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Looking at a few of Littelmann's papers, he seems to only apply root operators $f_\alpha$ for $\alpha$ a simple root. However, the definition seems to make perfect sense for $\alpha$ any positive root. (Indeed one can change the hyperplane that divides positive and negative roots and get a new set of simple roots corresponding to that choice.)

My question is: how much have the $e_\alpha$ and $f_\alpha$ been studied or used for nonsimple $\alpha$?
(This question also applies to other path models, like LS paths.)

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1 Answer 1

The question may be somewhat open-ended, but perhaps I can focus some aspects of it. First, I'm not sure which papers by Littelmann you've looked at, but probably the most definitive treatment of his operators occurs in his often-cited paper Paths and root operators in representation theory. Ann. of Math. (2) 142 (1995), no. 3, 499–525. Here and in related papers the objects of study are the finite dimensional irreducible representations of a semisimple Lie algebra over $\mathbb{C}$ or more generally the irreducible integrable representations of a symmetrizable Kac-Moody Lie algebra (for which there is a Weyl-Kac character formula).

In the classical case the representations are classified by their dominant highest weights relative to a fixed choice of positive (or simple) roots. But the computation of the character or weight multiplicities is complicated by the fact that a subweight can usually be reached in many ways from the highest weight by applying root vectors corresponding to negatives of simple roots. So you tend to get a lot of inefficient overcounting and cancellations. Littelmann's formal method is intended to streamline this way of thinking for all related problems by avoiding overcounts. Elegant formulas like Weyl's do involve all positive roots, but it's extremely inefficient in any computational approach to apply the root vectors for all possible negative roots.

While the definitions of Littelmann's root operators don't require simple roots, the applications do require a notion of dominant weight and hence a fixed choice of simple roots. Of course, any root can be "simple" in some set of positive roots, but nothing new is obtained classically. (On the other hand, defining a Kac-Moody algebra in general presupposes a choice of simple roots.)

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