# 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.

## 2 Answers

This is true. See "The partial order of dominant weights" by John Stembridge, 1998 (https://www.sciencedirect.com/science/article/pii/S0001870898917364). In particular look at Corollary 2.7. Alternatively, look at the second proof of Corollary 2.7 given there, due to Robert Steinberg, which the paper claims was communicated by one James Humphreys :)

• Thanks very much for reminding me about this paper, which I had totally forgotten. Looking back now at my Stembridge folder, I see that I kept some copies of email (but not all). There is a long email from John dated June 16, 1997, printed on metric paper when I was at INI that month. I don't recall exactly what I wrote to him, but I did keep a concise proof by Steinberg (typed by me using LaTeX and printed on similar paper). We shared a room at INI then, so we must have talked about the problem. – Jim Humphreys Mar 30 '18 at 18:06
• Yes, I guess I should've said explicitly that Steinberg's proof is (short and) uniform. On the other hand, Stembridge proves a lot more: he essentially gives a characterization of the positive roots that can be covering relations in the partial order of dominant weights. – Sam Hopkins Mar 30 '18 at 18:15

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

See, in particular, Proposition 1 and Lemma 3.