Talk:Grassmannian

Construction of a vector field

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)

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)

Schubert Cells

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)

• Milnor and Stasheff have a little more detail. Is there something in particular you'd like to see? Cell counts? Naming the cells? I put an example to give an idea for how people think about these things. There's also a more detailed account in Hatcher's vector bundles notes, on his webpage. Rybu (talk) 19:03, 25 October 2009 (UTC)

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)

Orientation

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)

• I guess the easy way to get at orientability is to compute the fundamental group. The fibration O_n --> G_{n,k} gives you that computation. In most cases the fundamental group is Z_2, and you can check relatively quickly how orientations are transported across that loop. I think the generator is rotation by \pi in a 2-dimensional subspace that intersects the given k-dimensional subspace in a 1-dimensional subspace. I believe this argument proves G_{n,k} is non-orientable provided k>1 and n-k>1. So the only possible orientable G_{n,k} is G_{n,1} and G_{n,n-1} which happens only when n is even, as you've pointed out. Rybu (talk) 19:42, 25 October 2009 (UTC)

Grassmanian as a metric space

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)

What kind of properties do you want to know about? Rybu (talk) 03:16, 6 August 2010 (UTC)

Are complex Grassmannian manifolds Kahler-Einstein? —Preceding unsigned comment added by 114.255.218.1 (talk) 13:02, 5 August 2010 (UTC)

You should be able to answer this question from this article and the Kahler manifold article. 1) Are Grassmannians complex submanifolds of $\mathbb {C} P^{n}$? 2) Are complex submanifolds of $\mathbb {C} P^{n}$ Kahler? Rybu (talk) 03:16, 6 August 2010 (UTC)

condescension please

Can a Grassmannian topology be described in terms more accessible to the layman? When someone referred me here I was hoping for something along the lines of "the cartesian product of a p-sphere and a q-sphere" — I didn't expect to have to learn about stabilizers and Plücker embeddings for what seems like it ought to be a simple concept. (Is that enough whining?) —Tamfang (talk) 04:01, 7 September 2010 (UTC)

Read the section about the Grassmannian as a homogeneous space. It includes an explicit metrization of the Grassmannian. Rybu (talk) 05:32, 7 September 2010 (UTC)

Does the metric tell me what its overall shape is? —Tamfang (talk) 08:27, 7 September 2010 (UTC)

I don't think there is any explicit description like you're hoping for. Grassmannians are, like projective spaces, fundamental geometric objects. You may find the Plücker embedding of Gr(2,4) to be a helpful thing to study. If you're familiar with Lie groups and their representations, then the homogeneous space approach should help you get a handle on things. Ozob (talk) 11:05, 7 September 2010 (UTC)

A projective space can at least be modeled by a sphere with antipodes identified. —Tamfang (talk) 23:48, 9 September 2010 (UTC)

What do you mean by "overall shape"? If you don't have a well-defined question in mind it's hard to expect a nice answer for it at Wikipedia. :) Rybu (talk) 17:07, 7 September 2010 (UTC)

I do have a well-defined purpose in mind: to generate orthogonal projections of n-dimensional uniform polytopes into k-space, maximizing the distance between the image of any j-facet and that of any non-incident (k-1-j)-facet. (Nearly all of the higher polytope images in Wikipedia are aligned for maximum symmetry instead.) This means – or at least can be approximated by – excluding some bounded pieces of the Grassmannian, and finding the 'points' furthest from the excluded pieces. A tractable mental image of the space that contains the exclusions would help!

After another look at operator norm, I have a slightly better idea of the shape of my problem; just enough to see a flaw in the most obvious approach, and not enough to see a practical way around the flaw. —Tamfang (talk) 23:48, 9 September 2010 (UTC)

The most common way of cutting up the Grassmannian is into Schubert cells. If you could arrange it so that the right thing to exclude was certain Schubert cells, then you might be able to get somewhere pretty easily. (If you don't know about Schubert cells, I recommend Fulton's Young Tableaux.) Ozob (talk) 11:12, 10 September 2010 (UTC)

This was interesting: http://everything2.com/title/Schubert+cell —Tamfang (talk) 22:52, 11 September 2010 (UTC)

Dimension of G_r,n

Perhaps it would be informative to add that dim(G_r,n)=(n-r)*r —Preceding unsigned comment added by 157.193.5.75 (talk) 15:22, 9 January 2011 (UTC)

More elementary treatment needed

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)

Which way round?

