# 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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### 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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### 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 ...
1 vote
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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 ...
1 vote
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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.,... 1 vote
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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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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)... 1 vote 1 answer 166 views ### 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}_{... 2 votes 1 answer 122 views ### 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 ... 2 votes 0 answers 131 views ### 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 ... 3 votes 1 answer 120 views ### 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 ... 4 votes 1 answer 166 views ### 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}... 8 votes 1 answer 208 views ### 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 ... 6 votes 2 answers 296 views ### 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 ...
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 ...
### 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 ...