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Questions tagged [schubert-calculus]

Schubert calculus is the study of flag varieties, which are quotients of algebraic groups (usually complex semisimple, but sometimes over the real numbers or even finite fields) by parabolic subgroups.

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Homology and Cohomology of Schubert varieties

Let $X$ be a Schubert variety , seen as a sub variety of the projective variety of complete flags (over the field $\mathbb R$ or $\mathbb C$) . Is it true that $H_n(X,\mathbb Z)\cong H^n(X,\mathbb Z)...
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Reference for Grassmann and Schubert varieties for Beginners .

I need some references to understand Grassmann and Schubert Variety as a beginner. I am looking for self-contained notes on these. Thanks.
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Schubert Cell Structure for some variety with prescribed bilinear form

We know the construction of Schubert cell complex structure for the case of Grassmann manifold $Gr_k(\mathbb{C}^n)$ i.e variety of all k- dimensional subspaces $V$ of $\mathbb{C}^n$. Consider that ...
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Unipotent action on flag variety

I am reading https://www.sciencedirect.com/science/article/pii/S138572587680008X and must be missing something obvious. The premise is that we are considering the fixed point set of a unipotent ...
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Poincare duality pairing matrix Grassmannian

I'm trying to prove that for $i+j = 2k(n-k)$ and $i = 2s$, $j = 2l$, ($i \neq j$, this is easy) the following Poincaré duality pairing holds: $$ H^i(\mathbb{G}(k, n)) \times H^{j}(\mathbb{G}(k, n)) \...
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$H^*(\mathbb{G}(k,n))$ free with basis the partitions that index the Schubert varieties [duplicate]

I have to proof the fact that $H^*(\mathbb{G}(k,n))$ is, as an abelian group, free with basis the partitions that index the Schubert varieties, but I'm having trouble doing it myself. Denote $\sigma_{...
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Every Schubert cycle a Chern class?

Consider the Grassmann variety $\mathbb{G}(k,n)$ and its Chow ring $A$. It is known that the classes of Schubert cycles form a $\mathbb{Z}$ basis of $A$. Is it known which of these Schubert cycles can ...
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Schubert class in the Grassmannian G(3,6)

How to compute the Schubert class $\sigma$$^2$$_2$$_1$ in the Grassmannian G(3,6)? I remember the result is $\sigma$$_3$$_3$ + 2$\sigma$$_3$$_2$$_1$ + $\sigma$$_2$$_2$$_2$.
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Singular Schubert Variety in $Fl_4(\mathbb C)$

Consider the Complete Flag Variety $Fl_4(\mathbb C)$ and its Schubert Variety given by the permutation $(1234) \to (3142)$, i.e. its highest dimension cell can be parametrized by $$\begin{bmatrix} ...
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How to prove Wielandt minimax formula?

The statements are as follows: Let $1\leqslant i_1<i_2<\cdots<i_k\leqslant n$ be integers. Define a partial flag to be a nested collection $V_1\subset V_2\cdots\subset V_k$ of subspaces of $\...
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Schubert variety associated to a flag of subspaces of a vector space

At the end of the following page : The associated Schubert variety of a flag of subspaces of a vector space. , the author says : Let $[W]\in X=X_{\underline i}$, by construction: \begin{equation} [W]=...
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Orthogonal Grassmannian

The orthogonal Grassmannian $OG(k,n)$ is the set of all isotropic $k$ dimensional subspaces of a $n$ dimensional subspaces with respect to a non-degenerate symmetric bilinear form. Is it isomorphic to ...
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Reference for flag varieties G/P

Is there a good reference for learning about flag varieties $G/P$? I'm already comfortable with the algebraic geometry and the example of Grassmannians, but I am not so comfortable with algebraic ...
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Understanding Schubert Varieties

I'm reading the paper "Chern Classes of Schubert Cells and Varieties" by Paolo Aluffi and Leonardo Constantin Mihalcea. I'm going through this paper rather slowly, but I'm a bit stuck on the second ...
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How can I get the matrix form for a schubert cell?

I am trying to learn how to understand the cell-decomposition for the grassmannian and am following these notes: http://www.math.drexel.edu/~jblasiak/grassmannian.pdf. On page 2 the author considers ...
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Is the quotient of standard parabolic subgroups isomorphic to a Schubert variety

Let $G$ be a reductive algebraic group over an algebraically closed field. Let $B \subseteq P_2 \subseteq P_1$ be a Borel subgroup and two parabolic subgroups. Then $P_1/P_2 \subseteq G/P_2$ is a ...
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$\mathbb P^1(\mathbb C)$ inside $Gr(2,4)$

Let $V$ be a vector space over $\mathbb C$ of dimension $4$ and consider the Grassmanian $Gr(2,V)$. Fix an index $I=(1,3)$ and a complete flag of $V=\langle e_1, e_2, e_3, e_4 \rangle$ given by $ F =\...
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Is the geometrical meaning of cup product still valid for subvarieties?

