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.

(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.