The Unapologetic Mathematician

Mathematics for the interested outsider

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<j 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

   

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