$z_{-\varpi_{1} + \varpi_{2}}$	$d$	$0$	$0$	$c$
$z_{-\varpi_{2} + \varpi_{3}}$	$0$	$1$	$0$	$0$
$z_{-\varpi_{3}}$	$0$	$0$	$1$	$0$
$z_{\varpi_{1}}$	$b$	$0$	$0$	$a$

The Corresponding SL2

$x_\alpha(t)=1+t\bar e_\alpha$ raises weights with pairing -1.

$x_{-\alpha}(u)=1+u\bar e_{-\alpha}$ lowers weights with pairing 1.

$n_\alpha(t)=x_\alpha(t)x_{-\alpha}(-t^{-1})x_\alpha(t)$ lifts the reflection across alpha.

$h_\alpha(a)=n_\alpha(a)n_\alpha(-1)$ acts diagonally by $z_\mu\mapsto a^{(\mu,\alpha^\vee)}z_\mu$.

Basis vector	Pairing	$x_\alpha(t)$	$x_{-\alpha}(u)$	$h_\alpha(a)$
$z_{-\varpi_{1} + \varpi_{2}}$	-1	$z_\mu\mapsto z_\mu+t z_{\varpi_{1}}$	$z_\mu\mapsto z_\mu$	$z_\mu\mapsto a^{-1} z_\mu$
$z_{\varpi_{1}}$	1	$z_\mu\mapsto z_\mu$	$z_\mu\mapsto z_\mu+u z_{-\varpi_{1} + \varpi_{2}}$	$z_\mu\mapsto a z_\mu$

Cartan Matrix

	$\alpha_1$	$\alpha_2$	$\alpha_3$
$\alpha_1^\vee$	2	-1	0
$\alpha_2^\vee$	-1	2	-1
$\alpha_3^\vee$	0	-1	2

Longest Weyl Element Representative

$w_0=s_{1}\cdot s_{2}\cdot s_{1}\cdot s_{3}\cdot s_{2}\cdot s_{1}$ in the displayed minuscule basis:

	$z_{-\varpi_{1} + \varpi_{2}}$	$z_{-\varpi_{2} + \varpi_{3}}$	$z_{-\varpi_{3}}$	$z_{\varpi_{1}}$
$z_{-\varpi_{1} + \varpi_{2}}$	$0$	$1$	$0$	$0$
$z_{-\varpi_{2} + \varpi_{3}}$	$-1$	$0$	$0$	$0$
$z_{-\varpi_{3}}$	$0$	$0$	$0$	$1$
$z_{\varpi_{1}}$	$0$	$0$	$-1$	$0$