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 "half-sum of positive 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?](https://mathoverflow.net/questions/80150/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.



Significance of half-sum of positive roots belonging to root lattice?
There is no single answer to this question. Here is one observation.


The element $z\_G=e^{2\pi i\rho^\vee}$, where $\rho^\vee$ is one-half the sum of the positive co-roots, is a canonical (independent of the choice of positive co-roots) element of $G$, fixed by every automorphism of $G$. If $G$ is simply connected $z\_G=1$ if and only if $\rho^\vee$ is in the co-root lattice. For whatever reason this $z\_G$ tends to come up. For a slightly more precise list of your cases see Bourbaki, Lie Groups and Lie Algebras, Chapters 7-9, Chapter IX, Section 4, Exercise 13.


For example the Frobenius-Schur indicator of a self-dual finite dimensional representation $V\_\lambda$ (telling whether the invariant form is orthogonal or symplectic) is $e^{2\pi i\langle\lambda,\rho^\vee\rangle}=\lambda(z\_\rho)$. So $\rho^\vee$ is in the co-root lattice if and only if every such representation of the simply connected group is orthogonal. See [loc. cit. Chapter 9, Section 7] and 
Math Overflow: [Is there a formula for the Frobenius-Schur indicator of a rep of a Lie group?](https://mathoverflow.net/questions/227869)


