Let $\Phi$ be a (crystallographic) root system and $\Phi^{+}$ a choice of positive roots, with $\Delta$ the corresponding choice of simple roots. So the root lattice of $\Phi$ is just $\mathbb{Z}\Delta$ ($=\mathbb{Z}\Phi^{+}=\mathbb{Z}\Phi$).

Then $\rho := \frac{1}{2}\sum_{\alpha \in \Phi^{+}}\alpha$ (the "positive half-sum of roots", a.k.a. the "Weyl vector") is a distinguished vector associated to $\Phi$ with great significance in representation theory; see e.g. the following Mathoverflow question: What is significant about the half-sum of positive roots?.

A priori, $2\rho \in \mathbb{Z}\Delta$, but in fact it may happen that $\rho \in \mathbb{Z}\Delta$.

Question: is there any representation-theoretic significance to whether we have $\rho \in \mathbb{Z}\Delta$?

With collaborators we have discovered some combinatorial phenomenon that is apparently related to whether $\rho \in \mathbb{Z}\Delta$, and we would like to relate this phenomenon more to representation theory.

It is not hard to work out exactly when $\rho \in \mathbb{Z}\Delta$:

• Type $A_{n}$: Have $\rho \in \mathbb{Z}\Delta$ iff $n$ is even.
• Type $B_n$: Always have $\rho \notin \mathbb{Z}\Delta$.
• Type $C_n$: Have $\rho \in \mathbb{Z}\Delta$ iff $n \equiv 0,3 \mod 4$.
• Type $D_n$: Have $\rho \in \mathbb{Z}\Delta$ iff $n \equiv 0,1 \mod 4$.
• Type $E_6$: $\rho \in \mathbb{Z}\Delta$.
• Type $E_7$: $\rho \notin \mathbb{Z}\Delta$.
• Type $E_8$: $\rho \in \mathbb{Z}\Delta$.
• Type $F_4$: $\rho \in \mathbb{Z}\Delta$.
• Type $G_2$: $\rho \in \mathbb{Z}\Delta$.

But I see no particular rhyme or reason to these root systems.