67 votes
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When can one continuously prescribe a unit vector orthogonal to a given orthonormal system?

$\def\RR{\mathbb{R}}$ This problem was solved by Whitehead, G. W., Note on cross-sections in Stiefel manifolds, Comment. Math. Helv. 37, 239-240 (1963). ZBL0118.18702. Such sections exist only in ...
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45 votes

When can one continuously prescribe a unit vector orthogonal to a given orthonormal system?

Unless I'm missing something, I think that the hairy ball theorem states precisely that you cannot do this when $n = 3$ and $k = 1$. I'm not sure what happens for other values of $n$ and $k$.
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22 votes

When can one continuously prescribe a unit vector orthogonal to a given orthonormal system?

The space of orthonormal $k$-frames in $\mathbb{R}^n$ is the Stiefel manifold $V(k, n) = SO(n)/SO(n - k)$. There is a natural $SO(k)$ action on $V(k, n)$ and the quotient is the oriented grassmannian $...
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20 votes
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Is there a lift of the q-Vandermonde identity to some geometric (motivic) identity for Grassmannians over $F_q$?

Assume $V$ is a vector space of dimension $m+n$, $M \subset V$ is a subspace of dimension $m$, and $N = V/M$. Let $p:V \to N$ be the projection. Consider the Grassmannian $X = Gr(k,V)$ and its ...
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17 votes

When can one continuously prescribe a unit vector orthogonal to a given orthonormal system?

Denoting the Stiefel manifold of orthonormal $k$-frames in $\mathbb{R}^n$ by $V(k,n)$ as in Michael Albanese's answer, what you are asking for is a section of the sphere bundle $$ S^{n-k-1}\to V(k+1,n)...
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17 votes
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Homology of the free loop space of a Grassmanian

The complex Grassmannian $Gr(2,4)$ can be realized up to homotopy as the homotopy fiber of the map $BU(2) \times BU(2) \rightarrow BU(4)$ which corresponds to the Whitney sum of two complex rank 2 ...
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16 votes
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What is the $\mathbb{Z}_2$ cohomology of an oriented grassmannian?

I was surprised to learn that the ring structure of $H^*({\rm Gr}^+(k,n);\mathbb{Z}_2)$ seems to be unknown, in general. The ring structure in the case $k=2$ is given in Korbaš, Július; Rusin, Tomáš, ...
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15 votes

Map of Grassmannians associated with a Veronese embedding

I don't know a reference, but here is a simple argument. Note that $G(2,V)$ (let me use linear notation) is a homogeneous space for $GL(V)$: $$ G(2,V) = GL(V)/P_2, $$ where $P_2$ is a parabolic. If $...
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14 votes

What are the automorphisms of a Grassmannian?

Automorphisms of Grassmannians, Michael J. Cowen (1989).
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14 votes
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Can one divide algebraic manifolds ? Make sense: $Gr(2,n)/ Gr(2,n+m) = P^{n-1}/P^{n+m-1} P^{n-2}/P^{n+m-2}$

Let me begin with a sketch of an answer to your question two, namely giving an interpretation of the cross-multiplied equality $$\text{Gr}(2,n)\mathbb{P}^{n+m-1}\mathbb{P}^{n+m-2}=\text{Gr}(2,n+m)\...
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14 votes
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Grassmannians on a vector space without metric

$\DeclareMathOperator\Gr{Gr}$I discussed this in my thesis. Lemma: Every tangent space of the Grassmannian is a tensor product $T_P \Gr(k)=P^* \otimes (E/P)$; these isomorphisms are invariant under ...
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13 votes
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Planes in Lagrangian Grassmannians

This is, indeed, true. To prove this, assume we have an embedding $\mathbb{P}^2 \to \operatorname{LGr}(V)$ (where $V$ is a symplectic vector space). Let $U \subset V \otimes \mathcal{O}$ be the ...
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12 votes

Conceptual algebraic proof that Grassmannian is closed in Plucker embedding

Here is the argument I have written up in my thesis. It was suggested to me by my advisor Jarod Alper. We use the fact that a proper monomorphism is a closed immersion (EGA IV, 18.12.6). Furthermore, ...
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12 votes

What is the amplituhedron?

