(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.

John began by discussing projective spaces. Let V be a d-dimensional vector space over a field F and $\mathrm{PG}(d-1,F)$ be the set of all subspaces of V with incidence given by symmetrised inclusion.  The elements of the projective space can be partitioned into types such that two elements are of the same type if they have the same dimension. A flag is a set of mutually incident subspaces while a chamber is a maximal flag. For example, if V has basis $e_1,e_2,\ldots, e_d$ then $\{\langle e_1\rangle,\langle e_1,e_2\rangle,\ldots,\langle e_1,\ldots,e_{d-1}\rangle\}$ is a chamber.  The group $\mathrm{GL}(d,F)$ acts on $\mathrm{PG}(d-1,F)$  such that the group Z of all scalars fixes each subspace setwise, inducing the group $\mathrm{PGL}(d,F)=\mathrm{GL}(d,F)/Z$. In this example we will usually work in the matrix group instead of the projective group. A Borel subgroup is a stabiliser of a chamber and so given our example of a chamber in $\mathrm{GL}(d,F)$, it is the subgroup B of all lower triangular matrices. The stabiliser of any flag is called a parabolic subgroup.

A frame is a set of 1-dimensional subspaces of V so that any k span a k-subspace. So $\{\langle e_1\rangle,\ldots,\langle e_d\rangle\}$ is a frame and the setwise stabiliser in $\mathrm{GL}(d,q)$ of this frame is the group N of all monomial matrices.  The pointwise stabiliser of the frame is the group T of all diagonal matrices. Then $N\cap B=T$ and $W=N/T\cong S_d$, the group of all permutation matrices. The group W is called the Weyl group of $\mathrm{GL}(d,F)$ and is the permutation group induced on the frame.

Alice Devillers then gave a brief introduction to chamber systems and Coxeter groups continuing with the projective space as a thematic example.

The chamber system of a projective space, or similarly of buildings, consists of a set of chambers (maximal flags) and a collection of equivalence relations $\sim_i$ on chambers, where i takes its values in the set of types of the building. Two chambers are $\sim_i$-equivalent if they agree in all elements except maybe in their element of type i.  The chamber system induced on an apartment of a building (that is, a frame in the projective space case) is a Coxeter system, that is the coloured Cayley graph of a Coxeter group $(W,S)$. In the  case of the projective space $\mathrm{PG}(n,K)$, we have $(W,S)=(S_{n+1},\{(1,2),(2,3)\ldots(n,n+1)\}$.  This is the Coxeter group of type $A_n$.

A Coxeter group is a group generated by involutions  $s_1,s_2,\ldots,s_n$  which can be presented solely by relations  of the type $(s_is_j)^{m_{ij}}=1$.  The  $m_{ij}$‘s form the entries of a matrix M, with $m_{ii}=1$ for all i, and $m_{ij}=\infty$ if there is no relation involving $s_i$ and $s_j$. We call M the Coxeter matrix. The information contained in M can also be visually encoded in a diagram: take one node i for each involution $s_i$, draw no edge if $m_{ij}=2$, and an edge with label $m_{ij}$ otherwise (the “3” labels are often omitted).  There was some lively debate as to whether S had to be a simple system to be able to define the diagram properly.

Let’s go back to projective spaces. A residue of a flag F is the set of all elements of type not  in $type(F)$ which are incident with all elements in F.  In terms of chamber systems, that means taking a chamber c containing F and the residue of F is the connected component, containing c, of the chamber system in which the equivalence relations $\sim_i$ for $i\in type(F)$ are dropped.

The projective space $\mathrm{PG}(n,F)$ has an associated diagram which has vertices the types, that is, 1-spaces, 2-spaces, …, $n$-spaces. We say a subspace is of type i if it has dimension i, and let $I=\{1,\ldots, n\}$.Two types  $i,j$ are not joined by an edge if in the residue of a flag of type $I\backslash\{i,j\}$, any element of type i is incident with every element of type j. If the residue of a flag of type $I\backslash\{i,j\}$ is a projective plane we join types i and j by a single edge. The diagram is the usual $A_n$ diagram which we get for the Weyl group $S_{n+1}$ of $\mathrm{GL}(n+1,F)$.

