# Questions tagged [grassmannians]

Grassmannians are algebraic varieties whose points corresponds to vector subspaces of a fixed dimension in a fixed vector space.

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### Is the Grassmannian of a Banach space an infinite dimensional manifold?

Grassmannian of complemented subspaces in a Banach space is a Banach manifold. This is explained for example in the thesis of Douady and is rather analogous to the finite-dimensional case.
I would ...

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### Product on cellular cochains of the real Grassmannian

The real Grassmannian $Gr(k,n)$ of $k$-planes in $\Bbb R^n$ admits a Schubert cell decomposition, with one cell for each Young diagram $\lambda$ of height $\leq k$ and width $\leq (n-k)$; the ...

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### A specific integration with Grassmann variables

I have recently read (for example, here) that this relation below is true
$$ \int dz \: e^{\frac{1}{2} \sum_{ij} z_i A_{ij} z_j} = Pf(\mathbf{A}),
$$
where $Pf(\mathbf{A})$ is the Pfaffian of an even ...

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### Given a subspace $U \subseteq S^d(V)$ of a particular form, does there always exist a complement of the form $S^d(W)$?

Let $V$ be a $\mathbb{C}$-vector space of dimension $N \geq 2$, let $d$ be a positive integer, let $l < N$ be a positive integer, and let $U \subseteq S^d(V)$ be a linear subspace of codimension $k=...

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### Given a subspace $U \subseteq S^d(V)$, does there always exist a complement of the form $S^d(W)$?

Let $V$ be a $\mathbb{C}$-vector space of dimension $N$, let $d$ be a positive integer, let $l \leq N$ be a positive integer, and let $U \subseteq S^d(V)$ be a linear subspace of codimension $k=\binom{...

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### What is the dimension of this subvariety of the Grassmannian?

Well, actually, what are the dimensions of the following two subvarieties of the Grassmannian. Let $N$ be a positive integer.
Let $V \subseteq \mathbb{C}^N$ be a linear subspace of dimension $N-k$ ...

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### Stability and simplicity of tangent sheaf of Grassmannian

Everything is over the complex numbers. Let $ X = \text{Gr}(k,n) $ be a Grassmannian variety and with tangent sheaf $ T_X $.
(1) Is $ T_X $ simple, i.e. is $\text{Hom} ( T_X, T_X) = \mathbb{C} $?
(2)...

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### What does $\mathbb E_V \max_{x \in V,\,\|x\|=1} x^T Ax$ evaluate to when $V$ is random $k$-dim suspace of $\mathbb R^n$ and $A$ is fixed psd matrix?

Let $G_{k,n}$ be the grassmannian of $k$-dimensional vector spaces of $\mathbb R^n$. By the Courant–Fisher characterization, the $k$th largest eigenvalue of an $n \times n$ psd matrix $A$ is given by
$...

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### Extensions for a short exact sequence on Grassmannians

$\DeclareMathOperator\Sym{Sym}\DeclareMathOperator\Ext{Ext}$Let us consider a $n$-dimensional complex vector space $V$ and denote by $G(k,n)$ the Grassmannian of $k$-planes in $V$. We use the ...

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### Extension of short exact sequence on orthogonal Grassmannians

We work over $\mathbb C$. Let $X=OG(k,V)$ be the orthogonal Grassmannian parametrizing the $k$-dimensional subspaces of $V$, isotropic with respect to a non-degenerate bilinear symmetric form $q$.
As ...

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### Projectivization of the normal bundle of $\mathbb P^4$ in the 10-spinor variety

Let $X=S_{10} \subset \mathbb P^{15}$ be the 10-dimensional spinor variety in its minimal embedding. Consider a $\mathbb P^4 \subset S_{10}$, hence we can define the normal bundle $N=N_{\mathbb P^4|S_{...

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### Generalized polynomials and a $4$-point cross-ratio function on the non-degenerate complex $4$-quadric

The Grassmannian $Gr_1(\mathbb{C}^2)$ is another name for $\mathbb{P}^1$. If one endows $\mathbb{C}^2$ with a complex symplectic form, or if one prefers (since this will allow us to generalize in ...

