Talk:Grassmannian

message: This argument that I wrote down for \chi G_{n,k} = \chi G_{n-1,k-1} + (-1)^k \chi G_{n-1,k} it seems to imply that \chi G_{n,k} \geq 0 for all n and k, and \chi G_{n,k} = 0 if and only if n is even and k is odd.  So G_{6,3} has a non-vanishing vector field.  Does anyone know how to construct such a vector field? Rybu 23:14, 8 April 2007 (UTC)
replies: text: Answering my own question, take the action of S^1 on \mathbb C^{n/2} \equiv \mathbb R^n.  This action descends to a fixed point free action on G_{n,k} provided n even and k odd. Rybu (talk) 00:52, 1 January 2010 (UTC)
thread_title: Construction of a vector field

message: The explanations on the Schubert cell look a little bit confusing to me. I think they need to be more explicit. Unfortunately, I didn't find anything in the few math books I have. --Lhead 23:56, 15 May 2007 (UTC)
replies: text: Well, I just came here looking for the naming of the cells, so I guess that's some kind of vote that it would be nice to have it here. Ctourneur (talk) 19:32, 20 May 2012 (UTC)
thread_title: Schubert Cells

message: I have a similar problem. Does anyone how to handle the orientation?
If G(n,k) are the k-dimensional subspaces of R^n I learned that G(n,k) is orientable for n even and non-orientable for n odd. Does anyone know a prove for that. I managed to find one for the n even case which has a lot of computation in it and seems a bit unelegant. But the prove for the n odd case seems to be wrong in the source I have. I only know a method for proving that a manifold is non-orientable which uses vector fields that I am not able to handle too. Can anyone help or at least give a hint for some good sources about that?  —Preceding unsigned comment added by Del the living manga (talk • contribs) 22:05, 15 October 2007 (UTC)
thread_title:  Orientation 

message: I'm very interested in the properties of the metric suggested in the article: does anyone have any reference on the 
subject? Hobbs&Calvin (talk) 15:56, 10 July 2010 (UTC)
replies: text: What kind of properties do you want to know about? Rybu (talk) 03:16, 6 August 2010 (UTC)

text: You should be able to answer this question from this article and the Kahler manifold article.  1) Are Grassmannians complex submanifolds of \mathbb CP^n? 2) Are complex submanifolds of \mathbb CP^n Kahler?  Rybu (talk) 03:16, 6 August 2010 (UTC)
thread_title:  Grassmanian as a metric space 

message: Perhaps it would be informative to add that dim(Gr,n)=(n-r)*r  —Preceding unsigned comment added by 157.193.5.75 (talk) 15:22, 9 January 2011 (UTC)
thread_title:  Dimension of G<sub>r,n</sub> 

message: This page uses a number of mathematical constructions that would normally only be familiar to someone who knows very well what a 
Grassmannian manifold is.  While this makes the page a useful reference for experts, it is completely useless for someone learning
about Grassmanians for the first time.  — Preceding unsigned comment added by 99.235.250.152 (talk) 21:01, 24 April 2013 (UTC)
thread_title: More elementary treatment needed

message: Some theoretical physicists are using the positive Grassmannian to calculate scattering amplitudes.  I'm not sure how to do the right things to make references properly fit the usual habits of pages here, though; can someone more wiki-fluent please add this to the Applications section ? Eddy 84.215.22.225 (talk) 19:56, 14 December 2013 (UTC)
replies: text: Done. I used a more recent and hopefully more readable reference. Hope that's OK. — Cheers, Steelpillow (Talk) 20:43, 14 December 2013 (UTC)
replies: text: Well, I think he's thinking about the amplituhedron. 67.198.37.16 (talk) 19:03, 9 July 2016 (UTC)
thread_title:  Applications 

message: There are a bunch of comments in the article about Grassmannians as manifolds or varieties, e.g., the recently added "In general they have the structure of a smooth algebraic variety."  I think it would be good to be careful here -- what about, say, Grassmannians over finite fields?  Not that every statement should come with the disclaimer that it is (or is not) valid only over R or C, but it would be good to keep this sort of thing in mind. --JBL (talk) 15:55, 2 January 2014 (UTC)
replies: text: The definition of smoothness for a variety is completely different than the definition of smoothness for a manifold, but it still corresponds (as much as possible) to our intuitive sense of what "smoothness" ought to mean, and when both definitions make sense, then they are equivalent.  Smoothness of varieties does not require any hypotheses on the ground field, and in particular it makes sense over a finite field; it means that the map to the ground field is a smooth morphism.  Ozob (talk) 23:08, 2 January 2014 (UTC)

text: Is that something which would be useful to cover in this article? i.e. that there are two kids of entity, each with a different kind of smoothness, and a Grassmannian may be either kind of entity? — Cheers, Steelpillow (Talk) 10:03, 3 January 2014 (UTC)
replies: text: The article already says that the Grassmannian is a non-singular algebraic variety over an arbitrary ground field (in the homogeneous space section), so I think this should be clear to the reader who has a background in these concepts.  Ozob (talk) 01:38, 4 January 2014 (UTC)
replies: text: Thank you. I guess I am thinking of the kind of reader (like myself) who does not have sufficient grounding but wants to know more. — Cheers, Steelpillow (Talk) 13:20, 4 January 2014 (UTC)
thread_title:  Smoothness etc. 