Next we had Sukru Yalcinkaya introduce some notions of algebraic groups. A linear algebraic group is a group which is also an algebraic variety (that is, the set of zeros of some set of polynomials with coefficients from some algebraically closed field K)  such that multiplication and inversion are given by polynomials. One example is $\mathrm{SL}(n,K)$ as it can be viewed as a subset of $K^{n^2}$ and consists of all matrices A which are zeroes of the polynomial $\mathrm{det}(A)-1$. We need to be a bit more cunning for $\mathrm{GL}(n,K)$. We can embed $\mathrm{GL}(n,K)$ in $\mathrm{SL}(n+1,K)$ by the map $A\mapsto X_A$, where $\displaystyle{X_A=\left(\begin{array}{cc} A&0\\ 0 &\mathrm{det}(A)^{-1}\end{array}\right)}$ and $\mathrm{GL}(n,K)$ is then all such block diagonal matrices $X$ in $\mathrm{SL}(n+1,K)$ which satisfy $\mathrm{det}(X)-1=0$.

Linear algebraic groups come with the Zariski topology (that is, the closed sets are the sets of zeros of ideals of $K[x_1,x_2,\ldots,x_r]$)  and so we have closed and open subgroups. A Borel subgroup is a maximal closed connected soluble subgroup and for $\mathrm{GL}(n,K)$ the standard one  is the subgroup of lower triangular matrices. A  maximal torus is a maximal abelian closed subgroup  consisting entirely of semisimple elements (again for $\mathrm{GL}(n,K)$ one is the diagonal matrices) and the subgroup N is the normaliser of a maximal torus. A parabolic subgroup is a subgroup which contains a Borel subgroup.

Next Sukru discussed root systems. Let $V=\mathbb{R}^n$ equipped with the usual inner product $(.,.)$. Given a vector $\alpha\in V$ we can define the reflection $\displaystyle{\sigma_\alpha:v\rightarrow v-\frac{2(\alpha,v)}{(\alpha,\alpha)}v }$. This reflection fixes the hyperplane orthogonal to $\alpha$ pointwise and maps $\alpha$ to $-\alpha$.  A root system is a spanning subset $\Phi$ of V such that

• $\Phi$ is invariant under $\sigma_\alpha$ for each $\alpha\in\Phi$.
• 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 $\alpha,\beta\in\Phi$.

The last condition is often left out and root systems satisfying the last condition are then called crystallographic root systems. The elements of $\Phi$ are called roots. A root system is called reducible if it can be written as the disjoint union of two root systems $\Phi_1,\Phi_2$ such that each element of $\Phi_1$ is perpendicular to each element of $\Phi_2$. It is called irreducible otherwise. The rank of the root system is the dimension of V.

Given a root system $\Phi$, a base  $\Delta$ of $\Phi$  is a subset of $\Phi$ which spans V and each element of $\Phi$ is a linear combination of the elements of $\Delta$ such that the coefficients are either all positive or all negative. This allows the roots to be split between positive and negative roots. The elements of $\Delta$ are called simple roots. Of course, the definition of positive roots and negative roots depends on the choice of the base.

The rank 2 root systems can be easily classified and we get $A_1\times A_1, A_2, B_2, C_2$ and $G_2$.  See here.  Given a root system $\Phi$, then the set $\{\frac{2}{(\alpha,\alpha)}\alpha\mid \alpha\in\Phi\}$ is also a root system and is called the dual system of $\Phi$. The root system $B_2$ is the dual of $C_2$.

Given a root system with base $\Delta$, we can draw the Coxeter graph which has vertices the simple roots. Two roots are connected by

• no edge if they are perpendicular,
• a single edge if the angle between them is $2\pi/3$,
• a double edge if the angle is $3\pi/4$,
• a triple edge if the angle is $5\pi/6$.

The Coxeter graph becomes a Dynkin diagram if on each multiple edge we put a >  pointing from  the longer root to the shorter root.