I'm wondering how to express certain subgroups of GL(E), E being a vector space of finite dimension n.
These subgroup are caracterised as :
let $ \sigma \in S_n $ , then $ G_{\sigma} $ is the elements M of GL(E) such that, for a certain base B, $ Mat_B M $ is triangular superior, and $ Mat_{\sigma \cdot B} M $ is also triangular superior.
Where, for $ B = (e_1, ..., e_n), \quad \sigma \cdot B = (e_{\sigma(1)}, ..., e_{\sigma(n)} ) $.
This is also caracterised by the fact that $ Mat_B M $ is triangular superior, and also null on positions (i,j) st $ i < j, \sigma(i) > \sigma(j) $.
Does anyone has an idea for a nice caracterisation of these groups that would not involve expressing their elements in matricial form?
To add some more context, these groups are the stabilisators of the orbits of the action of GL(E) on pairs of flags, $ f : (d, d') \rightarrow (f(d), f(d')) $ where each orbit is caracterised by a certain $ \sigma \in S_n $ .