Talk:Bruhat decomposition

disjointness
Isn't the Bruhat decomposition disjoint?

Does anyone know when G/H has a section? —Preceding unsigned comment added by 129.11.62.100 (talk) 14:23, 16 April 2008 (UTC)


 * I added the disjoint comment. G/H should very rarely have a group section, but I think often has a sheaf section.  Did you mean when does W=N/H have a section?  I think this is more common to have a group section, but still not assured.  Also it can depend on which of the groups with a (B,N) pair you mean (which of PSL, SL, GL, PGL for those of type A). JackSchmidt (talk) 15:17, 16 April 2008 (UTC)

Does the field need to be algebraically closed?
GLn(R) can be Bruhat-Decomposed. We proved it in class. GLn(R) is obviously not over an algebraically closed field (R), so is this wrong? —Preceding unsigned comment added by Negi(afk) (talk • contribs) 21:09, 28 April 2009 (UTC)


 * Counterexample: $$SO(3, \mathbb R)$$ does not have a Bruhat decomposition: All its Borel subgroups have dimension 1; a Weyl subgroup always has dimension 0; so $$ B W B$$ has dimension at most 2. But $$SO(3, \mathbb R)$$ has dimension 3. Notice though that $$SO(3, \mathbb C)$$ does have a Bruhat decomposition, evidently since $$SO(3, \mathbb C) \cong PSL_2(\mathbb C)$$, and we know that the latter admits one. Svennik (talk) 12:02, 20 November 2023 (UTC)

Vielbeins
So I was cleaning-up and expanding the article on vielbeins at the same time that I was reading this article, and I think some neurons mis-fired, as I could not help noticing the similarity. Can the similarity be made more explicit? So, vielbeins (and the special case of tetrad (general relativity)) are just a cute calculational technique to work in a "flat" coordinate system on the tangent space of some Riemannian manifold. Since its flat, everything is "easy" and all of the complicated parts are hidden in the vielbein, which is really just a fancy similarity transform or normalizer. So should I think of the Bruhat decomposition as this normalizer-like thing? How far can this analogy be pushed? Yes, of course this is a goofy question; I just felt compelled to blurt it out.

To add some details, one can write vielbeins for symmetric spaces i.e. G/H with H coming from an involution on G (which oh by the way is a permutation). I guess there's no torus unless we got something nilpotent hiding in there... but the point is that the G/H decomposes things into B,N-like pairs and I am too lazy/too stupid to work out any further details. I mean, every time I do work out the details for something, it always turns out to be "well-known" to the experts in the field. Sooo... 67.198.37.16 (talk) 18:14, 1 November 2020 (UTC)

It's not "Gauss-Jordan elimination"
It's wrong. Rather, the Bruhat decomposition is equivalent to the LPU decomposition. What people normally use in numerical computation is the PLU decomposition. They differ in where they place the P. But this makes all the difference! The LPU (Bruhat) is totally unusable for numerical computation: Consider a matrix $$M$$ whose top-left entry $$M_{11}$$ equals zero, and consider perturbing $$M_{11}$$ by a very small amount -- this changes the matrix P discontinuously! This means forward numerical stability is unattainable. Furthermore, the entry $$U_{11}$$ will balloon, causing $$U$$ to have a very large condition number. --Svennik (talk) 01:20, 11 November 2023 (UTC)