There is now an AMS Notices article whose title is the same as the title of this question (and which therefore may be enlightening to anyone on this page): http://www.ams.org/journals/notices/201802/...
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12 votes
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Cohomology of $G_3(\mathbb{R}^5)$

The cohomology groups of the Grassmann manifold are worked out in combinatorial terms in Luis Casian and Yuji Kodama's paper, http://arxiv.org/pdf/1309.5520v1.pdf; they make a conjecture at the end ...
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12 votes

q-Catalan numbers from Grassmannians

There are a few nice answers to related questions. Unfortunately none of them quite answers the question you asked. The $q$-Catalan number $\frac{1}{[n+1]_q}{ 2n \brack n}_q$ is the Hilbert series of ...
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12 votes
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Relation between the homotopy classes of maps on a torus, and maps on a sphere

One case in which you can establish a simple relationship is when $Y$ is a loop space. Suppose that $Y\simeq \Omega Z=\mbox{map}_*(S^1, Z)$. Then there is a bijection $[{\mathbb T}^d, Y]_*\cong [\...
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11 votes
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Steenrod operations on cohomology of grassmannians

As in Prasit's comment, the action of the Steenrod squares on the Stiefel-Whitney classes of any vector bundle are given by Wu's formula $$ Sq^i(w_j) = \sum_{t=0}^i \binom{j+t-i-1}{t} w_{i-t} w_{j+t}. ...
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11 votes

Vector bundles on Stein manifolds

As has been established in the comments, the answer to your question is yes. It is a special case of a general result known as the Oka principle which has been strengthened over time. The key is that $...
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11 votes
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Quotients of Grassmannians

I am writing this as an answer because the comments are already too long. In the following I am incredibly pedantic, because there seems to be endless possibility for confusion with the several ...
10 votes

integral or rational cohomology of real grassmannians

A good reference for the integral cohomology of $BO(k) = G_k(\mathbb{R}^\infty)$ is Brown, Edgar H., Jr. The cohomology of BSOn and BOn with integer coefficients. Proc. Amer. Math. Soc. 85 (1982), ...
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10 votes

Canonical bundle of the Lagrangian Grassmannian

Here is a group theoretical solution which makes it is easy to compute the canonical bundle of any generalized flag variety. The Lagrangian Grassmannian is the homogeneous space $G/P$ where $G=Sp(2n)$...
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10 votes

How few $k$-dimensional subspaces of $V$ are enough to have a complement to each $n-k$-dimensional subspace?

If $F$ is algebraically closed then $d(n,k)=k(n-k)+1$. For $W\subset V$ of dimension $k$, write $X_W\subset Gr(V,n-k)$ for the set of $n-k$-dimensional subspaces of $V$ that intersect $W$ non-...
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10 votes
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Schubert calculus expressed in terms of the cotangent space of the Grassmannians

The tangent space to the Grassmanian corresponds to the following representation of $U(r)\times U(n-r)$, call it $\rho$: it is the $r\times (n-r)$ matrices, with $U(r)$ acting on the left and $U(n-r)$ ...
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10 votes

Why are Lagrangian subspaces in a symplectic vector space interesting?

In symplectic linear algebra this is perhaps not completely clear. However, if one passes to symplectic geometry then the Lagrangean submanifolds indeed play a dominant role. This is in some sense ...
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10 votes
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Is it possible that the GHKK canonical basis for cluster algebras is the Lusztig/Kashiwara dual canonical basis?

I think there is good reason to think the answer is "no". In rank 2, the theta basis agrees with the greedy basis (arXiv:1508.01404). Greedy basis elements are indecomposable positive elements (see ...
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9 votes

rational cohomology of finite real grassmannian

I could not find the explicit formulas in the Algebraic models book (they seem to only do infinite Grassmannians and Stiefel varieties) or Mimura-Toda (they do the complex and symplectic case but not ...
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9 votes
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Is there a geometric interpretation of skew Schur functions?

This is discussed in Stanley's paper Some combinatorial aspects of the Schubert calculus. Corollary 3.7 says that under the natural isomorphism given by the Borel presentation of $H^*(G/P)$ which ...
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9 votes

Is it possible that the GHKK canonical basis for cluster algebras is the Lusztig/Kashiwara dual canonical basis?

Quoting from Geiss, Christof; Leclerc, Bernard; Schröer, Jan, Preprojective algebras and cluster algebras., ...
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9 votes
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Is the representation of $GL_n(\mathcal{O})$ in functions on Grassmannian multiplicity free?

Yes, this is due to Hill: Hill, Gregory, On the nilpotent representations of (GL_ n({\mathcal O})), Manuscr. Math. 82, No. 3-4, 293-311 (1994). See especially Corollary 3.2. This was generalised and ...
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