This article currently writes the Grassmannian of k-spaces through the origin in V as Gr(k, V). Mathworld's article writes it as Gr(V, k). Which if either is correct? If both orderings have comparable status then the article needs to explain this. — Cheers, Steelpillow (Talk) 13:31, 23 September 2013 (UTC)

It's simply a convention. Some authors choose to do it one way and others choose to do it another way. (In particular, Gr(s, t) is ambiguous: It could mean s-dimensional subspaces of a t-dimensional space or the other way around, and I believe I've seen it both ways.) I wouldn't object to adding a short note on notation. Ozob (talk) 20:30, 24 September 2013 (UTC)

Thanks. Would it be useful to remark that k < V and this can be used to determine which the author means? — Cheers, Steelpillow (Talk) 10:27, 25 September 2013 (UTC)

I presume you mean k < n, where n = dim V.

It's not exactly true that k < n. The Grassmannian is a point if k = n and empty if k > n. These are uninteresting cases, and they're unlikely to show up in practice. But an author talking about Grassmannians in general might want k to be arbitrary. While maybe some of his statements are trivial they're not wrong, and allowing k to be arbitrary might make it clear that there are no special cases. (I'm reminded of a phrase that Milnor uses in "Topology from the Differentiable Viewpoint" in his introduction to the proof of Sard's theorem: "The cases where n ≤ p are comparatively easy. We will, however, give a unified proof which makes these cases look just as bad as the others.") Ozob (talk) 18:47, 25 September 2013 (UTC)

Thank you again. That's just what I meant. — Cheers, Steelpillow (Talk) 20:21, 25 September 2013 (UTC)

dimensionalities

The lines through a point in some n-dimensional vector space V do not form P(V), but the projective space of one dimension lower. One might write this as P(n−1). Mathworld describes P(V) as the lines in Gr (V+1, 1). Are we forbidden to subtract 1 from V while Mathworld may add it? Either way, the example in our lead must be clear that its P is one dimension lower than its V. — Cheers, Steelpillow (Talk) 10:24, 25 September 2013 (UTC)

P(V) by definition has dimension one less than the dimension of V. See projectivization.

The notation P(V−1) is nonsense because you can't subtract a number from a vector space. The notation V+1 is just as much nonsense for just the same reason. As far as I can see, Mathworld doesn't even use these notations. Whereas n + 1 and n − 1 both make complete sense. Ozob (talk) 18:41, 25 September 2013 (UTC)

You know, it helps if one takes care to properly read the sources one is discussing. My apologies for wasting your time. And thank you yet again for your patience. — Cheers, Steelpillow (Talk) 20:32, 25 September 2013 (UTC)

Applications

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)

Done. I used a more recent and hopefully more readable reference. Hope that's OK. — Cheers, Steelpillow (Talk) 20:43, 14 December 2013 (UTC)

Well, I think he's thinking about the amplituhedron. 67.198.37.16 (talk) 19:03, 9 July 2016 (UTC)

Smoothness etc.

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)

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)

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)

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)

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)

Name of variables in the introduction

I changed k to r in the introduction, as it is used in the rest of the article, to make especially the second paragraph (For r = 1, ...) easier to understand. 84.59.99.184 (talk) 12:47, 29 June 2014 (UTC)

More description needed for equivalence of Gr(2,3) and Gr(1,3)

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)

homology, homotopy groups

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)

Formatting etc.

In a recent series of edits, KarlJacobi 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 Gr_r(V), generally formatted using the template {{math}}. 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)

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 Gr_r(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 Gr_r(V) (also commonly denoted Gr(r,V)) is... and be done with it.Tazerenix (talk) 02:33, 8 December 2020 (UTC)

whole vector space unspecified?

JayBeeEll, 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)

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)

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)

The only reasonable alternative that comes to mind is "... k-dimensional subspaces of an n-dimensional vector space." But given that the article up to that point has made V explicit, I think that might confuse the reader. Ozob (talk) 15:02, 24 May 2021 (UTC)

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)

What about something like "Notations for the Grassmannian vary between authors; notations include Gr_k(V), Gr(k, V), Gr_k(n), or Gr(k, n) to denote the Grassmannian of k-dimensional subspaces of an n-dimensional vector space V" 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 Gr_k(V) being equivalent to Gr(k, V), and some authors using Gr_k(n) or Gr(k, n) for the same Grassmannian, where n is the dimension of V." --JBL (talk) 18:35, 24 May 2021 (UTC)

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)

Ok, great, I will implement it. --JBL (talk) 20:51, 24 May 2021 (UTC)

Needs a section on history

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