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A friend and I are writing a paper that uses the BGG resolution of $L(\lambda)$ (where $\mathfrak g$ is a semisimple complex Lie algebra, $\lambda \in P^+$ is a dominant integral weight, and $L(\lambda)$ the simple finite module with highest weight $\lambda$).

A point is unclear for us : one needs to choose appropriate signs for each edge $w \to w'$ in the Bruhat graph. It's probably easy but we can't see why the resulting complex is independent of the choice of the signs. Of course the maps will be different, but we expect that two different choices of signs give isomorphic complexes.

There is a discussion of unicity of signs in Humphrey's book "Representations of Semisimple Lie Algebras in the BGG Category $\mathcal O$" chapter 6. It seems to me the conclusion is that the signs $\sigma(w',w)$ can be actually chosen in $\{-1,1\}$. I don't understand why it clarifies the uniqueness question, i.e why two different choices of such signs give the same complex up to isomorphism.

I've probably misunderstood the argument or I don't see something obvious, but in any case I would appreciate clarification about it. Thanks in advance !

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    $\begingroup$ In a slightly different setup, unicity is also discussed in a recent preprint by Mazorchuk-Mrden: arxiv.org/pdf/1907.04121.pdf where it follows from a strong Koszulity property called "balanced". $\endgroup$ – Julian Kuelshammer Oct 15 '19 at 17:24
  • $\begingroup$ @JulianKuelshammer : Thanks for your comment ! I believe what I want is theorem 33, however the authors seems to work in a different setting. Do you think their argument still apply in my case ? $\endgroup$ – Nicolas Hemelsoet Oct 15 '19 at 22:54
  • $\begingroup$ @Nicolas: This is a complicated subject, given the earlier distinction between strong and weak versions of BGG resolution. I'll give it some more thought, but meanwhile please check the list of revisions on my homepage or on the AMS bookstore page. (There is for example a modification in the proof method for Theorem 6.8 at the bottom of page 118.) Nowadays most publishers avoid corrected reprints, preferring the cheaper print-on-demand technology using the original plates. $\endgroup$ – Jim Humphreys Oct 15 '19 at 23:44
  • $\begingroup$ @NicolasHemelsoet : In the preprint mentioned by JulianKuelshammer, the uniqueness theorem applies to category O, both regular and singular blocks. (However, it does not apply directly to parabolic versions of category O). $\endgroup$ – Rafael Mrđen Oct 16 '19 at 6:57
  • $\begingroup$ @RafaelMrđen : Dear Rafael, this totally answer my question, could you post it as an answer please ? $\endgroup$ – Nicolas Hemelsoet Oct 16 '19 at 7:27
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Theorem 33 in the preprint [1] gives the uniqueness of BGG resolutions (= direct sums of Verma modules resolving a simple module) in category $\mathcal{O}$, both in regular and singular blocks. (However, this does not apply directly to parabolic versions of category $\mathcal{O}$).


[1]: Mazorchuk-Mrđen: BGG complexes in singular blocks of category $\mathcal{O}$, https://arxiv.org/abs/1907.04121

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