# Difference of adjacent dominant weights is a root?

The basic set-up here makes sense in the theory of abstract root systems if one brings (integral) weights into the picture, but it may be more natural to think about the classical characteristic 0 theory of finite-dimensional representations of a semisimple Lie algebra. (This question was recently raised by a colleague, who had observed that it was true in type $A$ and was tempted to go on in case-by-case fashion.)

First fix a simple system of roots in an irreducible root system, hence notions of positive roots and dominant weights. Recall the usual partial ordering of weights: $\mu \leq \lambda$ means that $\lambda -\mu$ is a sum of positive roots. There is some evidence that the following statement is always true:

$(*)$ Suppose $\lambda, \mu$ are dominant weights. If $\lambda > \mu$ and there is no intermediate dominant weight, then $\lambda - \mu$ is a single root (not necessarily simple, of course).

Is $(*)$ true for all root systems, and if so is there a reference?

[UPDATE] Jantzen's recent short paper here has an affirmative case-by-case answer in 2.1-2.2. Although his paper deals with modular representations of semisimple algebraic groups, this part of the proof only concerns root-and-weight computations. (However, he does rely on the classification of irreducible root systems, so it's still a question whether there is a uniform proof.)

P.S. The argument due to Steinberg in Stembridge's 1998 paper seems to answer this last question, as pointed out by Sam Hopkins.

A more precise description of positive roots $\gamma$ such that $\mu=\lambda-\gamma$ can be found in the paper