Phan systems study group III

Sukru  lead this week’s discussion and worked through the root systems of type A_n, B_n, C_n, D_n. Before covering the maths I should first mention that Sukru feels that we should come up with a better title for the study group as he doesnt think that it is very reflective of what we are actually doing.  We are yet to agree on a new one so I have stuck with the old one.

Recall from the first study group post that a root system is a finite spanning subset \Phi of E=\mathbb{R}^n\backslash\{0\}  equipped with a positive definite symmetric bilinear form (.,.) such that

  • \Phi is invariant under \sigma_\alpha for each \alpha\in\Phi. (\sigma_\alpha is the reflection through the hyperplane orthogonal to \alpha.)
  • given \alpha\in\Phi, the only scalar multiples of \alpha in \Phi are \alpha and -\alpha.
  • \frac{2(\alpha,\beta)}{(\alpha,\alpha)} is an integer for all \alpha,\beta\in\Phi (crystallographic condition).

We begin with a general construction. Let B=\{e_1,\ldots,e_n\} be a basis for E=\mathbb{R}^n. Let I be the \mathbb{Z}-lattice of B and let \Phi be the set of all vectors in I which have certain length or lengths (two lengths). This set is finite, spans E and does not contain 0  (as we don’t allow vectors of length 0.)  The certain lengths are  chosen so that the only scalar multiples of \alpha in \Phi are \pm\alpha and the squared lengths divide 2. Note that for all \alpha,\beta\in I we have (\alpha,\beta)\in\mathbb{Z}. Since each \sigma_\alpha preserves I (by the crystallographic condition) and preserves lengths it follows that \sigma_\alpha  preserves \Phi. Hence \Phi is a root system.

Type A_n: Choose E\leqslant \mathbb{R}^{n+1} to be the subspace orthogonal to e_1+e_2+\cdots +e_{n+1}. Let I'=I\cap E and define \Phi=\{\alpha\in I'\mid (\alpha,\alpha)=2\}.  Then

\Delta=\{e_1-e_2,e_2-e_3,\ldots,e_n-e_{n+1}\}\subset\Phi

and in fact \Phi=\{\mp(e_i-e_j)\mid i\neq j\} and we can take \Delta  to be a set of simple roots. (check the first study group post for definition of simple roots).  Let W=\langle \sigma_\alpha\mid\alpha\in\Phi\rangle=\langle \sigma_\alpha\mid \alpha\in\Delta\rangle and set \alpha_i=e_i-e_{i+1}. Then

\displaystyle{\sigma_{\alpha_i}(e_i)=e_i-\frac{2(e_i-e_{i+1},e_i)}{(e_i-e_{i+1},e_i-e_{i+1})}(e_i-e_{i+1})=e_{i+1}}.

Also, \sigma_{\alpha_i}(e_{i+1})=e_i. Thus we can define an isomorphism \varphi: W\rightarrow S_{n+1}, \sigma_{\alpha_i}\mapsto (i,i+1).

The Coxeter graph is

An

Type B_n: Let E=\mathbb{R}^n and

\Phi=\{\alpha\in I\mid (\alpha,\alpha)=1 \text{ or }2 \} =\{\pm e_i\}\cup\{\mp (e_i\pm e_j) \mid i\neq j\}.

Then a set of simple roots is \alpha_1=e_1-e_2, \alpha_2=e_2-e_3,\ldots, \alpha_{n-1}=e_{n-1}-e_n, \alpha_n=e_n.  In particular, we can write e_i=(e_i-e_{i+1})+\ldots+(e_{n-1}-e_n)-e_n. Clearly the first n-1 are long roots and \alpha_n is a short root.

Let W=\langle \sigma_\alpha\mid\alpha\in\Phi\rangle=\langle \sigma_\alpha\mid \alpha\in\Delta\rangle. Then \sigma_{e_i}=-e_i and corresponds to a signed permutation which changes the sign of i and fixes all the other j. In fact V=\langle\sigma_{e_i}\rangle\vartriangleleft W.  Moreover, W/V\cong \langle\sigma_{\alpha_i}\mid\alpha_i=e_i-e_{i+1}\rangle\cong S_n. Thus W\cong C_2^n\rtimes S_n.

The Coxeter graph is

Bn

Type C_n, n\geq 3: Let E=\mathbb{R}^n and

\Phi=\{\alpha\in I\mid (\alpha,\alpha)=4 \text{ or } 2\}=\{\pm 2 e_i\}\cup\{\mp(e_i\pm e_j) \mid i\neq j\}.

This is the dual of B_n so we get the same Weyl group. A set of simple roots  now has n-1 short roots and 1 long root.

Type D_n, n\geq 4: Here E=\mathbb{R}^n and

\Phi=\{\alpha\in I\mid (\alpha,\alpha)=2\}=\{\pm (e_i\pm e_j)\mid i\neq j\}.

A set of simple roots is \alpha_1=e_1-e_2,\alpha_2=e_2-e_3,\ldots,\alpha_{n-1}=e_{n-1}-e_n,\alpha_n=e_{n-1}+e_n.

As seen in type A_n, for i=1,2,\ldots,n-1, we have \sigma_{\alpha_i}(e_i)=e_{i+1} and \sigma_{\alpha_i}(e_{i+1})=e_i. Thus \langle \sigma_{\alpha_i}\mid 1\leq i\leq n-1\rangle \cong S_n. Now

\displaystyle{ \sigma_{\alpha_n}(e_n)=e_n-\frac{2(e_{n-1}+e_n,e_n)}{(e_{n-1}+e_n,e_{n-1}+e_n)}(e_{n-1}+e_n)=-e_{n-1} }

and

\displaystyle{ \sigma_{\alpha_n}(e_{n-1})=e_{n-1}-\frac{2(e_{n-1}+e_n,e_{n-1})}{(e_{n-1}+e_n,e_{n-1}+e_n)}(e_{n-1}+e_n)=-e_n }

Hence \sigma_{\alpha_n} induces the signed permutation of \{\pm 1,\ldots, \pm n\} which interchanges n and n-1 and multiplies both by -1. Thus W=\langle \sigma_{\alpha_i}\rangle includes all the changes of signs which change an even number of signs. Hence W\cong C_2^{n-1}\rtimes S_n.

The Coxeter graph is

Dn

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One thought on “Phan systems study group III

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  1. In your general construction, ‘the squared lengths divide 2’ is too strong a condition, and is not satified by C_n. Instead it should just be ‘such that \frac{2(\alpha,\beta)}{(\alpha,\alpha)}\in \mathbb{Z}‘.

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