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### Dimension of the Grassmannian of lines of an hyperplane section

Let $X$ be an isotropic Grassmannian, $Pic(X)=\mathbb Z$ (for example $X$ is a projective space or a quadric hypersurface). Consider a global section $s \in \Gamma(X,L)$, where $L$ is the generator of ...

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### Globally generated sheaf arising from orthogonal Grassmannian

We will use the Grothendieck's notation, according to the book of Hartshorne.
Let us consider a finite dimension $\mathbb C$-vector space $V$, with a non-degenerate symmetric bilinear form $q:V \times ...

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### Vector bundle associated to orthogonal flag

Let $V$ be a $(2n+1)$-dimensional complex vector space endowed with a non-degenerate symmetric bilinear form $q:V\times V\to \mathbb C$.
Fix the notation:
$$
OG(n-1,n,V):=\{W_{n-1}\subset W_n\subset ...

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### Global sections of a vector bundle over $OG(2,7)$

Let us work over $\mathbb C$, using the Grothendieck projectivization $\mathbb P():=Proj(Sym())$.
Consider a $7$-dimensional vector space $V$ endowed with a symmetric non-degenerate bilinear form $q:V ...

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### Grassmannians on a vector space without metric

$\DeclareMathOperator\Gr{Gr}$Let $E$ be a real, finite dimensional vector space of dimension $n$. Let $\Gr(k)$ be the set of linear subspaces of dimension $k$ of $E$. I am wondering what structures ...

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### Unsplitting sequence of vector bundles

Let $V$ be a $n$-dimensional complex vector space. Using Grothendieck's notation, we define the Grassmannian $G(k,V)$ as the space of $k$-quotients of $V$ or, equivalently, as
$$
G(k,V)=\{ \mathbb P W ...

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### Tautological bundle and its dual

Let $X=\mathbb G(2,V)$ be the Grassmannian of $2$-planes in $V=\mathbb C^n$. We denote by $\mathcal S$ the tautological bundle on $X$.
In a paper there is written that "since $\mathcal S$ is a ...

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### Projective bundles on quadrics

Let us fix the setup over $\mathbb C$. Let $V$ be a $n$-dimensional vector space endowed with a non-degenerate symmetric bilinear form $q: V \times V \to \mathbb C$.
We have that
$$
Q^{n-2}=OG(1,V)=\{\...

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### Invariant measure on affine charts of complex Grassmannian

Consider the complex Grassmannian $U(n)/U(k)\times U(n-k)$ with it's $U(n)$-invariant measure. The affine chart corresponding to $i_1, \ldots, i_k$ is given by $n\times k$ matrices for which the ...

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### Quadrics into Grassmannian as zero locus of a section

Let $V$ be a $\mathbb C$-vector space of dimension $n+2$ with a symmetric bilinear non-degenerate map $q: V \times V \to \mathbb C$. We define
$$
G(k+1,V):=\{\mathbb PW \subset \mathbb PV : \dim W=k+1\...

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### Tangent bundle for orthogonal and isotropic Grassmannians

We will work over $\mathbb C$. Let us consider a $n$-dimensional vector space $V$, then we define the $k$-th Grassmannian as
$$
\mathbb G(k,V):=\{W \subset V : \dim W=k\}.
$$
Then consider a non-...

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### Embedding of co-oriented subspaces into positive Grassmannian

$\def\R{\mathbb{R}}$Let $P_1$, $P_2$, $P_3$ be three $m$-dimensional subspaces in $\R^n$. With a slight abuse of notation they will also denote the ortho-projectors on the respective subspaces. We ...

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### Does the rational normal curve embedding extend as a mapping from the "bulk" to some bigger ambient space?

The complex projective line $\mathbb{P}^1(\mathbb{C})$ can be identified with the sphere at infinity of hyperbolic $3$-space, modeled say by the Poincare open $3$-ball in $\mathbb{R}^3$ (the sphere at ...

