Stack Exchange Network

Stack Exchange network consists of 174 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

Visit Stack Exchange

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

7
votes
1answer
187 views

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^*(\...
44
votes
4answers
2k views

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 ...
0
votes
0answers
37 views

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}^{...
6
votes
0answers
80 views

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, ...
1
vote
1answer
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,\...
2
votes
0answers
109 views

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 $...
2
votes
0answers
40 views

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:...
0
votes
0answers
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'...
2
votes
0answers
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 ...
3
votes
2answers
352 views

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 ...
2
votes
0answers
65 views

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 ...
6
votes
1answer
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 ...
14
votes
1answer
402 views

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: $$ \binom{ a }{ b}_{\!\!q} $$ ...
1
vote
1answer
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 $U$....
2
votes
1answer
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)...
5
votes
1answer
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 ...
4
votes
3answers
167 views

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}:=...
2
votes
0answers
54 views

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 ...
11
votes
3answers
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 ...
7
votes
0answers
189 views

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 ...
11
votes
1answer
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 ...
10
votes
2answers
214 views

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 ...
0
votes
0answers
149 views

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 ...
3
votes
1answer
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 ...
1
vote
0answers
112 views

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 ...
3
votes
0answers
45 views

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}^...
2
votes
1answer
195 views

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-...
14
votes
3answers
411 views

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,...
5
votes
0answers
107 views

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 ...
6
votes
0answers
89 views

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 ...
2
votes
0answers
79 views

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[...
4
votes
1answer
221 views

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 ...
2
votes
1answer
125 views

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 ...
0
votes
1answer
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 ...
2
votes
1answer
262 views

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 $G(k,V)$?
5
votes
2answers
243 views

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 $...
9
votes
2answers
830 views

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 ...
4
votes
1answer
388 views

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 ...
1
vote
0answers
99 views

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 ...
3
votes
0answers
247 views

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 ...
0
votes
0answers
88 views

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 ...
14
votes
3answers
422 views

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 ...
3
votes
1answer
68 views

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 ...
0
votes
0answers
49 views

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 $\...
3
votes
0answers
121 views

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 ...
4
votes
1answer
321 views

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 ...
0
votes
1answer
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]...
4
votes
1answer
141 views

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 ...
1
vote
1answer
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 ...
1
vote
0answers
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]...