# The Unapologetic Mathematician

## Construction of A-Series Root Systems

Starting from our setup, we construct root systems corresponding to the $A_n$ Dynkin diagrams.

We start with the $n+1$-dimensional space $\mathbb{R}^{n+1}$ with orthonormal basis $\{\epsilon_0,\dots,\epsilon_n\}$, and cut out the $n$-dimensional subspace $E$ orthogonal to the vector $\epsilon_0+\dots+\epsilon_n$. This consists of those vectors $v=\sum_{k=0}^nv^k\epsilon_k$ for which the coefficients sum to zero: $\sum_{k=0}^nv^k=0$. We let $J=I\cap E$, consisting of the lattice vectors whose (integer) coefficients sum to zero. Finally, we define our root system $\Phi$ to consist of those vectors $\alpha\in J$ such that $\langle\alpha,\alpha\rangle=2$.

From this construction it should be clear that $\Phi$ consists of the vectors $\{\epsilon_i-\epsilon_j\vert i\neq j\}$. The $n$ vectors $\Delta=\{\alpha_i=\epsilon_{i-1}-\epsilon_i\}$ are independent, and thus form a basis of the $n$-dimensional space $E$. This establishes that $\Phi$ spans $E$. In particular, if $i we can write

$\displaystyle(\epsilon_i-\epsilon_j)=(\epsilon_i-\epsilon_{i+1})+(\epsilon_{i+1}-\epsilon_{i+2})+\dots+(\epsilon_{j-1}-\epsilon_j)$

showing that $\Delta$ forms a base for $\Phi$.

We calculate the Cartan integers for this base

$\displaystyle\frac{2\langle\epsilon_{j-1}-\epsilon_j,\epsilon_{i-1}-\epsilon_i\rangle}{\langle\epsilon_{i-1}-\epsilon_i,\epsilon_{i-1}-\epsilon_i\rangle}=\langle\epsilon_{j-1}-\epsilon_j,\epsilon_{i-1}-\epsilon_i\rangle$

For $i=j$ we get the value ${2}$; for $j+1$ or $j-1$ we get the value $-1$; otherwise we get the value ${0}$. This clearly gives us the Dynkin diagram $A_n$.

Finally, the reflections with respect to the $\alpha_i$ should generate the entire Weyl group. We must verify that these leave the lattice $J$ invariant to be sure that we have a root system. We calculate

\displaystyle\begin{aligned}\sigma_{\alpha_i}(v)&=v-\frac{2\langle v,\alpha_i\rangle}{\langle\alpha_i,\alpha_i\rangle}\alpha_i\\&=v-\langle v,\alpha_i\rangle\alpha_i\\&=v-(v^{i-1}-v^i)(\epsilon_{i-1}-\epsilon_i)\\&=v-(v^{i-1}\epsilon_{i-1}+v^i\epsilon_i)+(v^i\epsilon_{i-1}+v^{i-1}\epsilon_i)\end{aligned}

That is, it swaps the coefficients of $\epsilon_{i-1}$ and $\epsilon_i$, and thus sends the lattice $J$ back to itself, as we need.

We can also see from this effect that any combination of the $\sigma_{\alpha_i}$ serves to permute the $n+1$ coefficients of a given vector. That is, the Weyl group of the $A_n$ system is naturally isomorphic to the symmetric group $S_{n+1}$.

March 2, 2010 Posted by | Geometry, Root Systems | 4 Comments