Buildings, geometries and algebraic groups study group XI

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.


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