# 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.

70 questions
Filter by
Sorted by
Tagged with
20 views

### Schubert condition $\dim(X\cap E^{i+N-1})\geq i$ for $1\leq i\leq q$ with $X\subset E^{N+q}$ gives $qN-i$ cycle?

Let $X$ be a q-dimensional linear space through $0$ in $E^{q+N}$ where $q+N$ is the dimension of ambient vector space. Let $E^0\subset E^1\subset \dots\subset E^{q+N}$ be the filtration of ambient ...
145 views

### Faulty calculation in Schubert calculus: the number of lines on the intersection of two planes.

As a easy test of the correctness of the results of Schubert calculus I tried to solve the following problem with the symbolic method. Q: How many lines are there lying on the intersection of two ...
71 views

### Generically transversally 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 $G=G(k,V)$...
40 views

15 views

### which mehod can easily design index of Grassmannian and its k and n or any function for this in Macaulay2 and how to convert poset to this index?

how to know k and n and its index of Grassmannian? which mehod can easily design index of Grassmannian and its k and n or any library or function for this in Macaulay2? is there any library or ...
61 views

90 views

### 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 ...
295 views

94 views

### 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]=...
389 views

### 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 ...
208 views

### 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 ...
53 views

### 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 ...
77 views

### 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 ...
135 views

### 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 ...
93 views

96 views

521 views

### 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 ...
320 views

### 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 $Fl_3(\mathbb C^3)$.
150 views

### 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 ...
55 views

### 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 ...
254 views

### 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 ...
### 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 ...
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)}^...