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.


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 :)

  • $\begingroup$ 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. $\endgroup$ – Jim Humphreys Mar 30 '18 at 18:06
  • $\begingroup$ 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. $\endgroup$ – 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.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.