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Let $V$ be a real vector space and let $\Phi \subset V$ be an irreducible root system of rank $r \geq 2$. Let $\Phi_1, \Phi_2 \subset \Phi$ be irreducible root subsystems (that is, root systems in their own right) of ranks $r_1, r_2 \ge 1$, respectively. I'm interested in the rank of the root subsystem $\Phi_1 \cap \Phi_2$.

In the case of interest, the rank of $\Phi_1$ is large: $$ \mathrm{rank} (\Phi_1) = r-1, $$ and $\Phi_2 = w \Phi_1 \subset \Phi$ arises as the image of $\Phi_1$ under a Weyl group element $w \in W$. Let $\Delta$ be a basis of $\Phi$. We additionally assume that $\Phi_1$ is a parabolic subsystem of $\Phi$, that is, a basis of $\Phi_1$ is given by $\Delta \smallsetminus \{\alpha\}$ for some $\alpha \in \Delta$. I'd like to determine whether, for all $w \in W$, the inequality $$ \mathrm{rank}(\Phi_1 \cap w\Phi_1) \ge r - 2 $$ holds. I know that it is true, if $\Phi$ is of type $A_n \, (n \geq 2)$ and that it is false in general if $\Phi_1$ is not assumed to be irreducible.

I'd be grateful for any hint or reference, or for a better understanding of why the statement fails, in case it is not true. Thank you.

user6453
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  • Why is it true in type $A_{\ge 2}$? – Torsten Schoeneberg Nov 07 '25 at 04:59
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    Thank you for your question, I've added an assumption to the original question: that $\Phi_1$ is a parabolic subsystem. Let me now answer your quesiton. Since $\Phi_1$ is irreducible, $\Delta \smallsetminus {\alpha}$ is connected and $\alpha$ is one of the end notes of $\Delta$. It suffices to check that $\mathrm{rank}(\Phi_1 \cap w \Phi_1) \geq r-2$ on a set of representative of the double quotient $W_1 \backslash W / W_1$, where $W_1 = \langle s_{\beta} : \beta \in \Delta \smallsetminus {\alpha} \rangle$. In the case of $A_n$, ${1, s_{\alpha}}$ is such a set. – user6453 Nov 08 '25 at 01:51
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    Let $\gamma \in \Delta$ be the unique neighbor of $\alpha$. A set of simple roots of $s_{\alpha} \Phi$ is $$ s_{\alpha} (\Delta \smallsetminus {\alpha}) = {\gamma - \frac{2 \langle \alpha, \gamma \rangle}{\langle \alpha, \alpha \rangle} \alpha} \cup \Delta \smallsetminus {\alpha, \gamma}. $$ In particular, $\Phi_1$ and $s_{\alpha} \Phi_1$ share the simple roots $\Delta \smallsetminus {\alpha, \gamma}$, which contains $r - 2$ many elements. – user6453 Nov 08 '25 at 01:51

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