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### Trivial rational solution of a system of hyperplanes

Let us consider a vector space $ V $ over $ \mathbb{Q} $ of dim $6$. We denote all the two dimensional subspace in $ V $ by $ G(2,6) $ (The Grassmanian variety). One can define a map $ p $ from $ G(2,...

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### Cohomology ring of grassmannian and Pieri rule

I am sorry if this question is not for mathoverflow. I asked the same question on stackexchange (https://math.stackexchange.com/questions/4203667/cohomology-ring-of-grassmannian-and-pieri-rule), but I ...

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### Define an inner product between p-blades so that 0= completely orthogonal and 1=completely overlapping for their subspaces

$\DeclareMathOperator\span{span}$Denote two $p$-blades $\nu=v_1\wedge \dots \wedge v_p$ and $\omega=w_1\wedge \dots \wedge w_p$ $\in \bigwedge^p X$, where $X$ is an inner product space. How to define ...

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### Trivial subbundle of universal bundle on the Grassmannian $\mathbb{G}(k,n)$

I was looking at the following paper by Tango:
https://projecteuclid.org/journals/journal-of-mathematics-of-kyoto-university/volume-14/issue-3/On-n-1-dimensional-projectlve-spaces-contained-in-the-...

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### Closure of a smooth algebraic variety in a Grassmannian

Let $V,W$ be finite dimensional real vector spaces. I have a set $Z\subseteq V\times \mathbb{R}^*\times Gr(W)$, where $Gr$ is the Grassmannian manifold.
I know that $Z$ is closed smooth submanifold. ...

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### Planes in Lagrangian Grassmannians

Let $LG(h,2h)$ be the Lagrangian Grassmannian of subspaces of dimension
$h$ of a complex vector space of dimension $2h$.
For instace, $LG(1,2)=\mathbb{P}^1$, and $LG(2,4)\subset\mathbb{P}^4$ is
a ...

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### Subvarieties of Lagrangian Grassmannians

Let $LG(n,2n)$ be the Lagrangian Grassmannian parametrizing Lagrangian subspaces (so of dimension $n$) of $\mathbb{C}^{2n}$. Then $LG(n,2n)\subset G(n,2n)$, where $G(n,2n)$ is the Grassmannian of ...

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### Fiber product in the proof of the representability of the Grassmaniann functor

In Görtz-Wedhorn's book on algebraic geometry, chapter 8, there is a lemma necessary to prove that the Grassmaniann functor is a representable functor in the category of functors $Sch^{op} \to Sets$ - ...

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### Reference request: Détailed explanation why the Grassmannian scheme represents the Grassmannian functor

Similar questions have been asked on this site, including by myself, but none of these have been given a satisfying answer.
The question is: Why does the Grassmannian scheme represent the Grassmannian ...

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### Definition of tautological vector bundle [closed]

Could you please give a detailed definition (or construction)of tautological vector bundle of Grassmannian over arbitrary base scheme? Thank you in advance!

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### Jumping conics in Grassmannians

Let $Gr(1,n)$ be the Grassmannian of lines in $\mathbb{P}^n$, and $f:\mathbb{P}^1\rightarrow Gr(1,n)$ a morphism of degree two. The pull-back $f^{*}S$ of the tautological bundle $S$ on $Gr(1,n)$ ...

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### 3-secant lines of a projective curve

Consider a smooth projective curve $C\subset\mathbb{P}^n$. Let $G(1,n)$ the Grassmannian of lines of $\mathbb{P}^n$. The variety $S_2(C)\subset G(1,n)$ parametrizing lines that are secant to $C$ (i.e.,...

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### number of maxima, minima, and saddle points on a Grassmannian

I have a positive, smooth function on a Grassmannian. I am wondering whether there exists any relation between the numbers of maxima, minima, and saddle points. At least on a 2-sphere, we have the ...

