# 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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### The conditions to determine whether multivector $\Lambda\in\wedge^k V$ is decomposable

In Section 5, Chapter 1 of the famous book "Principles of algebraic geometry" by Griffiths and Harris, there are two equivalent conditions to determine whether a multivector $\Lambda\in\...

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### Algebraic description of Gauss map

Let $X \subset \mathbb{P}^n$ a smooth connected projective variety of
dimension $k$ over complex numbers. In classical literature
there is a Gauss map $\mathcal{G}: X \to \mathbb{G}(k, n)$ which
...

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### Quadric contain tangent variety of a curve in $\mathbb{P}^5$

Let $Q^4 \subset \mathbb{P}^5$ a smooth quadric over $\mathbb{C}$
which is via Pluecker map isomorphic
to Grassmannian of lines $\mathbb{G}(1,\mathbb{P}^3)$
in $\mathbb{P}^3$.
Consider following ...

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### Ideals whose alebraic variety is a singleton

I do not work in algebra, so i apologize in advance if there are some unclear/wrong sentences. Let us consider the ring $\mathbb{C}[X_1,\ldots,X_q]$ of polynomials in $q$ variables. For an ideal $I$ ...

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### Image of $H^0(C,\omega_C-x)$ in $G(g-1,H^0(C,\omega_C))$

Let $C$ be an algebraic curve over $\mathbb{C}$ and $\omega_C$ be its canonical bundle. We may assume that $C$ has genus $g\geq2$. Let $x\in C$ be an arbitrary point.
Question: What is the image of $...

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### Differential of a specific morphism to a Grassmannian

This is a problem that's been bugging me for some time, and therefore I've decided to ask it here. Let $X$ be a smooth projective (irreducible) variety over an algebraically closed field of ...

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### Universal hyperplane section and nondegeneracy of general hyperplane section

I have a question about Exercise 18.11 In Harris' book Algebraic Geometry, on page 231:
Give a proof of the nondegeneracy of the general hyperplane
section of an projective irreducible nondegenerated ...

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### The dimension of the Grassmannian cohomology ring $H^*(\mathrm{Gr}_{n,d})$ and the fundamental $\frak{sl}_n$-representation $V_{\pi_d}$

The vector space dimension of the cohomology group of the $2$-plane Grassmannian $\mathrm{Gr}_{2,n}$ is given by the number of tuples $(\lambda_1,\lambda_2)$ satisfying
$$
n - 2 \geq \lambda_1 \geq \...

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### Intersection of schubert varieties

Let $L_1$ and $L_2$ $\in$ $\mathbb{P}^4$ be two planes that intersect in exactly one point $Q$. Let $P_1 \in L_1$, $P_2 \in L_2$ points, such that $P_1 \neq Q \neq P_2$. Using the duality theorem, ...

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### How to perform this Gaussian matrix integral?

I'm reading this book The supersymmetric method in random matrix theory and applications to QCD. In page 302-303, the author calculate the following integral
$$
Z=\int d\psi \, dH \, P(H)\exp\left(-\...

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### Fundamental domain for two Grassmannians

Let $\pi_1, \pi_2$ be two $k$-dimensional subspaces of $\mathbb R^n$. Using elements of the orthogonal group $O(n)$, how much can we simplify $\pi_1, \pi_2$? Certainly there always exists $A \in O(n)$ ...

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### The geography of models of Fano varieties

This question aims to compute ${\rm Vol}(-K_X-tD)$ where $X$ is a $\mathbb{Q}$-factorial Fano variety of dimension $n$ and $D$ is a nonzero effective divisor on $X$. This volume is positive when $0\le ...

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### A Plücker coordinate matroid

Let $V$ be an $n$-dimensional vector space over a field $F$. Let
$\mathrm{Gr}(V,d)$ be the set (Grassmann variety) of all
$d$-dimensional subspaces of $V$. We can regard
$\mathrm{Gr}(V,d)$ as a subset ...

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### Relation between $3$-term Plücker relations and more than $3$-term Plücker relations

$\DeclareMathOperator\Gr{Gr}$Let $\Gr(k,n)$ be the Grassmannian variety of $k$-planes in an $n$-dimensional vector space. The coordinate algebra $\mathbb{C}[\Gr(k,n)]$ is generated by Plücker ...

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### Does the space of hyperplanes in the Grassmannian have a name?

A way of defining the Grassmannian $Gr(k,n)$ is to consider the space of $k\times n$ matrices mod $GL(k)$ transformations on the rows. I'm interested in the space of $k\times 2n$ matrices mod $GL(k)$ ...

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### Non-trivial extension and tangent bundle isotropic Grassmannian

Let $V$ be a $2n$-dimensional vector space endowed with a nondegenerate skew-symmetric form $q:V \to V^\vee$. We define the isotropic Grassmannian to be
$$
X:=G_q(k,V)=\left\{[W] \in \mathbb P \left( \...

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### Reference request & more: compute vector bundles for homogeneous $G$-varieties

We work over the field of complex numbers $\mathbb C$.
Let $G$ be a simple linear algebraic group and let $P,Q$ be standard maximal parabolic subgroups of $G$ containing the same Borel subgroup $B$. ...

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### Generalize spinor bundles over orthogonal Grassmannians

We will work over $\mathbb C$ and the notation will be coherent with the paper of Ottaviani (see [Ott]).
Consider a $n$-dimensional quadric hypersurface $Q_n \subset \mathbb P^{n+1}$. We have ...

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### Transition maps between coordinate charts on the Grassmann manifold

Let $\mathbf{Gr}_{n,k}$ be the manifold of $k$-dimensional subspaces of $\mathbb{R}^n$, and let $\mathbf{col}$ be the map that takes a matrix in $\mathbb{R}^{n\times k}$ to its columnspace. The map
\...

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### Riemannian geometry of Grassmannian bundles

The Grassmannian bundle of a vector bundle $E$ is a smooth manifold where each fiber over the base space is replaced by the Grassmannian (of specified rank) of the fiber. I am interested in defining a ...

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### Counting non-zero Gramians of Grassmanians over finite field

In case of $\mathbb{F}_{2}$, we can obtain the number of all reduced row echelon forms (so called Grassmannians) for some m$\times$n full rank matrices by the following gaussian polynomial,
$$
\binom{...

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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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### 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. ...