Weight lattices

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia
aram@unimelb.edu.au

Last update: 19 September 2012

$G{L}_n\left(ℂ\right)$

$$T=\{\left(\begin{array}{ccc}{x}_1& & 0\\ & \ddots & \\ 0& & x_n\end{array}\right)\mid x_i\in {ℂ}^{\times }\}$$

Define

$$\begin{array}{c}{X}^{{\epsilon }_i}:T⟶{ℂ}^{\times }\text{by}{X}^{{\epsilon }_i}\left(\begin{array}{ccc}{x}_1& & 0\\ & \ddots & \\ 0& & x_n\end{array}\right)=X_i,\\ h_{{\epsilon }_i^{\vee }}:{ℂ}^{\times }⟶T\text{by}{h}_{{\epsilon }_i^{\vee }}\left(t\right)=\left(\left(\begin{array}{ccccccc}1& & & & & & 0\\ & \ddots & & & & & \\ & & 1& & & & \\ & & & t& & & \\ & & & & 1& & \\ & & & & & \ddots & \\ 0& & & & & & 1\end{array}\right)\right)\text{.}\end{array}$$

Then

$$\begin{array}{ccc}{𝔥}_{\mathbb{Z}}^{*}& =& \{{\left(X^{{\epsilon }_1}\right)}^{{\lambda }_1}\dots {\left(X^{{\epsilon }_n}\right)}^{{\lambda }_n}\mid {\lambda }_1,\dots ,{\lambda }_n\in \mathbb{Z}\}\\ {𝔥}_{\mathbb{Z}}& =& \{{\left(h_{{\epsilon }_1^{\vee }}\right)}^{{\lambda }_1}\dots {\left(h_{{\epsilon }_1^{\vee }}\right)}^{{\lambda }_n}\mid {\lambda }_1,\dots ,{\lambda }_n\in \mathbb{Z}\}\\ N\left(T\right)& =& \left\{\genfrac{}{}{0ex}{}{n\times n\text{matrices with exactly one nonzero}}{\text{entry on each row and each column}}\right\}\\ W_0& =& \raisebox{1ex}{$N\left(T\right)$}\!\left/ \!\raisebox{-1ex}{$T$}\right.=\{wT\mid w\in S_n\}\text{where}\\ S_n& =& \left\{\text{permutation matrices}\right\}\text{.}\end{array}$$

$S{L}_n\left(ℂ\right)$

$$T=\{\left(\begin{array}{ccc}{x}_1& & 0\\ & \ddots & \\ 0& & x_n\end{array}\right)\mid x_1\dots x_n=1\}$$

With $X^{{\epsilon }_i}:T\to {ℂ}^{\times }$ and $h_{{\epsilon }_i^{\vee }}:{ℂ}^{\times }\to T_{G{L}_n}$ as in ???

$$\begin{array}{ccc}{𝔥}_{\mathbb{Z}}^{*}& =& ⟨X^{{\epsilon }_1},\dots ,X^{{\epsilon }_n}\mid X^{{\epsilon }_1}\dots X^{{\epsilon }_n}=1⟩\\ {𝔥}_{\mathbb{Z}}& =& ⟨h_{{\epsilon }_1^{\vee }-{\epsilon }_2^{\vee }},\dots ,h_{{\epsilon }_{n-1}^{\vee }-{\epsilon }_n^{\vee }}⟩=\{h_{{\lambda }_1{\epsilon }_1^{\vee }+\dots +{\lambda }_n{\epsilon }_n^{\vee }}\mid {\lambda }_1+\dots +{\lambda }_n=0\}\\ N\left(T\right)& =& \{\begin{array}{c}n\times n\text{matrices with}\\ \text{(a) exactly one nonzero entry in each row and each column}\\ \text{(b) product of the nonzero entries is 1}\end{array}\end{array}$$

$PG{L}_n\left(ℂ\right)$

$$T=\{\left[\begin{array}{ccc}{x}_1& & 0\\ & \ddots & \\ 0& & x_n\end{array}\right]\mid \left[\begin{array}{ccc}{x}_1& & 0\\ & \ddots & \\ 0& & x_n\end{array}\right]=\left[\begin{array}{ccc}\lambda x_1& & 0\\ & \ddots & \\ 0& & \lambda x_n\end{array}\right]\text{for}\lambda \in {ℂ}^{\times }\}$$

Then with $X^{{\epsilon }_i}:T_{G{L}_n}\to {ℂ}^{\times }$ and $h_{{\epsilon }_i^{\vee }}:{ℂ}^{\times }\to T_{G{L}_n}\to T_{PGL}$

$$\begin{array}{ccc}{𝔥}_{\mathbb{Z}}^{*}& =& \{x^{\lambda }=x^{{\lambda }_1{\epsilon }_1+\dots +{\lambda }_n{\epsilon }_n}\mid {\lambda }_i\in \mathbb{Z},{\lambda }_1+\dots +{\lambda }_n=0\}\\ {𝔥}_{\mathbb{Z}}& =& ⟨h_{{\epsilon }_1^{\vee }},\dots ,h_{{\epsilon }_n^{\vee }}\mid h_{{\epsilon }_1^{\vee }},\dots ,h_{{\epsilon }_n^{\vee }}=1⟩\text{.}\end{array}$$

For $S{L}_2$

$$\begin{array}{cc}{𝔥}_{\mathbb{Z}}^{*}& ε2 ε1 \end{array}$$

Identify points on diagonal lines.

The 2-fold cover of $\mathbb{Z}\text{–span}\{{\epsilon }_1-{\epsilon }_2\}$ is very visible.

$$\begin{array}{cc}{𝔥}_{\mathbb{Z}}& ε2∨ ε1∨ \end{array}$$ $${𝔥}_{\mathbb{Z}}=\mathbb{Z}\text{–span}\{{\epsilon }_1^{\vee }-{\epsilon }_2^{\vee }\}\text{.}$$

$S{L}_3$

$$ε2+ε3 ε1+ε3 ε1+ε2 ε1+ε2+ε3 ε2 ε3 ε1$$

The weight lattice of $S{L}_3$ as a 3–fold cover of

$$\text{span}\{{\epsilon }_1-{\epsilon }_2,{\epsilon }_2-{\epsilon }_3\}=\{{\lambda }_1{\epsilon }_1+{\lambda }_2{\epsilon }_2+{\lambda }_3{\epsilon }_3\mid {\lambda }_1+{\lambda }_2+{\lambda }_3=0\}\text{.}$$

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