# Reference for endomorphisms preserving a flag.

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.

• 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 $V$. – Eric Wofsey Sep 25 '18 at 23:25
• 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. – rschwieb Sep 25 '18 at 23:30
• 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. – Alan Muniz Sep 25 '18 at 23:32
• Do you know any "citeable" reference for this? – Alan Muniz Sep 25 '18 at 23:37
• 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. – Joppy Sep 26 '18 at 0:16