Fix an irreducible root system $\Phi$ with rank $r$ and a root base $\Delta$ (we only care type ADE). Call a disjoint union $\Phi = \Phi_{1}\sqcup\dotsb\sqcup\Phi_{s}$ a decomposition of $\Phi$ if each $\Phi_{k}$, for $k = 1,\dotsc,s$, is a root subsystem of $\Phi$ and it satisfies $\Phi_{k} = \Phi\cap\operatorname{span}_{\mathbb{R}}\Phi_{k}$. For a good decomposition $(s;\Phi_{k},k = 1,\dotsc,s)$, denote also $r_{k} := \operatorname{rank}\Phi_{k}$
For example, Denote by $\mathfrak{h}$ the Cartan algebra of $\Phi$, then $\Phi\subset\mathfrak{h}^*$. Let $L\subset\mathfrak{h}$ be a two dimensional subspace which doesn't lie in the kernel of any root in $\Phi$. Hence, the restriction of each root $\theta\in\Phi\subset\mathfrak{h}^{*}$ to $L$ has a one-dimensional kernel denoted by $l_{\theta}$. The choice of subspace $L$ induces a map: \begin{equation*} \begin{aligned} \mu_{L} : \Phi &\to \mathbb{P}(V), \\ \theta & \mapsto [l_{\theta}]. \end{aligned} \end{equation*} Because $\Phi$ is a finite set, the image of the map $\mu_{L}$ is also a finite set written as $\mathrm{Im}\,\mu_{L} = \{p_{1},\cdots,p_{s}\}$. Hence there are decomposotions defined as : \begin{equation*} \label{eq_Root_decomposition} \begin{aligned} \Phi &= \Phi_{1} \sqcup \Phi_{2} \sqcup \cdots \sqcup \Phi_{s} \end{aligned} \end{equation*} where $\Phi_{k} := \mu_{L}^{-1}\{p_{k}\}$, for $k = 1, \ldots, s$. Each $\Phi_{k}$ is a root subsystem of $\Phi$. The decomposition induced by $L$ is a decomposition.
Due to lemmas in Chapter 4–6 in Bourbaki's book "Lie Groups and Lie Algebras", we can prove $r_{k}<r$, for $ k = 1,\dotsc,s$. By the existence of the maximal root, it's easy to show that the lower bound of: $$ r+1 \leq \sum_{k = 1}^{s}r_{k}\leq \frac{1}{2}\sharp\Phi.$$
And
$$3\leq s\leq \frac{1}{2}\sharp\Phi.$$
For $\Phi$ is of Type ADE, I obtain a better lower bounds that $\sum_{k = 1}^{s}r_{k} \geq 2r-1$ and it is sharp for Type A. But for the number of root subsystems $s$, I have not found the best lower bounds.
My questions are
(1) Is this a new phenomenon? because I can't find any reference about the decomposition of root system. I meet this decomposition in geometric problem about the Kleinian singularity $\mathbb{C}^{2}/\Gamma$.
(2) For root system $\Phi$ of type ADE, when considering all the decompositions of $\Phi$, Can we find the best lower bound of the number $s$ and the sum $\sum_{k = 1}^{s}r_{k}$ ? (A related question asked also by me is there)
e.g. By straightforward computation, we can show for type $A_{2}$ and $A_{3}$, $s\geq 3$ with sharpness.
(2') Moreover, for the reason from geometry, we need to find the best lower bound for the sum $$r_{m_{1}} + r_{m_{2}} + 2\sum_{k\neq m_{1},m_{2}} r_{k}$$, where $r_{m_{1}},r_{m_{2}}$ are the two largest number in $r_{k}$'s in that decomposition induced by some two dimensional subspace $L$.
The following two questions concern the relation with other objects
(3) The Dynkin diagram plots the relations of simple roots, is there any diagram plotting all the roots ? I.e. can we associate to each root a vertex and link two roots if their inner product is non-zero ? It seems complicated, how could we study this diagram through combinatorial method ?
(4) In this paper, the author shows there is a 'zeta' function associated with a root system. So are there some correspondence between the decomposition of root system and the decomposition of zeta function ?
I am very grateful if someone could offer me some references or give me some advice to attack this problem.
