Example 3

Strict triangular group.

Let $[1, n]$ be the set of integers $m$ such that $1 \leq m \leq n$. Let $J$ be a subset of $[1, n] \times [1, n]$ satisfying the following conditions: 1) if $(i, j) \in J$ then $i < j$; 2) if $(i, j) \notin J$ then, for every integer $k$ such that $i < k < j$, at least one of the two pairs $(i, k)$ and $(k, j)$ does not belong to $J$.

Let $T_J$ be the set of matrices $Z = (z_{ij})_{1 \leq i \leq n, 1 \leq j \leq n}$ with elements in $K$, such that $z_{ii} = 1$, and $z_{ij} = 0$ if $i \neq j$ and $(i, j) \notin J$. This is a closed subset of $\mathbf{GL}(n, K)$. The mapping $Z \mapsto (z_{ij})_{(i, j) \in J}$ is a homeomorphism of $T_J$ onto $K^s$ (where $s$ denotes the number of elements of $J$). If $Z' = (z'_{ij}) \in T_J$, then $Z'Z = (z''_{ij})$ with

whence $Z'Z \in T_J$. If $T_J$ is identified with $K^s$, then the mapping $Z \mapsto Z'Z$ (for fixed $Z'$) is identified with an affine mapping, and its determinant is 1, as one sees by ordering the pairs $(i, j) \in J$ lexicographically and applying the following lemma: