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

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I'm reading about spinor bundles in Kuznetsov's paper https://arxiv.org/pdf/math/0512013. On page 17 he states that the orthogonal Grassmanian $\mathsf{OGr}(m,2m)$ with respect to a non-degenerate ...
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I try to study a specific operation of pullback and pushforward related to flag varieties. Let $Y=Gr(r-1,2r-1)$ be the variety of $r-1$ subspaces in a vector space of dimension $2r-1$. I also have $Y'=...
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Formulas for the Riemannian exponential map $\exp_{P}$ and its differential at a given tangent vector $\Delta\in T_PGr(n,p)$ $(d\exp_P)_{\Delta}$ can be computed for the Grassmann manifold $Gr(n,p)$ (...
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$\newcommand\O{O}%In case \\\$\operatorname O\\\$ might be acceptable, just change this to `\DeclareMathOperator\O{O}` \DeclareMathOperator\tr{tr}$Let $\O_n$ be the orthogonal group of $n \times n$ ...
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I have been thinking about derived categories of homogeneous spaces a little bit lately, especially $\mathrm{D}^{\mathrm{b}}(\mathrm{Gr}(k,V))$. Some foundational results are proven in a paper of ...
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Jacobi fields on a Riemannian manifold can be expressed using the differential of the exponential map, given an initial value of the field $J(0)$ and its derivative $D_t J(0)$. Is it also possible to ...
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An integral point $P$ of a Grassmannian $Gr(k,n)$ is a $k$-dimensional subspace such that $P \cap \mathbb{Z}^n$ is a rank $k$ sublattice of $\mathbb{Z}^n$. Its height $H$ is given by the determinant ...
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I have been recently studying the construction of the Hilbert scheme from the book 'Deformations of algebraic schemes' by Sernesi and I am stuck with one of the very first steps. Let me expose the ...
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The question is complementary to this question. How to construct pointless real forms of Grassmannians $\operatorname{Gr}(k,n)$? For which $k$ and $n$ do they exist? Any reference will be very helpful....
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Let $q$ be a power of a prime $p$, $\Bbb F_q[\theta]$ the analog of $\Bbb Z$ in characteristic $p$, and $\Bbb C_\infty$ the analog of $\Bbb C$ in characteristic $p$. A lattice of rank $r$ in $\Bbb C_\...
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Related to this question but much simpler. I am looking for a reference to a description of the cluster algebra of finite type C in terms of generators and relations. Specifically, I am interested in ...
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In my research, I came across the following strange family of varieties by trying to construct smooth varieties with a given Poincaré polynomial. Given $n\in \mathbb{N}$ and a tuple $(a_1, \ldots, a_n)...
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A basic question about finite type cluster algebras. Let $A$ denote the cluster algebra of type $\mathrm C_{n-1}$, $n\ge 3$. I will view $A$ as a $\mathbb Z$-algebra with the $n^2$ generators $\Delta_{...
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Let $X\subseteq \mathbb{P}_{\mathbb{C}}^n$ be a smooth hypersurface of degree $d\ge 4$, and consider the Grassmanian $G=\operatorname{Gr}(3,n+1)$ parametrizing the ($2$-)planes inside $\mathbb{P}_{\...
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Let $\mathcal{C}$ be a triangulated category, and let $\mathcal{B} \subset \mathcal{C}$ be an admissible subcategory. The admissibility condition gives rise to two semi-orthogonal decompositions $\...
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I decompose $\mathbb{R}^n=\mathbb{R}^p\times \mathbb{R}^q$ for some $p,q\in \mathbb{N}$. Let $B\subseteq \mathcal{L}(\mathbb{R}^p,\mathbb{R}^q)$ be the closed unit ball with center $0$ (with respect ...
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Let $\mathcal{T}$ be a triangulated category, and let $\mathcal{S} \subseteq \mathcal{T}$ be an admissible subcategory. Recall, this means that the inclusion $\mathcal{S} \hookrightarrow \mathcal{T}$ ...
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The functor $k_{U(d)}$ of the set of isomorphism classes of numerable principal $U(d)$-bundles is represented by $BU(d)$ the Milnor construction of the classifying space. The restriction of $k_{U(d)}$ ...
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In my research, one can find as a special case the following family of varieties. Fix integers $0<k<n$ and let $G=Gr(k,n)$ be the Grassmannian of $k$-planes in an $n$-dimensional vector space $V$...
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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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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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$\DeclareMathOperator\GL{GL}\DeclareMathOperator\Gr{Gr}$Let $\mathbb{F}$ be a non-Archimedean local field. Let $\mathcal{O}$ be its ring of integers. The natural representation of the group $\GL_n(\...
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I asked a similar question in PhysicsSE but it seems more like a mathematical issue, so I post here in a more refined form. I am not confident if the below description of the problem makes sense. ...
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Let $G := G(k,V)$ denote the Grassmannian of $k$-linear subspaces in a $\mathbb{C}$-vector space $V$ of dimension $n$. Let $S$ denote the tautological bundle over $G$. There is a canonical map on ...
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Let $d>0$ and $r_1,r_2>0$ such that $r_1+r_2 < d$. Consider two (say, closed) balls $B_1,B_2$ in $\mathbb{R}^m$ having radii $r_1,r_2$ and whose centers are at distance $d$. Let $C$ be the ...
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$\DeclareMathOperator\SO{SO}\DeclareMathOperator\Gr{Gr}$I came across a problem that could be formulated in term of Grassmannians, I would be very glad to have your opinion about it. I have an ...
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Let $V$ be a $\mathbf{C}$-vector space of dimension $n$, and consider the Grassmannian $G:=Gr(2, V)$ of 2-dim subspaces of $V$. Then we have the tautological subbundle $E\subset V\otimes \mathcal{O}_G$...
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$\DeclareMathOperator\Gr{Gr}$The Grassmnnian variety $\Gr(k,n)$ is the set of $k$-dimensional subspaces of $\mathbb{C}^n$. The coordinate ring $\mathbb{C}[\Gr(k,n)]$ is generated by Plucker ...
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Consider the grassmannian $\mathrm{Gr}(2,5)$. We know there is an embedding of $\mathrm{Gr}(2,5)$ into $\mathbb{P}^9$ by using the 10 Plucker coordinates, and they satisfy 5 Plucker relations. And, so ...
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In a book I am reading (Leon Simon, Lectures on Geometric Measure Theory) at some point the author claims that if a certain property $(P)$ holds for almost every $n$-plane $\pi\subset \mathbb{R}^{n+k}$...
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The Schur functions are symmetric functions which appear in several different contexts: The characters of the irreducible representations for the symmetric group (under the characteristic isometry). ...
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This is a cross-post from this other question that I asked ~1 month ago in the mathematics forum, with no reaction. I am still stuck on this, looking for references or approaches to proofs. I hope I ...
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The Schur polynomials $$s_\lambda(x_1, \ldots, x_n) = \frac{|x_i^{\lambda_j+n-j}|_{1\le i,j\le n}}{|x_i^{n-j}|_{1\le i,j\le n}}$$ naturally appear as polynomial representatives for Schubert classes in ...
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In Section 5, Chapter 1 of the famous book "Principles of algebraic geometry" by Griffiths and Harris, there are two equivalent conditions to determine whether a multivector $\Lambda\in\...
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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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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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Let $C$ be an algebraic curve over $\mathbb{C}$ and $\omega_C$ be its canonical bundle. We may assume that $C$ has genus $g\geq2$. Let $x\in C$ be an arbitrary point. Question: What is the image of $...
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This is a problem that's been bugging me for some time, and therefore I've decided to ask it here. Let $X$ be a smooth projective (irreducible) variety over an algebraically closed field of ...
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I have a question about Exercise 18.11 In Harris' book Algebraic Geometry, on page 231: Give a proof of the nondegeneracy of the general hyperplane section of an projective irreducible nondegenerated ...
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The vector space dimension of the cohomology group of the $2$-plane Grassmannian $\mathrm{Gr}_{2,n}$ is given by the number of tuples $(\lambda_1,\lambda_2)$ satisfying $$ n - 2 \geq \lambda_1 \geq \...
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Let $L_1$ and $L_2$ $\in$ $\mathbb{P}^4$ be two planes that intersect in exactly one point $Q$. Let $P_1 \in L_1$, $P_2 \in L_2$ points, such that $P_1 \neq Q \neq P_2$. Using the duality theorem, ...
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I'm reading this book The supersymmetric method in random matrix theory and applications to QCD. In page 302-303, the author calculate the following integral $$ Z=\int d\psi \, dH \, P(H)\exp\left(-\...
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Let $\pi_1, \pi_2$ be two $k$-dimensional subspaces of $\mathbb R^n$. Using elements of the orthogonal group $O(n)$, how much can we simplify $\pi_1, \pi_2$? Certainly there always exists $A \in O(n)$ ...
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This question aims to compute ${\rm Vol}(-K_X-tD)$ where $X$ is a $\mathbb{Q}$-factorial Fano variety of dimension $n$ and $D$ is a nonzero effective divisor on $X$. This volume is positive when $0\le ...
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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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$\DeclareMathOperator\Gr{Gr}$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 ...
Jianrong Li's user avatar
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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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Let $V$ be a $2n$-dimensional vector space endowed with a nondegenerate skew-symmetric form $q:V \to V^\vee$. We define the isotropic Grassmannian to be $$ X:=G_q(k,V)=\left\{[W] \in \mathbb P \left( \...
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We work over the field of complex numbers $\mathbb C$. Let $G$ be a simple linear algebraic group and let $P,Q$ be standard maximal parabolic subgroups of $G$ containing the same Borel subgroup $B$. ...
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We will work over $\mathbb C$ and the notation will be coherent with the paper of Ottaviani (see [Ott]). Consider a $n$-dimensional quadric hypersurface $Q_n \subset \mathbb P^{n+1}$. We have ...
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