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

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### What is the $\mathbb{Z}_2$ cohomology of an oriented grassmannian?

Let $\operatorname{Gr}(k, n)$ and $\operatorname{Gr}^+(k, n)$ denote the unoriented and oriented grassmannians respectively.
The $\mathbb{Z}_2$ cohomology of the unoriented grassmannian is
$$H^*(\...

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

Let $1 \leq k < n$ be natural numbers. Given orthonormal vectors $u_1,\dots,u_k$ in ${\bf R}^n$, one can always find an additional unit vector $v \in {\bf R}^n$ that is orthogonal to the preceding ...

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### Geometric intuition of the dimension of Grassmannians and flag manfolds [migrated]

I wish to understand geometrically (not just algebraically) why the dimension of the Grassmanian $G(k,n)$ is $k(n-k)$ and the dimension of a flag manifold $F(k_{1},k_{2},...,k_{n},N)$ is $\sum_{i=1}^{...

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### another extremal property of regular polygons

$\newcommand{\R}{\mathbb{R}}\newcommand{\D}[1]{\Delta_{#1}}\newcommand{\set}[1]{\{#1\}}\newcommand{\abs}[1]{\lvert#1\rvert}\newcommand{\E}{\mathbb{1}}$
In 1984 S.D.Berman, a Soviet mathematician, ...

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187 views

### fiber of a map into Grassmanian

Suppose $R\subset K=K_0\supset K_1\supset K_2\supset...\supset K_{n-1}\supset K_n=\{0\}$ are all vector spaces with $\dim R\cap K_i=r_i$ where $r_i$ are some fixed numbers. Suppose $O\subset Gr(r_0,\...

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### Chebyshev-like Problem for Plucker Coordinates

$\newcommand{\R}{\mathbb{R}}\newcommand{\D}[1]{\Delta_{#1}}\newcommand{\set}[1]{\{#1\}}\newcommand{\abs}[1]{\lvert#1\rvert}$
Let $n=2d+1$ be an odd integer, let $Gr(2,n)$ denote the Grassmmanian over $...

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### Ranks of cycles with base field coefficients as a generalization of ranks of multivectors?

This must be probably only reference request since I am inclined to believe that I am asking about something well known but just cannot pin down appropriate keywords for searching.
The starting point:...

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120 views

### Orthogonal Grassmanians: cases where $\text{OG}( \mathbb{P}^1 , Q) \not \simeq \mathbb{P}^3$

Let $Q = \{ q(x_0, \dots, x_4) = 0 \}$ be a quadric-threefold over a field $k$. Are there cases where the orthogonal Grassmanian $\text{OG}( \mathbb{P}^1 , Q)$ is not a copy of $\mathbb{P}^3$?
Here'...

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73 views

### What is the symmetry group of the totally nonnegative Grassmannian $Gr_{tnn}(k,n)$?

What is the symmetry group of the totally nonnegative Grassmannian $Gr_{tnn}(k,n)$? [The latter consists of those elements of the Grassmannian that can be represented by $k \times n$-matrices all of ...

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### Milnor's proof of cohomology of BO(n)

In Milnor/Stasheff characteristic classes there is the proof that $H^*(BO(n);\mathbb{Z}_2)$ is the polynomial ring on the first n Stiefel-Whitney classes. I understand the part that the latter ring is ...

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### About anti-commutation of gauge charged Fermionic quantum fields

[Please correct anything I might say wrong in what follows!]
For everything that follows I am thinking in the context of a supersymmetric QFT. Hence I guess everytime I say "spacetime" it needs to be ...

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289 views

### 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's start from a little bit far.
Basic probability theory - chain rule reads:
$$ P(AB) = P(A)P(B|A)$$
Example: consider n+m balls, where n - white balls, m - black balls,
consider A - first ...

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### Is there a lift of the q-Vandermonde identity to some geometric (motivic) identity for Grassmannians over $F_q$?

The q-Vandermonde identity reads:
$$ \binom{m + n}{k}_{\!\!q} =\sum_{j} \binom{m}{k - j}_{\!\!q} \binom{n}{j}_{\!\!q} q^{j(m-k+j)} $$
The q-binomial coefficients:
$$ \binom{ a }{ b}_{\!\!q} $$
...

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111 views

### Can Schubert cells be defined, set theoretically, by less equations then the standard ones?

Let $V = \mathbb{C}^n$ with basis $e_1,\dots,e_n$, and $U = \langle e_1,\dots,e_k\rangle$. Let
$$\Sigma(U)=\{\sigma\in Gr(V,2)\mid \sigma\in U \}$$
be the Schubert cell of $2$-planes contained in $U$....

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155 views

### Linear sections of $Gr(V,2)$

Let $V$ be a vector space, and consider $G=Gr(V,2)\subset \mathbb{P}^N$ embedded via the Plucker embedding. Let $W\subset \mathbb{P}^N$ be a linear subspace. I want to find the class $[W\cap G]\in A(G)...

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178 views

### Ideal of the Spinor variety $S^{10}\subset\mathbb{P}^{15}$

The ideal of the $10$-dimensional Spinor variety $S^{10}\subset\mathbb{P}^{15}$ is generated by $10$ quadrics.
Does anyone know a reference where these 10 quadratic equations are written down ...

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### Subset of $G_1(\mathbb{R}^n)$ having a line in common with every hyperplane of $G_{n-1}(\mathbb{R}^n)$

I am currently working on some problems related to Grassmann manifolds and eventually come to the following question.
Let $S$ be a subset of $G_1(\mathbb{R}^n)$ such that any element of $G_{n-1}:=...

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### evolution of Grassmannians along geodesic line

Let $p_0$, $p_1$ be two $n \times 2$ orthonormal matrices that represent two points on the real $Gr_{2,n}$, i.e. two 2-d subspaces in $\mathbb{R}^n$. Let $p(t): [0,1] \rightarrow Gr_{2,n}$ be a ...

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351 views

### Geometrically quantizing real Grassmannians

It seems that the Grassmannian of oriented 2-dimensional planes in $\mathbb{R}^n$
$$ \mathrm{Gr}(n,2) = \frac{\mathrm{SO}(n)}{\mathrm{SO}(n-2) \times \mathrm{SO}(2)} $$
has a symplectic structure ...

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### Stabbing disks in space, or: Galactic alignment

I have a collection of $n$ unit-radius disks in $\mathbb{R}^3$, whose centers are
random within a sphere of radius $R>1$, and which are each oriented randomly.
I'd like to find a line $L$ that ...

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**1**answer

183 views

### Is there a geometric interpretation of skew Schur functions?

Consider the cohomology ring of the Grassmannian of k-planes in complex n-space. It has a standard presentation as a quotient of the ring of symmetric functions. In this presentation, the Schur ...

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### Orbits of $GL(n, \mathcal{O})$ on pairs of linear subspaces over non-Archimedean local fields

Let $F$ be a non-Archimedean local field. Let $\mathcal{O}$ be its ring of integers. Let $Gr_{i,n}$ denote the Grassmannian of $i$-dimensional linear subspaces in $F^n$.
Can one describe ...

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### Number of Generators of the Cohomology Ring of the Grassmannians

For complex projective space, its cohomology ring has $1$ generator. Extending up to the first Grassmannian which is not a projective space, that is, Grass$(4,2)$, a direct investigation shows that it ...

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548 views

### Quotients of Grassmannians

Let $G=SL_n(\mathbb C)$ and $T$ be a maximal torus. Then the Grassmannians $Gr(r,n)$ and $G(n-r,n)$ are isomorphic. Now for the left action of the torus on each of them can we say that the GIT ...

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### Algebra Invariants of Schubert Calculus

For the Grassmannian Gr[N,k] of $k$-planes in $\mathbb{C}^N$, the cohomology ring $H^*(Gr[N,k])$ is a much studied object in an area called Schubert calculus. As a complex algebra, $H^*(Gr[N,k])$ is ...

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### Linear dependence on the Grassmannian

If I have $k+1$ linearly dependent points on the Grassmannian $G_{k-1,n-1}$ (I am using projective dimension), what can I say on the corresponding $(k-1)$-subspaces of the projective space $\mathbb{P}^...

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### Disconnecting the Lagrangian Grassmannian

Let $(V, \omega)$ be a symplectic vector space of dimension 2n. This has a Lagrangian Grassmannian $\Lambda(V)$ of Lagrangian subspaces of $V$. Now consider the following subvariety: Fix a half-...

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### Schubert calculus expressed in terms of the cotangent space of the Grassmannians

Let $T^*_{\mathbb{C}}(Gr_{n,r})$ denote the cotangent space of the Grassmannian of $r$-planes in $\mathbb{C}^n$. Moreover, let $\Lambda^\bullet$ denote the exterior algebra of $T^*_{\mathbb{C}}(Gr_{n,...

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### Fubini--Study Orthogonality for Schubert Calculus

Consider the following points:
$\bullet$ Let ${\cal Harm}(n,d)$ denote the harmonic forms of the de Rham complex of the Grassmannian $Gr_{\mathbb{C}}(n,d)$ with respect to the Riemannian metric ...

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### Sections of “forgetful” projections between flag manifolds

Given a subset $S\subseteq\{1,\cdots,n\}$ there is an associated flag manifold $F(S)$. Whenever $A\subseteq B$ there is a "forgetful" projection $F(A)\leftarrow F(B)$ (in fact I think its fibers are ...

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### Partial Flag Varieties and Quotients of Symmetric Polynomials

$\def\Q{\mathbf Q}\DeclareMathOperator{\Gr}{Gr}$First, consider a Grassmannian $\Gr(k, N)$ of $k$-dimensional subspaces in an $N$-dimensional space. It is known that its cohomology ring is
$$H_k=\Q[...

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**1**answer

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### Vector bundles on Grassmannians

Let $Gr(k,n)$ be the Grassmannian of $k$-dimensional vector subspaces $H^k$ of an $n$-dimensional vector space $V$.
Let us fix an $h$-dimensional vector subspace $\Gamma\subset V$ with $h\leq k$, and ...

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**1**answer

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### Exterior powers of $Sym^p T$ over Gr(k,n)

Let G=Gr(k,n) the Grassmannian of $k$-dimensional subspaces of $\mathbb{C}^n$ and denote by $T$ the (rank $k$) tautological bundle over $G$, and by $Sym^p T$ its $p$-th symmetric power. Is there any ...

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153 views

### How can Kernel functions make a Grassmann manifold into an Euclidean vector space?

I'm trying to read a paper called "Graph Embedding Discriminant Analysis on Grassmannian Manifolds for Improved Image Set Matching" and I came across a sentence that confused me (the last one):
A ...

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**1**answer

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### Sheaf cohomology of the universal sub and quotient bundles of the Grassmannian

How do we compute the sheaf cohomology of the universal subbundle and universal quotient bundle of the Grassmannian $G(k,V)$?

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### Decorated permutations and subset permutations

Decorated permutations are defined as permutations where the fix-points come in two colors (say $\overline{\cdot}$ and $\underline{\cdot}$). For example, the 16 decorated permutations of length 3 are
$...

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### Relation between the homotopy classes of maps on a torus, and maps on a sphere

In modern condensed matter physics, one is often interested in the homotopy classes of mappings from a $d$-dimensional torus $$\mathbb{T}^d=\underbrace{S^1\times\ldots \times S^1}_d$$
(corresponding ...

