Exercise 18, § 2

We use the hypotheses and notation of no. 2. Let $V_i$ be the vector subspace of $k^n$ generated by $e_1, \ldots, e_i$ (for $1 \leq i \leq n-1$).

a) Show that, for any subset X of S, the subgroup G_X (no. 5) consists of the elements $g \in G$ such that $g(V_i) = V_i$ for all $i$ such that $s_i \notin X$.

b) Let I be the building associated to a Tits system $(G, B, N, S)$ (Exerc. 10 and 12). Show that the map $j$ that associates to the point $gG^{(i)}$ of I (where $G^{(i)}$ denotes the subgroup $G_{S-\{s_i\}}$ of G, in other words the stabilizer of $V_i$) the vector subspace $g(V_i)$ is a bijection from I to the set $\mathscr{B}$ of vector subspaces of $k^n$ that are $\neq \{0\}$ and $\neq k^n$, compatible with the action of G.

c) If E is a vector space, a flag of E is a set of vector subspaces of E that is totally ordered by inclusion. Show that elements $a_1, \ldots, a_k$ of I belong to the same facet of I if and only if $\{j(a_1), \ldots, j(a_k)\}$ is a flag of $k^n$.

d) Show that G acts doubly-transitively on the set of 1-dimensional vector subspaces of $k^n$. Suppose that $n \geq 2$ and let $N_1$ be the subgroup of G generated by the element that interchanges $e_1$ and $e_2$ and leaves fixed the other $e_i$. Show that $(G, G^{(1)}, N_1)$ is a Tits system whose Weyl group is of order 2.

e) Assume that $k$ is commutative. Set

Show that $N'/T'$ can be identified with $\mathfrak{S}_n$ and that $(G', B', N', S)$ is a Tits system (argue as in no. 2).