# 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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### Spherical functions in the space of functions on real Grassmannians

Let $G=O(n)$ be the orthogonal group. Let $K=S(O(k)\times O(n-k))$ be the subgroup of $O(n)$. Then the pair $(G,K)$ is symmetric, and the homomogeneous space $G/K$ is the Grassmannian of $k$-...
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### Grassmannian containing tangent variety of a curve

We work over $k=\mathbb{C}$. We consider the the Grassmanian $G(2,4)$ of lines in $\mathbb P^3$ which we embed by Plücker into $\mathbb P^5$. It is basic that under this embedding $G(2,4)$ is ...
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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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### 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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### 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 ...
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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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