Buildings, geometries and algebraic groups study group X

This week Sükrü led the discussion which focussed on root systems, algebraic groups and the Curtis-Tits Theorem.

Recall from an earlier week that given {\alpha\in V=\mathbb{R}^n} and a positive definite bilinear form {(.,.)} on {V}, we can define the reflection

\displaystyle \sigma_{\alpha}:\beta\mapsto \beta-\frac{2(\alpha,\beta)}{(\alpha,\alpha)}\alpha

Then a root system is a subset {\Phi} of {V} such that

  1. {\langle \Phi\rangle=V} and {0\notin \Phi},
  2. If {\alpha\in\Phi} then the only scalar multiples of {\alpha} in {\Phi} are {\pm \alpha},
  3. {\sigma_{\alpha}(\Phi)=\Phi} for all {\alpha\in\Phi},
  4. {2(\alpha,\beta)/(\alpha,\alpha)\in\mathbb{Z}} for all {\alpha,\beta\in\Phi}.

A root system is called irreducible if it cannot be written as a disjoint union of {\Phi_1} and {\Phi_2} such that {\langle \Phi_1\rangle \perp \langle \Phi_2\rangle}.

A fundamental system is then a subset {\Pi} of {\Phi} which is a basis for {V} and such that each element of {\Phi} can be written as a linear combination of the elements of {\Pi} such that the coefficients are either all nonnegative integers or all nonpositive integers.

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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.

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Phan systems study group

(with help from John Bamberg, Alice Devillers, and Sukru Yalcinkaya.)

We have recently started a study group here with the rather ambitious aim of exploring the links between diagram geometries, buildings, Phan systems, groups of Lie type and algebraic groups.  It is very informal, on a Friday afternoon and often involves several people at the blackboard during the hour and plenty of questions. We are also trying to do it via examples. We have decided that it would be a good idea to keep of record here of what we cover. If things go really well one of us will try to blog directly during the study group instead of taking notes. Before we can do so I need to give a brief overview of what we have covered so far.  Future posts should be more fleshed out. We are mainly working from Don Taylor’s book (The Geometry of Classical Groups), Humphreys book (Reflection Groups and Coxeter Groups), Carter’s book (Simple Groups of Lie Type), Abramenko and Brown’s book (Buildings) and Garrett’s book (Buildings and Classical Groups). Comments and corrections are more than welcome.

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