Convex PBW bases

Question

Given a reduced expression for the longest word $w_0$ in the Weyl group of $\mathfrak{g}=\mathfrak{n}^+\oplus\mathfrak{h}\oplus{n}^-$, one obtains a convex ordering on the set of positive roots, $\beta_1<\ldots<\beta_N$. Using the braid group action we get (divided powers of) root vectors $E_{\beta_i}^{(n_i)}$ in $U_q(\mathfrak{n}^+)$ and the set of elements of the form $$E(\underline{n})=E_{\beta_N}^{(n_N)}\cdots E_{\beta_1}^{(n_1)},\;\;\;\underline{n}=(n_N,\ldots,n_1)\in\mathbb{Z}^N_{\geq0}$$ forms a basis for $U_q(\mathfrak{n}^+)$.

I'm wondering if anyone can point me to an elementary proof of the following fact (the shorter the better):

$$\overline{E(\underline{n})}=\sum_{\underline{m}}\lambda_{\underline{n}\underline{m}}E(\underline{m})$$ where $\lambda_{\underline{n}\underline{n}}=1$ and $\lambda_{\underline{n}\underline{m}}=0$ if $\underline{m}>\underline{n}$ with respect to some total ordering. (I think the lexicographic ordering on $\mathbb{Z}_{\geq0}^N$ or maybe the opposite ordering works, but if not I'd like to know what does).

By elementary, I mean that I want to avoid the geometry of quiver varieties. Also, I know that Leclerc has a proof of this fact in his paper on quantum shuffles, but I'm hoping for something less involved.

Answer 1

If by Leclerc, you mean his paper "Dual canonical bases, quantum shuffles and q-characters", then his Lemma 37 is not for a general convex order, but only for those that come from Lyndon words. And if you mean a different Leclerc paper, I'd like to know which you have in mind.

I know two proofs of this fact.

One relies on Proposition 1.9 in Lusztig's "Braid group action and Canonical Bases", together with standard facts about the T_i's, where standard means that you can read about them in Lusztig's book. From here, now that Lusztig has done the hard work for us, it is not hard (but may be a pain to typeset a proof). You might want to look at Beck and Nakajima's paper "Crystal Bases and two-sided cells of Quantum Affine algebras". In particular Section 2.6 is useful since it contains a restatement of Lusztig's result with all the relevant definitions on the same page, and then later on, (Prop 3.36) they prove an analogous upper triangularity result in the affine case under consideration.

EDIT(20/12/12): Another proof of this fact is to consider the dual picture, where the uppertriangularity is equivalent to bar-invariance of the root vectors together with the Levendorskii-Soibelman formula. Originally here I claimed that the uppertriangularity of the bar involution is a Corollary of Theorem 3.1 of my own paper "Finite Dimensional Representations of Khovanov-Lauda-Rouquier Algebras I: Finite Type." Since the proof of said theorem invokes the Levendorskii-Soibelman formula, I cannot call this an independent proof. There is a discussion of this in arXiv:1210.6900.

I would be happy to see a quiver variety proof, the only one I know of is Prop 7.9 in Lusztig's "Canonical Bases arising from Quantized Enveloping Algebras", where, as in Leclerc, not all convex orders are considered.