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### Sheaf cohomology of Grassmannian G(2,4) with values in twisted tautological bundles over an arbitrary field

Let k be an arbitrary field. Let $G(2,4)_k$ be the Grassmannian of 2-planes in 4-space over that field. Let $\mathcal{E}$ be the tautological quotient bundle on the Grassmannian. I am trying to ...

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### Grassmannian cluster algebra of infinite type has no trees in its mutation class

The question is why the statement in the title is true (is it?).
To elaborate, recall that Grassmannian cluster algebra, according to Scott`s paper Grassmannians and Cluster Algebras, is the cluster ...

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### What's known about the matroid induced by the Plücker coordinates of the representation of a matroid?

Let $M$ be a linear matroid with ground set $E$ and independent subsets $\mathcal I$, represented by $\rho: E \rightarrow V$.
This induces a map
$$
\hat\rho: \mathcal I \rightarrow \mathbf P(\Lambda V)...

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### Smoothness of moduli spaces of stable maps

If $X$ is a projective variety the moduli space of stable maps $\overline{M}_{0,0}(X,\beta)$ is a normal variety with finite quotient singularities.
Can the pairs $(X,\beta)$ such that $\overline{M}_{...

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### Unitary orbits on the Grassmann manifold of 2-planes in complex affine space

The unitary group, $U(n)$, acts transitively on the Grassmann manifold $X = Gr(2, C^n)$. The isotropy group is $H = U(2)\times U(n-2)$, i.e. the group elements leaving some $x$ fixed. What are the ...

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### Finding the decorated permutation of a non-reduced plabic graph

This is a question about Postnikov's theory of positroids and plabic graphs. The short version is
If we have an non-reduced plabic graph $G$, how do we look at the alternating strands and read off ...

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### Varieties in the $\mathit{GL}(V)$-module $S_{\lambda}V$

It is known that, given a complex vector space $V$ of dimension $\mathit{dim}(V)=n+1$, all irreducible representations for the group $\mathit{GL}(V)$ are parametrized by Young tableaux $\lambda$.
For ...

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### Describe $\mathcal{N}_{G(\mathbb{P}^1,\mathbb{P}^k)\mid G(\mathbb{P}^1,\mathbb{P}^n)}$ [from MSE]

Note: This question came from MSE, but since I've received some useful observations I posted it here. Post on MSE
Consider $1 \leq k < n$ positive integers, and denote by $G(\mathbb{P}^k,\mathbb{P}...

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### Groups that act transitively on $\mathrm{Gr}(k,\Bbb R^n)$ but not transitively on $\mathrm{Gr}(k+1,\Bbb R^n)$

Is it known for which $n, k\in\Bbb N$ there exists a matrix group $\Gamma\subseteq\mathrm{GL}(\Bbb R^n)$ that
acts transitively on $\mathrm{Gr}(k,n)$, i.e., on the $k$-dimensional subspaces of $\Bbb ...

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### Cohomological behavior of the embedding $Gr(3,5)\to Gr(3,6)$

The following question is particularly interesting for me:
Does the natural map $Gr(3,5)\to Gr(3,6)$ induce a surjection
$$H^4(Gr(3,6),\mathbb{Z})\to H^4(Gr(3,5),\mathbb{Z})?$$
Here $Gr(k,n)$ means ...

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### The locus of lines intersecting with another fixed line on a Fano threefold

Let $Y$ be an index $2$, degree $5$, Picard number $1$ Fano threefold, i.e $Y$ is a linear section of Grassmannian $\operatorname{Gr}(2,5)$. Let $\Sigma(Y)$ be the Hilbert scheme of lines on $Y$, it ...

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### Subgroup of $PGL(n(n-1)/2, \mathbb K)$ preserving the grassmannian $Gr(2, n)$

How can we determine the subgroup of $PGL(\wedge^2 \bar{\mathbb Q}^n)$ which preserves the grassmamnnian $Gr(2, n)$ embedded as a projective variety in $\wedge^2(\bar{\mathbb Q}^n)$ via the Plucker ...