It is known that cup product is Poincaré dual to the intersection. I'm referring to the following fact: if $X$ is a closed, oriented smooth manifold and $A, B$ are transverse-intersecting oriented ...
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$GL_n(F)$ acts on the flag variety

I have the following 2-part question as a homework assignment... Let $F$ be a field and $n\in \mathbb{Z}_{\geq 1}$. A full flag in the vector space $F^n$ is a chain of subspaces \begin{align} \{...
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Intersection of Schubert cycles

I want to compute intersection of the Schubert cell $\sigma_{(3,0)}$ with all the cells $\sigma_{a_1, a_2}$ in the grassmanian $G(2,5)$. I am not sure I am doing correctly but I can't see my mistake. ...
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Why is the Complete Flag Variety an algebraic variety?

Let $V$ be a $\mathbb C $ - vector space of dimension $n$. Let's consider the set $Fl(n)$ of all the complete flags $F_{\bullet}$ $$F_1 \subset F_2 \cdots \subset F_n$$ where the $F_i$ are subspaces ...
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Understanding the cohomology ring of the Grassmannian

Some background first: I'm trying to understand the solution of some enumerative geometry problems, such as proving that a smooth cubic contains $27$ lines. I know that this becomes easier once one ...
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Betti numbers for the isotropic grassmannian

I want to know if there is some type of combinatorial formula for computing the Betti numbers of the isotropic grassmannian $IG(r,2n)$ for $r\leq n$. I'm thinking of this as the homogenous space $G/P$...
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What is a general linear subspace?

When it comes to defining a general plane with respect to a line in $\mathbb R^3$, I can think of this definition as: take any plane not containing the line. Reading Fulton's "Young Tableau" I can't ...
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Schubert and Grassmann varieties

Can anybody please suggest nice reference which will have lots of examples and counter examples for studying Grassmann varieties in particular Schubert variety?
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Locus of tangent lines to a smooth curve of degree $d$ and genus $g$

Suppose $C\subseteq\mathbb{P}^3$ is a smooth curve of degree $d$ and genus $g$ (let's say we are working over $\mathbb{C}$). Let $T(C)$ be the locus of tangent lines to $C$. In other words, $$ T(C) = \...
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W action of Schubert varieties

Let $X(w)$ be a Schubert variety in $G/B$, where $G$ is a semisimple algebraic group and $B$ is a Borel in $G$. Then for $v \in W$, what is $v X(w)$ ? Is it same as $X(vw)$ ?
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Spherical Schubert Variety

I am studying Schubert variety and I came across a problem understand a particular detail. Let $G$ be a reductive group, and $\mu\in X_{\bullet}(T)$ a coweight i.e. $\mu\in Hom(\mathbb{G}_m,T)$, ...
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Number of fixed points of torus action over partial flag variety

Consider $g\in U(n)$ and $t\in T$, where $T$ is the diagonal maximal torus in $U(n)$. Some common manifolds may be obtained as quotients of the $U(n)$ like the complex grassmannian, $Gr(k,n)=U(n)/U(...
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Finding cohomology group of open dense subset of Schubert variety

Let $Y=Gr_{m}(\mathbb{C}^n)$ be the Grassmannian of $m$-plane inside $\mathbb{C}^n$. Let $X$ and $X'$ be two Schubert varieties inside $Y$ such that $X'\subset X$ and $dim(X')<dim(X)$. Let $Z=X\...
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Singular Locus of a Schubert variety

I am trying to compute the singular locus of the schubert variety $X_w$ in $G_{2,7}$ where $w=(4,7) \in I_{2,7}$. Following the notation in the book "The Grassmannian Variety: Geometric and ...
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Smoothness of Schubert Variety

Consider the Schubert variety $X(s_3s_2s_1s_4s_3s_2)$ in $SL_5/P_2$, where $P_2$ is the maximal parabolic corresponding to the simple root $\alpha_2$. In one line notation this permutation can be ...
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Applying the divided difference operator

This question is about divided difference operators. How do I perform $\partial _2$ or $\partial_3$ on $x_1^2x_2$? $\partial_i$ is defined as $\frac{p-r_i.p}{x_i-x_{i+1}}$, where $r_i$ is the ...
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The associated Schubert variety of a flag of subspaces of a vector space.