message: I changed  to  in the introduction, as it is used in the rest of the article, to make especially the second paragraph (For , ...) easier to understand. 84.59.99.184 (talk) 12:47, 29 June 2014 (UTC)
thread_title:  Name of variables in the introduction 

message: Where the equivalence of Gr(2,3) and Gr(1,3) is mentioned, it should be emphasized in more detail, why this should be the case?
Even if each plane is characterized by a perpendicular line, these lines do not lie in a plane, and just translations will not bring them to one (Imagine tangential planes to a sphere embedded in three spatial dimensions). While for Gr(1,3) the lines will lie in plane(after translations, imagine a tangents to a circle embedded in 3D). Therefore the equivalence of the two Grassmannians is not totally clear. — Preceding unsigned comment added by 14.139.128.21 (talk) 07:22 6 April 2016 (UTC)
: "While for Gr(1,3) the lines will lie in a plane" no, certainly not.  The map "send each line to its perpendicular plane and vice-versa" really is an isomorphism between the two Grassmannians. --JBL (talk) 13:43, 6 April 2016 (UTC)
thread_title: More description needed for equivalence of Gr(2,3) and Gr(1,3)

message: A discussion of the homology and homotopy groups would be nice. A quick search gives some incomplete discussions: stack-exchange infinite grassmanians stack exchange fundamental groups grassmanian manifolds and some summary of  these given here would be nice.  Relationships to the homotopy groups of spheres?  67.198.37.16 (talk) 18:50, 9 July 2016 (UTC)
thread_title:  homology, homotopy groups 

message: In a recent series of edits,  began the project of unilaterally changing the formatting and notation in the article, which I have reverted (per  MOS:STYLERET and MOS:FORMULA).  We have had an unproductive exchange on user-talk, but it would make more sense for the substantive issue to be discussed here.  At the moment, the article uses all three notations Gr(r, V), Gr(r, n), and Grr(V), generally formatted using the template .  Also, the indices r and k are used roughly equally often.  Is this a problem?  If it is a problem, should usage be standardized in the article?  If so, to what?  --JBL (talk) 20:35, 22 November 2020 (UTC)
replies: text: Gr(r,V) and Gr(r,n) are not necessarily incompatible. The latter is shorthand for the "standard Grassmannian" Gr(r, Kn) for what ever field you decided to use, which is common enough to have a special shorthand. You can do the same notational shortcut with Grr(n) also, and frankly this is probably slightly better notation as it clearly separates the parameter specifying the dimension of the subobject from the dimension of the vector space. It is not so hard to remember "dimension of subspace comes first" of course, but once one starts working with Grassmannians of quotients this mnemonic fails and the subscript is a bit more notationally robust. The article should pick one convention for consistency (and also one index out of r or k for the subspace dimension) but one could write: The Grassmannian Grr(V) (also commonly denoted Gr(r,V)) is... and be done with it.Tazerenix (talk) 02:33, 8 December 2020 (UTC)
thread_title:  Formatting etc. 

message: , could you elucidate what you meant by no, it's the whole vector space that's unspecified in this notation?  Given the dimension of a vector space and its underlying field, up to isomorphism, a vector space is fully specified.  Ergo, since the dimension is specified, only the underlying field can be said to be unspecified.  I'm not even sure that this is the meaning, since in most notations of this type, the underlying field (when not indicated) is assumed to be the reals.  —Quondum 13:38, 24 May 2021 (UTC)
replies: text: In the notation Gr(k, V), I specify a vector space, not a vector space up to isomorphism.  In the notation Gr(k, n), I do not specify a vector space, I only specify it up to isomorphism.  In both notations, the field is specified only implicitly.  So the distinction being drawn here between the notations must be about whether the vector space is specified, not whether the field is specified.  (It is another question whether drawing this distinction is valuable to the reader.) --JBL (talk) 13:46, 24 May 2021 (UTC)
replies: text:  I see what you're saying with this.  Though I would expect that this kind of distinction is usually made by the context and not any particular variant of the notation, so making this claim of the notation would need confirmation (my perception is that even the most precise mathematicians are careless about such notational details, but rely instead on a local definition of what they mean by whatever notation they use).  My own reaction is very much in line with your parenthetical comment: should we be making such a statement at all?  My instinct would be 'no'.  —Quondum 14:47, 24 May 2021 (UTC)
replies: text: The only reasonable alternative that comes to mind is "... -dimensional subspaces of an -dimensional vector space." But given that the article up to that point has made  explicit, I think that might confuse the reader.  Ozob (talk) 15:02, 24 May 2021 (UTC)
replies: text: That wording sounds decent to me.  Confuse the reader?  Maybe I'm confused, but my impression is that different authors use this type of variant to mean exactly the same thing, their choice between them being whatever they prefer.  For example, one author might define V as an n-dimensional vector space and then Gr(k, V), and another might just define Gr(k, n) directly.  In particular, I cannot picture an author using both notations together, one referring to V, and the other to "all isomorphs of V".  —Quondum 15:13, 24 May 2021 (UTC)
replies: text:  What about something like "Notations for the Grassmannian vary between authors; notations  include ,  , , or  to denote the Grassmannian of -dimensional subspaces of an -dimensional vector space " that sidesteps this issue by not drawing a distinction?  Another possibility, more closely mirroring what's already there, would be "Notations vary between authors, with  being equivalent to , and some authors using  or  for the same Grassmannian, where  is the dimension of ."  --JBL (talk) 18:35, 24 May 2021 (UTC)
replies: text:  I like the first of those two – it reads far more naturally and says what needs to be said.  —Quondum 18:58, 24 May 2021 (UTC)
replies: text:  Ok, great, I will implement it.  --JBL (talk) 20:51, 24 May 2021 (UTC)
thread_title:  whole vector space unspecified? 

message: Yes, there should be a section on history, including a link to Grassmann the person. Ishboyfay (talk) 00:07, 25 April 2023 (UTC)
thread_title:  Needs a section on history