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0
votes
1answer
84 views

expressing in terms of sum of (double) schubert polynomial

It is well known that Schubert polynomials form a basis for the polynomial ring $\mathbb{Z}[x_1,x_2,x_3,...]$. I am interested in knowing how to express a particular polynomial into sum of Schubert ...
4
votes
1answer
159 views

Parametrization of Schubert varieties in isotropic Grassmannians by partitions

Let $X=\mathbb{G}_Q(l,p)$ be the isotropic Grassmannian, where $l\leq p-2$. Let $q=p-l$. Let $W^P$ be the set of minimal length representatives. Let $\tilde{\mathcal{Q}}(l,p)$ be the set of partition ...
4
votes
1answer
136 views

Is the upper boundary of a Schubert variety Cartier?

On $G/B$, the divisor $\bigcup_\alpha X_{r_\alpha}$ is Cartier (where $X_w := \overline{B_- w B}/B$, and $\alpha$ varies over simple roots), not least because $G/B$ is smooth. Is the same true for ...
2
votes
1answer
193 views

Looking for a canonical (matroid polytope) subdivision of the hypersimplex

A matroid polytope is the convex hull of the indicator vectors of the bases of a matroid, and a matroid polytope subdivision (MPS) is a polyhedral subdivision of a matroid polytope whose cells are ...
6
votes
0answers
125 views

Are Schubert varieties for Kac-Moody groups cut out by linear equations?

Let $G$ be a reductive group, and let $X$ be a partial flag variety for $G$. Then it is known that for any projective embedding of $X$, that the equations scheme-theoretically cutting out a Schubert ...
3
votes
4answers
249 views

Bruhat order and Schubert cycles

I am looking for a good (textbook) reference for the basic fact (due to Chevalley) that for every semisimple Lie group $G$ (without compact factors) with Weyl group $W$, the Bruhat order on $W$ ...
4
votes
1answer
126 views

How to make Schubert calculus into a Hopf (actually a PCH) algbera?

The parallels between the formulas in Schubert calculus and in the theory of the representations of symmetric groups (par Geissinger-Zelevinsky) are so apparent (e.g. Giambelli formula), that one must ...
3
votes
1answer
344 views

A cohomology computation request.

The short: Let $X= \{(x,y,z) \in \mathbb{C}^* \times \mathbb{C} \times \mathbb{C} \, |\, yz-x\neq 0\}$ Compute $H^*_c(X)$ (say with $\mathbb{C}$-coefficients). The long: Unless I messed something ...
3
votes
1answer
165 views

“Degree” of a Fano Scheme of a projective variety

Consider subschemes $F$ of the Grassmannian $\mathbb{G}(k,n)$ satisfying the condition that each point of $\mathbb{P}^n$ is contained in only finitely many of the $k$-planes in $F$. Does this give us ...
8
votes
1answer
378 views

Schubert varieties which admit small resolutions of singularities

I am looking for an (incomplete) list of partial flag varieties for which all Schubert cells admit small resolutions of singularities. This is interesting, for many reasons. My motivation is, that a ...
4
votes
1answer
194 views

Minimal relative Schubert modules

I am trying to better understand the definition of certain objects called minimal relative Schubert modules. My primary reference is Chapters 1 and 2 of Wilberd van der Kallen's Lectures on Frobenius ...
3
votes
0answers
394 views

Schubert varieties of flag variety , perverse sheaves

The set of Schubert varieties in a flag variety is in one-to-one correspondence with elements of the Weyl group via left cells. There is also some relation between products of Schubert varieties and ...
11
votes
1answer
454 views

Do Richardson varieties have rational singularities in arbitrary characteristic?

The title basically asks the question. I'll review the relevant terminology and explain what I have and haven't found in the literature. Let $G$ be a reductive group. Let $v \leq w$ be elements of ...
2
votes
1answer
199 views

Can we see the geometric realization of $U_q(sl_2)$'s relations as Schubert Conditions?

In Nakajima's Geometric construction of algebras(pages 3-7), he uses the subalgebra of the convolution algebra of $Gr(k^N)\times Gr(k^N)$ invariant under $GL_N$ action to construct $U_q(sl_2)$. To do ...
9
votes
2answers
656 views

Expository treatment of Schubert Cells Paper

I was wondering about the paper by Bernstein, Gel'fand, and Gel'fand on Schubert Cells. This paper is fairly old(and often cited) so I figured someone must have represented this material. In ...
7
votes
1answer
733 views

What functor does a Schubert variety represent?

I'm inspired by Yuhao's question. The functor that takes a scheme S to the set of k-dimensional vector subbundles of C^n x S (understanding "subbundle" to mean that the quotient by it is another ...
7
votes
2answers
627 views

Richardson varieties over finite fields

Let me start with some background to set the notation before I ask my question. Let G be a semisimple algebraic group over some algebraically closed field K, and suppose we have fixed a Borel ...