Let $V$ be a vector space and $W_1 \subsetneq W_2 \subsetneq ... \subsetneq W_\ell \subsetneq V $ a flag of subspaces. The associated Schubert variety is defined as : $ \Omega ( W_{ \bullet } ) = \{ \...
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Schubert decomposition of a Grassmannian

I'm going through Sheldon Katz's Enumerative Geometry and String Theory, and a few things regarding the Grassmannian $G(2,4)$ (lines in projective space) are bothering me: How can I compute the ...
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Schubert cell decomposition and full flags

I am looking for a self-contained basic theory of Schubert-cells through finding the decomposition of the full flag $Fl_3(\mathbb C^3)$.
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Schubert Cells of Flags

I have been reading on these notes Undergraduate Lectures on Flag Varieties and I need some explanations on two things: In page 3, how he modefied the matrices in the "Second Attempt" In the same ...
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Dimension of a linear space in $\mathbb{P}^{n}$ (Schubert Calculus)

I am reading about Schubert calculus and have come across this definition of a linear space: A linear space $L$ in $\mathbb{P}^{n}$ is defined as the set of points $P = (p(0), p(1), \ldots, p(n))$ of ...
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Schubert calculus on Grassmannians

Can anyone please suggest me some notes or books where I can read about Schubert calculus? I am studying Grassmannian varieties so I would like to understand how to use this tool, in particular with ...
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Integral homology of real Grassmannian $G(2,4)$

I would like to compute $\pi_1$ and the integral homology groups of the real Grassmannian $G(2,4)$. (This is a question on an old qualifying exam.) The hint for the computation of $\pi_1$ is to put a ...
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Two questions about Schubert calculus and Schur functions.

I am reading the file. I have a question on pae 28. How to prove that $[X_{\{2,4\}}] = S_{(1)} = x_1 + x_2 + \cdots$ and $S_{(1)}^4 = 2 S_{(2,2)} + S_{(3,1)} + S_{(2,1,1)}$? I tried to verify $S_{(1)}^...
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Schubert calculus and number of lines satisfying some properties.

I am reading the file. I have a question on pae 18. It is said that: Given a line in $\mathbb{R}^3$, the family of lines intersecting it can be interpreted in $G(2, 4)$ as the Schubert variety $$ X_{\{...
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Representation-theoretical reasons for positivity of product of two Schubert polynomials?

In the Wikipedia article on Schubert polynomials there is a claim that there are representation-theoretical reasons for the product of two Schubert polynomials to have nonnegative coefficients when ...
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Coordinates on a Richardson variety

I'm looking for convenient coordinates to use to describe the intersection of a Schubert cell $X^\circ_\lambda$ and an opposite Schubert cell $\Omega^\circ_\mu$ in a Grassmannian $G(k,n)$. Describing ...
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About the Chern class of infinite complex Grassmannian

I learned that any characteristic class of rank-$k$ complex vector bundles on paracompact spaces is determined bijectively by a cohomology class in $H^*(Gr_k^\infty(\mathbb C))$, the cohomology ring ...
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Chern classes of tautological bundle over the Grassmannian G(2,4)

I've the following problem: I know how to calculate Chern classes of the tautological bundle over the Grassmannian $G=G(2,4)$ using the Schubert calculus. If I am right, the Chern character should ...
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Correspondence of Grassmannian cells

I have shown that the correspondence $f: X \rightarrow \mathbb{R} \oplus X$ defines an embedding of the Grassmannian $\mathrm{Gr}_{k}(\mathbb{R}^{n+k})$ into $\mathrm{Gr}_{k+1}(\mathbb{R} \oplus \...
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dimension of a subspace of a flag variety

Let $X$ be a topological space. If $X = \bigcup U_\alpha$ is an open covering of $X$ then $$\dim X = \sup_\alpha \dim U_\alpha.$$ Now suppose that $X = \coprod U_\alpha$, i.e., $X$ is the disjoint ...
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Grassmannian, Plucker coordinates

In which books can I find something about the grassmannian and the plucker coordinates ?
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Schur functors as spaces of “flag tensors”?

Consider the following construction: for a vector space $V$, define $W \subseteq \bigwedge^2 V \otimes V$ by $W = \langle\ \alpha \otimes v : v \in \text{Span}(\alpha) \ \rangle$, that is, $W$ is ...