# 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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### Multiplication formula in Grassmannian cluster categories

Grassmannian cluster categories are studied in A categorification of Grassmannian cluster algebras and Cluster categories from Grassmannians and root combinatorics. The category $CM(B_{k,n})$ of Cohen-...
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### Confusion over spin representation and coordinate ring of orthogonal Grassmannian

This is a copy from MSE where the question did not attract much attention. I'm working over $\mathbb{C}$ here. Let $G=\mathrm{SO}(2n+1)$ be the odd orthogonal group, and $P$ be the maximal parabolic ...
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### Hilbert series of special linear sections of Grassmannian $Gr(2,n)$

Consider the Grassmannian $\operatorname{Gr}(2,n)$. I want to know Hilbert series of $H_1 \cap H_2 \dots \cap H_m \cap \operatorname{Gr}(2,n)$ in the Plücker embedding of $\operatorname{Gr}(2,n)$, ...
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### The group of global sections of the automorphism bundle of the tangent bundle on a Grassmannian

Let $X={\rm Gr}(k,n)$ denote the Grassmannian of $k$-dimensional subspaces in ${\Bbb C}^n$. We regard $X$ as an algebraic variety over $\Bbb C$. Let ${T_X} \to X$ denote the tangent bundle on $X$. For ...
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### What is the name of the real form corresponding to the quaternionic symmetric space?

Let $G$ be a compact simple Lie group. Choose a system of positive roots, and let $\mathrm{SU}(2) \subset G$ correspond to the highest root, and $\mathbb{Z}/2 \subset \mathrm{SU}(2)$ the centre. The ...
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### Parametrization of Plucker Equations for Grassmannian $Gr(2,n)$ by ordered quadruples

As David Speyer pointed out in a comment to this MO question, Number of Plücker relations for a Grassmannian, 'many people mean specific lists of relations when they say "the Plucker relations"'. ...
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### What is an orthogonal form?

I am reading the article: Algebro - Geometric applications of Schur S- and Q-polynomials On page 179, they said about an orthogonal form $\psi$ on $W$. There is no definition of an orthogonal form on ...
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### Degree of varieties swept out by linear spaces (Eisenbud & Harris' 3264 and All That)

I have a couple of questions on statements from Harris' and Eisenbud's lecture "3264 and All That" at page 145, Section 4.2.3: Varieties swept out by linear spaces. The contents of 4.2.3 answers ...
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### Radon transform on complex Grassmannians

Let $Gr_{i,n}$ denote the Grassmannian of complex linear $i$-dimensional subspaces in the Hermitian space $\mathbb{C}^n$. Let $1\leq i<n/2$. Consider the Radon transform between space of functions ...
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### Is the representation of $GL_n(\mathcal{O})$ in functions on Grassmannian multiplicity free?

Let $\mathbb{F}$ be a local non-Archimedean field. Let $\mathcal{O}\subset \mathbb{F}$ be its ring of integers. Let $GL_n(\mathcal{O})$ be the (compact) group of $n\times n$ invertible matrices with ...
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### Generically intersecting Schubert cycles

I have a question about the the proof of Pieri's formula from Harris' and Eisenbuds's lecture "3264 and All That"on page 146. Before the proof we use this terminology (see page 139): let $G=G(k,V)$...
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### Sheaf cohomology of the complement of a schubert variety

Let $k$ be a field, $d,n \in \mathbb{N}$ and denote by $Gr(d,n)$ the Grassmannian, which parameterizes the $d$-dimensional linear subspaces of $n$-dimensional $k$-vector space, considered as a ...
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### Homology of the free loop space of a Grassmanian

Is there any reference for calculation of the rational homology of the free loop space $H_*(\mathcal{L}Gr(k,n),\mathbb{Q})$ of a complex Grassmanian? More precisely, I am interested in computing ranks ...
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### Why are Lagrangian subspaces in a symplectic vector space interesting?

A subspace in a symplectic vector space could be one of two extremes: either symplectic (meaning the form is nondegenerate there) or Lagrangian. Or it could be something between the two, meaning a ...