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**1**answer

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### Applications of Schubert calculus

Schubert calculus is a venerable field in mathematics where the object of study is the cohomology ring of the Grassmannians. Since it has been around for over a hundred years one might wonder if any ...

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### How to interpret the Grassmannian $G(\mathbb P, \mathbb P^m)$ as a moduli space of stable maps

Definition Let $X$ be a nonsingular (complex)projective variety. A morphism $f$ from a $n$-pointed nodal curve $\Sigma$ to $X$ is a stable map if every genus 0 contracted component of $\Sigma$ (where ...

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### Tangent space of Grassmannians on Mukai's book

Let $L\in Pic^{2d}C$ be a line bundle on a curve $C$ such that $d\geq 2\,g(C)$, $Mat_{N}(H^{0}(L))$ the set of $N\times N$ matrices with entries in $H^{0}(L)$, $Mat_{N,1}(H^{0}(L))$ the subset of ...

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### resolution tautological sheaf projective dual $G(3,6)$

I consider the Plücker embedding of $G(3,6)$ in $\mathbb{P}^{19}$. I denote by $X \subset {\mathbb{P}^{19}}^*$ the projective dual to $G(3,6) \subset \mathbb{P}^{19}$. The variety $X$ is a quartic ...

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

Let $n$ and $k$ be nonnegative integers such that $k\leq n$. Let $F$ be a field, and let $V$ be an $n$-dimensional $F$-vector space. A set $\mathcal{S}$ of $k$-dimensional subspaces of $V$ is said to ...

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### Lifting Upper Semi-Continuous Functions To Grassmannians

Let $V$ be a vector space.
Suppose we are given some upper semi-continuous map $\pi:V\rightarrow \bigcup_{k\le d} Gr(V,d)$, i.e, for any $x\in V$ we specify some subspace $\pi(x) \subseteq V$ of ...

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### Continuous Functions On Grassmannans under containment restrictions

Let $V$ be a vector space. Suppose that for a $x\in V$, we are given some subspace of dimension no more than d (e.g., the kernel of some operator defined on V, which varies smoothly with x), call it $\...

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### Degree of the discriminant curve of conic bundle associated to a cubic threefold

Let $X\subset \mathbb P^4_{\mathbb C}$ be a smooth cubic hypersurface and $l_0\subset X$ a generic line. It is known that the resolution of indeterminacies of the projection $X\dashrightarrow \mathbb ...

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### Canonical bundle of the Lagrangian Grassmannian

I work through the paper On branched coverings of some homogeneous space of Kim and Manivel and I came across the definition of the canonical bundle of the Lagrangian Grassmannian $\mathbb{LG}_n$, the ...

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**1**answer

61 views

### The degree of the locus of lines admitting an osculating plane

Let $Q\subset \mathbb P_k^3$ be a smooth quadric. Its Fano variety of lines $F(Q)$ is the union of two disjoints lines of $G(2,4)$ and according to the answer to this question the locus $F_{osc} =\{[l]...

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### Relative tangent bundle of a twisted Grassmann variety

Assume $S$ is a scheme over $\mathbb{C}$ (as nice as you want), $\mathcal{E}$ is a locally free $\mathcal{O}_S$-module and $\mathcal{A}$ is a coherent $\mathcal{O}_S$-algebra, locally free of finite ...

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115 views

### Locus of osculating planes

Let $Q\subset \mathbb P^n$ ($n\geq 3$) be a smooth quadric. Let us denote $$Z=\{ [P]\in G(3,n+1), \exists l\subset Q\mathrm{\ line\ s.t.\ } 2l\subset Q\cap P\}$$ the locus of osculating planes. Is ...

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115 views

### Exact sequence for the (twisted) universal bundle of incidence correspondence

Let $p:P\rightarrow G(2,n+1)$ be the universal $\mathbb P^1$-bundle over the grassmannian. Let us denote $I=P\times_{\mathbb P^n}P$ the incidence correspondence whose generic point is of the form $([l]...