TITLE: Where does the canonical basis differ from the KLR basis?
QUESTION [14 upvotes]: The question implicitly asked in Ben Webster's question is: Does the canonical basis of Uq(n+) agree with the basis coming from categorification via Khovanov-Lauda-Rouqier algebras?
Thanks to Shunsuke Tsuchioka's answer to the very same question, the answer is No.
One can now ask for more, and ask for an explicit example where these two bases disagree. So that's what I am going to do here. 
In case someone brings it up, I'm disqualifying Example 3.25 of Khovanov-Lauda I on the grounds that it doesn't count (the algebra isn't quite defined "correctly" in characteristic zero (it doesn't match the geometry), and weird stuff is to be expected outside of finite type for KLR algebras in positive characteristic). I'd prefer to work in finite type please.
Of course Tsuchioka's answer provides us with an upper bound on where to look.

REPLY [3 votes]: Since this question was first asked, nontrivial decomposition numbers for Quiver Hecke algebras (=KLR algebras) have been discovered in finite type. Thus there is an alternative family of examples where the canonical and KLR basis disagree, if one is willing to accept ones algebras living over a field of positive characteristic.
Known examples are not always explicated in the literature right now. The historically first example is due to Geordie Williamson - see http://arxiv.org/abs/1212.0794 or http://arxiv.org/abs/1210.6900.