Dear all,

Consider a flag $V=V_1\supset V_2\supset \cdots \supset V_k\supset V_{k+1}=\{0\}$ of a vector space $V$ over a field of $p$ elements. Let $I$ be a subset of the index set $\{1,2,...,k\}$.

(1) I am working at the subgroup of $GL(V)$ consisting of all linear automorphisms $h$'s of $V$ such that $h$ preserves the flag and its induced isomorphism $\overline {h}: V_i/V_{i+1}\rightarrow V_i/V_{i+1}$ is the identity for every $i\in I$. This subgroup is a subgroup of the parabolic subgroup associated to the flag. When $I=\{1,2...,k\}$ then this subgroup is exactly the unipotent radical of the parabolic subgroup. My question is:

What is a good name for this subgroup? Is **uni-parabolic** a good name?

(2) Let $\varepsilon_1,...\varepsilon_s$ be a basis of $V$ such that $\varepsilon_{s-s_i+1},...,\varepsilon_s$ is a basis for $V_i$. We consider the subgroup of $GL(V)$ consisting of all linear automorphisms $h$'s of $V$ such that $h$ preserves the flag and its induced isomorphism $\overline {h}: V_i/V_{i+1}\rightarrow V_i/V_{i+1}$ is a permutation on the basis $\{[\varepsilon_{s-s_i+1}],...,[\varepsilon_s]\}$ of $V_i/V_{i+1}$ for every $i\in I$. Another question is:

What is a good name for this subgroup? Is **symmetry-parabolic** a good name?

Thank you very much for your help.

isthe standard name. – Amritanshu Prasad Mar 31 '13 at 15:33