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.

Let ${G}$ be an automorphism group of a chamber system ${\mathcal{C}}$ which is transitive on chambers and fix a chamber ${c}$. Let ${P_i}$ be the stabiliser in ${G}$ of the set of all chambers which are ${i}$-adjacent to ${c}$. This is the stabiliser in ${G}$ of the flag consisting of all the elements of ${c}$ other than the element of type ${i}$. The group ${P_i}$ is called a maximal parabolic but it should be noted that it is minimal by inclusion amongst the parabolic subgroups. Given types ${i,j}$ with ${i\neq j}$ we have ${P_i\cap P_j}$ is the stabiliser of ${c}$, that is a Borel subgroup ${B}$. If ${\mathcal{C}}$ is connected then ${G=\langle P_i\mid i \in I\rangle}$ where ${I}$ is the set of all types. Moreover, Tits’ Lemma states that if ${\mathcal{C}}$ is simply 2-connected then ${G}$ is the 2-amalgam of the maximal parabolics, that is, ${G}$ is generated by the ${P_i}$ and the only relations in ${G}$ are those which occur in the subgroups ${\langle P_i,P_j\rangle}$ for ${i\neq j}$.

For example, in the projective space ${\mathrm{PG}(n,F)}$ we can take

$\displaystyle c=\{\langle e_1\rangle,\langle e_1,e_2\rangle, \ldots, \langle e_1,e_2,\ldots,e_n\rangle\}$

Then for ${G=\mathrm{SL}(n+1,F)}$ we have ${P_1=Stab_G(\langle e_1,e_2\rangle, \ldots, \langle e_1,e_2,\ldots,e_n\rangle)}$ and ${G=\langle P_1,P_2,\ldots,P_n\rangle}$

Let ${\Delta}$ be a spherical building, that is, one whose Coxeter system ${(W,S)}$ is finite. Since it is finite it has a longest word ${w_0}$. We say that two chambers ${c,d\in\Delta}$ are opposite if ${\delta(c,d)=w_0}$, where ${\delta}$ is the distance defined in the building (see this earlier discussion), that is, if ${c}$ and ${d}$ are as far apart in the building as possible. We can then define the opposite geometry ${\mathrm{Opp}(\Delta)}$ whose chambers are the pairs ${(c,d)}$ such that ${c,d}$ are opposite chambers in ${\Delta}$. We define adjacency by ${(c_1,d_1)\sim_i (c_2,d_2)}$ if and only if ${c_1\sim_i c_2}$. Thus ${\delta(c_1,c_2)=s_i}$, where ${s_i}$ is the element of the fundamental system ${S}$ corresponding to type ${i}$. Then ${\delta(c_2,d_1)=s_iw_0}$ and as ${\delta(c_2,d_2)=w_0}$ we must then have ${\delta(d_1,d_2)=w_0s_iw_0}$. However, ${d_1}$ and ${d_2}$ must be adjacent chambers and so ${\delta(d_1,d_2)=s_j\in S}$ for some ${j}$. Hence we can define a map ${\mathrm{op}}$ on the type set ${I}$, such that ${s_{\mathrm{op}(i)}=w_0s_iw_0}$. In particular, ${d_1\sim_{\mathrm{op}(i)} d_2}$. This map gives rise to a symmetry of the Dynkin diagram.

For the Coxeter system of type ${A_n}$, ${\mathrm{op}}$ induces the reflection of the Dynkin diagram. For ${C_n, F_4, E_7}$ and ${E_8}$, ${\mathrm{op}}$ is the identity. For ${D_n}$, if ${n}$ is even then ${\mathrm{op}}$ is the identity while if ${n}$ is odd ${\mathrm{op}}$ interchanges the two tail nodes of the diagram. Similarly for ${I_m}$, if ${m}$ is even ${\mathrm{op}}$ is the identity while for ${m}$ odd it interchanges the two nodes of the diagram.

Mühlherr and Abramenko have proved that ${\mathrm{Opp}(\Delta)}$ is simply 2-connected if there are no rank 2 residues isomorphic to the generalised polygons ${B_2(2)}$, ${G_2(2)}$, ${G_2(3)}$ or ${{}^2F_4(2)}$. This result is analogous to the Curtis-Tits Theorem discussed last week so we must have needed the field condition that if there were double bonds in the diagram then the field was not equal to ${\mathrm{GF}(2)}$.

We will demonstrate this by looking at the projective space ${\mathrm{PG}(n,F)}$. Let

$\displaystyle c=\{\langle e_1\rangle,\langle e_1,e_2\rangle, \ldots, \langle e_1,e_2,\ldots,e_n\rangle\}$

and let

$\displaystyle d=\{\langle e_{n+1},e_n,\ldots,e_2\rangle,\langle e_{n+1},\ldots,e_3\rangle,\ldots,\langle e_{n+1}\rangle\}$

be an opposite chamber. In general, if ${c=\{U_1,U_2,\ldots,U_n\}}$ and ${d=\{U_1',U_2',\ldots,U_n'\}}$ where ${U_i,U_i'}$ have dimension ${i}$, we have that ${c}$ is opposite to ${d}$ if and only if ${V=U_i\oplus U_{n-i}'}$.

Let ${G=\mathrm{SL}(n+1,F)}$, which acts on ${\mathrm{Opp}(\Delta)}$. Then the Borel subgroup ${Stab_G((c,d))}$ is the subgroup of all diagonal matrices, that is, what is usually referred to as a maximal torus ${T}$. Moreover, the minimal parabolic ${P_1=Stab_G(\text{the set of chambers 1-adjacent to} (c,d))}$ is equal to the subgroup of all matrices of the form

$\displaystyle \begin{pmatrix} *&* & 0 &\ldots &0\\ *&* & 0 &\ldots &0\\ 0&0 & * &0\ldots&0\\ & & &\ddots & \\ & & & &*\end{pmatrix}$

and so is isomorphic to ${\mathrm{SL}(2,F)\circ T}$. Hence it is like one of the copies of ${\mathrm{SL}(2,F)}$ from last week. Since ${\mathrm{Opp}(\Delta)}$ is simply 2-connected we have ${G}$ is the 2-amalgam of the minimal parabolics ${P_i}$.

The study group will be having a break for a couple of weeks as various key participants will be away.