On Tuesday, Şükrü gave a talk in our groups and combinatorics seminar on material related to last week’s discussion. The slides of his talk are available on our seminars page and include a definition of a Phan system.

Alice Devillers led the discussion in the study group on Friday to give a buildings perspective on last week’s discussion by Şükrü.

Recall that a chamber system consists of a set of chambers such that two chambers are ${i}$-adjacent if they agree in all but their elements of type ${i}$. See also this earlier discussion. We say that a chamber system is simply 2-connected if every closed path can be reduced to the trivial path bya sequence  replacing any subpath lying in a rank 2 residue by another path in the same residue, that is, any closed path is 2-homotopy equivalent to the trivial path. Buildings are simply 2-connected chamber systems.

Alice led the discussion again this week.  At the end of the session we realised that this will be the last study group for the year due to various people going away for holidays/conferences/meetings in the lead up to christmas and of course the school’s christmas party. We will resume sometime early in the new year.

We first began by discussing the comments on the post on last week’s study group.

Recall from last week how given a building with apartments isomorphic to the Coxeter complex $(W,S)$, we can define the W-distance on chambers. We look again at the chamber system we obtain from the Fano plane.

Let $s_1$ denote differ by a point and $s_2$ denote differ by a line. Let x be the chamber whose line is the orange line and point is the bottom left hand corner of the diagram and y be the chamber whose line is the red line and point is the top of the diagram. Then $\delta(x,y)=s_1s_2=w$. There are four chambers which are neighbours of y. These are $z_1$, which has line the red line and point the point in the middle of the line, $z_2$ which has line the pink line and point the top point,  $z_3$ which has line the orange line and point the top point, and $z_4$ which has line the red line and point the bottom right hand corner.  Then $\delta(y,z_1)=s_1$ and $\delta(x,z_1)=s_1s_2s_1=ws_1$. Also $\delta(y,z_2)=\delta(y,z_3)=s_2$ and $\delta(x,z_3)=s_1=ws_2$ while $\delta(x,z_2)=s_1s_2=w$.  Moroever, $\delta(x,z_4)=ws_1$. Thus we have the property that if $z$ is a chamber such that $\delta(y,z)=s\in S$ then $\delta(x,z)\in\{w,ws\}$ and if $\ell(ws)=\ell(w)+1$,  (where $\ell(g)$ is the length of a reduced expression for g in terms of elements of S), then $\delta(x,z)=ws$.

This motivates the following definition of a building: A building of type $(W,S)$ is a pair $(C,\delta)$ with C a set (whose elements are called chambers) and $\delta:C\times C\rightarrow W$, such that for $x,y\in C$ with $\delta(x,y)=w$ the following hold:

• $w=1$ if and only if $x=y$
• If $z$ is a chamber such that $\delta(y,z)=s\in S$ then $\delta(x,z)\in\{w,ws\}$ and if $\ell(ws)=\ell(w)+1$, then $\delta(x,z)=ws$.
• If $s\in S$ then there exists $z\in C$ such that $\delta(y,z)=s$ and $\delta(x,z)=ws$.

This week Alice is leading the discussion and the topic is buildings.

We begin with an example. Let $W=S_4$ and $S=\{(12),(13),(34)\}$. Also let V be a 4-dimensional vector space with basis $e_1,e_2,e_3,e_4$. We can form a tetrahedron with vertices labelled by $\langle e_1\rangle,\ldots,\langle e_4\rangle$. We place vertices at the midpoint of each edge (these each represent the 2-dimensional subspace spanned by the two 1-spaces at either end of the edge) and a vertex at the midpoint of each face (these represent the 3-spaces spanned by the three 1-spaces at each corner of the face). This gives us a simplicial complex whose 0-simplices are the vertices, the 1-simplices are the lines between 0-simplices, and the 2-simplices are the parts of faces surrounded by the 1-simplices. Now W acts on V by permuting the basis elements. Let I be the 2-simplex surrounded by the vertices $\langle e_1\rangle,\langle e_1,e_2\rangle,\langle e_1,e_2,e_3\rangle$. We can associate each of these three vertices with the subgroup of W generated by the two elements of S which fix the associated subspace. That is, $\langle e_1\rangle$ is associated with $\langle (23),(34)\rangle$, $\langle e_1,e_2\rangle$ is associated with $\langle (12),(34)\rangle$  and $\langle e_1,e_2,e_3\rangle$ is associated with $\langle (12),(23)\rangle$. The 1-simplices joining these vertices are associated with the intersections of the subgroups associated with the end vertices, while we associate I with the identity subgroup. Now for each other simplex we can associate a coset of the appropriate simplex in I.

In general, given a Coxeter system $(W,S)$ we can form the Coxeter complex which is the poset on the set $\{\langle S'\rangle w\mid S'\subseteq S,w\in W\}$ and $B\leqslant A$ if and only if $A\subseteq B$. There was much discussion as to whether or not S needs to be a simple system but this wasn’t resolved.

A building is a simplicial complex $\mathcal{B}$ which can be expressed as a union of subcomplexes $\mathcal{A}$, called apartments, such that

• (B0) each apartment is a Coxeter complex;
• (B1) for any pair of simplices $A,B$ there exists an apartment $\Sigma\in\mathcal{A}$  containing both;
• (B2) If $\Sigma,\Sigma'\in\mathcal{A}$ both contain simplices $A,B$ then there is an isomorphism $\Sigma\rightarrow\Sigma'$ fixing $A$ and $B$ pointwise.

Taking $A,B=\varnothing$ and applying (B2) implies that all apartments are isomorphic.

(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.