# Dominance relation among Cartan matrices implies containment of root systems: Is this known?

Suppose $A$ and $A'$ are symmetrizable (generalized) Cartan matrices, in the sense of Kac's book Infinite-dimensional Lie algebras. Say $A$ dominates $A'$ if every entry of $A$ has weakly greater absolute value than the corresponding entry of $A'$. Write $\Phi(A)$ for the Kac-Moody root system associated to $A$. I can prove the following:

Proposition. If $A$ dominates $A'$ and $\Phi(A)$ and $\Phi(A')$ are constructed with the same ordered set of simple roots, then $\Phi(A')\subseteq\Phi(A)$.

1. We need the full root system here, including the imaginary roots (if there are any).
2. Yes, this looks like it is false. For example, there is no containment relation among the root systems of types $A_2$ and $B_2$, right? Right, if we embed them into the same vector space so that the two associated Euclidean metrics coincide. But wrong if we do what the proposition says. A useful rephrasing of the proposition: Choosing simple roots any way we please, the set of simple root coordinates of roots in $\Phi(A')$ is contained in the set of simple root coordinates of roots in $\Phi(A)$.

So, the question (as already asked in the title): Is this known?

On the one hand, the proof is fairly simple, using the Serre relations, so maybe it's known. On the other hand, it may not be very Lie-theoretically natural (for example, I don't think there is a corresponding subalgebra relationship), so maybe it's not known. It is very combinatorially natural and is connected to some interesting facts about the weak order on finite Coxeter groups and to some interesting phenomena surrounding dominance relations on exchange matrices (in the sense of cluster algebras).

• Just added the "lie-algebras" tag, hoping someone who works on Kac-Moody Lie algebras might have seen this result somewhere. – Nathan Reading Feb 2 '18 at 16:09