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Let $V$ be a dimension $n>0$ vector space over $\mathbb{C}$ and let $$ V=V_1 \supset V_2 \supset \dots \supset V_r \supset V_{r+1} = \{0\} $$ be a (not necessarily complete) flag of subspaces.

Is there any description in the literature for the space $$ \{ \phi \in \mbox{End}(V) \mid \phi(V_i) \subset V_i, \, 1\leq i \leq r \} $$ of endomorphisms of $V$ that preserve the flag?

I'm searching for references that deal with its Lie algebra structure and matrix description. I'm mostly interested in the particular case if we replace $\mbox{End}(V)$ by $\mathfrak{sl}(V)$.

References for particular cases are wellcome as well.

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  • $\begingroup$ What sort of description do you want? You can give a very simple explicit description in terms of matrices if you pick an appropriate basis for $V$. $\endgroup$ – Eric Wofsey Sep 25 '18 at 23:25
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    $\begingroup$ To get you started, the endomorphism ring of a complete flag is isomorphic to the ring of upper triangular matrices. I think adapting this description is what Eric is alluding to. $\endgroup$ – rschwieb Sep 25 '18 at 23:30
  • $\begingroup$ Yes, it seems it will be a "block-upper-triangular-ish" matrix. I seek as much info as one can get for these Lie algebras. I'll edit to specify the Lie algebra nature. $\endgroup$ – Alan Muniz Sep 25 '18 at 23:32
  • $\begingroup$ Do you know any "citeable" reference for this? $\endgroup$ – Alan Muniz Sep 25 '18 at 23:37
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    $\begingroup$ If you take the subgroup $P \subseteq \mathrm{GL}(V)$ which preserve a flag, this is called a parabolic subgroup of $\mathrm{GL}(V)$, and these are well-studied. The homogeneous space $\mathrm{GL}(V) / P$ is called the flag variety. There is a nice introduction to these in Fulton's Young Tableaux. $\endgroup$ – Joppy Sep 26 '18 at 0:16

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