Questions tagged [schubert-varieties]

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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 ...
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10 votes
1 answer
1k 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 ...
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9 votes
2 answers
697 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 ...
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2 answers
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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 ...
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8 votes
1 answer
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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 ...
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7 votes
0 answers
164 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 ...
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5 votes
4 answers
490 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 $W$ ...
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5 votes
1 answer
264 views

Intersection cycle in a product of Grassmannians

Let $G(k,n)$ denote the Grassmiannian of $k$-planes in $\mathbb C^n$. Let's define $$ I_j =\{ (\Lambda_1,\Lambda_2 ) \in G(k,n) \times G(l,n) \, | \, \dim(\Lambda_1 \cap \Lambda_2) \geq j \}. $$ These ...
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  • 504
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1 answer
230 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 ...
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5 votes
1 answer
145 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 ...
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1 answer
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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 ...
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4 votes
1 answer
181 views

May Schubert cell intersection with opposite big cell polynomial count?

Let $SL(n)$ be algebraic group defined over finite field $\mathbb{F}_{p^n}$, $B$ be Borel subgroup consist of upper triangular matrices and $T$ be maximal torus consist of diagonal matrices. Let $W$ ...
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4 votes
1 answer
257 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 ...
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3 votes
1 answer
435 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 ...
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3 votes
1 answer
370 views

Schubert varieties and Young diagrams

In his book Young Tableaux, Fulton asserts, in Exercise 9.4.18 on p. 152, that the Schubert variety $\Omega_{\lambda}$ is defined by the conditions $\text{dim}(V \cap F_{n+i- \lambda_{i}}) \geq i$ for ...
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  • 319
3 votes
1 answer
208 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 ...
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3 votes
1 answer
330 views

Properties of a general element of the intersection of two Schubert cycles

We have the following lemma: Lemma Let $\Sigma_a(\mathcal{V}),\Sigma_b(\mathcal{W})$ be two Schubert cycles defined relative to transverse flags $\mathcal{V}$ and $\mathcal{W}$. If $\Lambda \in \...
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3 votes
0 answers
92 views

Richardson variety over arbitrary field

Let $G$ be a split semisimple algebraic group. Let $B$ and $T$ be borel subgroup ,maximal torus respectively. In this case the Weyl group $W(B,T)$ is a constant finite group scheme. Let $P$ be a ...
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3 votes
0 answers
145 views

Trivial morphism between local cohomology groups

I have two questions concerning morphism between local cohomology groups which I think are related. Let $G$ be a reductive group with Weyl group $W$ and $B \subset G$ a Borel. Let $X=G/B$ be the flag ...
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0 answers
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Kazhdan-Lusztig polynomials and the defect of a Bruhat interval

Let $(W,S)$ be a Coxeter system with length function $\ell$ and $T=\bigcup_{w\in W}wSw^{-1}$. Set $N(u,v):=\{t\in T: u< tu \le v\}$, $\overline{\ell}(u,v):=|N(u,v)|$, $\ell(u,v):=\ell(v)-\...
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3 votes
0 answers
117 views

Non-generic intersections of Schubert varieties?

Let $G$ be a linear algebraic group, $B$ a Borel subgroup, $P$ a parabolic subgroup containing $B$, and $W$ the Weyl group. For $w \in W$, the Schubert variety $X_w^P$ is the closure of the Schubert ...
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3 votes
0 answers
487 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 ...
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2 votes
1 answer
173 views

On a criterion for rational-smoothness of Schubert varieties and an ambiguity of the taking the ambient Algebraic group to be simply connected or not

In the paper: Pattern Avoidance and Rational Smoothness of Schubert Varieties, Sara C. Billey, Advances in Mathematics 139, 141-156(1998), https://www.sciencedirect.com/science/article/pii/...
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2 votes
1 answer
133 views

Can Schubert cells be defined, set theoretically, by less equations then the standard ones?

Let $V = \mathbb{C}^n$ with basis $e_1,\dots,e_n$, and $U = \langle e_1,\dots,e_k\rangle$. Let $$\Sigma(U)=\{\sigma\in Gr(V,2)\mid \sigma\in U \}$$ be the Schubert cell of $2$-planes contained in $U$....
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2 votes
1 answer
347 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 ...
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2 votes
1 answer
252 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 ...
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  • 4,726
1 vote
0 answers
100 views

Bruhat decomposition and standard Frobenius

Let $G$ be a linear algebraic group define over $\overline{\mathbb{F}_p}$, consider it as a subgroup of $\operatorname{GL}(n)$. Let $F_p$ be the standard Frobenius. Let $B$ and $Q$ be an $F_p$-stable ...
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1 vote
0 answers
50 views

Is the complement of an $\epsilon$-neighborhood an affinoid open polydisc in a flag variety?

$\newcommand{\rig}{\mathrm{rig}}$Let $K$ be non-Archimedean local field of characteristic zero with non-Archimedean norm $|\,\,|$ (assumed to be normalised), ring of integer $\mathcal{O}_K$ and $C$, ...
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  • 921
1 vote
0 answers
83 views

Spaces intersecting a plane non-trivially in $G(3,6)$

I want to understand the Schubert variety $\Sigma\subseteq G(3,6)$ representing 3-dim subspaces intersecting a given 2-dim subspace non-trivially. Is it smooth? How to describe $det(T_{\Sigma})$?
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1 vote
1 answer
416 views

On some notations and notions of a paper on smoothness of Schubert varieties by Lakshmibai and Sandhya

I am reading the paper Criterion for smoothness of Schubert varieties in $\mathrm{Sl}(n)/B$ by V Lakshmibai and B Sandhya; Proc. Indian Acad. Sci. (Math. Sci.), Vol. 100, No. 1, April 1990, pp. 45-52. ...
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1 vote
1 answer
184 views

Coefficients of the monomials appearing in a Schubert polynomial

It is known that the coefficients of the monomials appearing in a Schubert polynomial are always positive. My question is: Is it always true that at least one such coefficient must be $1$? If that is ...
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0 votes
1 answer
177 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 ...
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  • 849
0 votes
1 answer
84 views

In Type $A$, if the Bruhat graph of an element $w$ in the Weyl group is regular, then to show that $l(w)=$ # $ \{\alpha \in R^+| s_{\alpha} \le w\}$

I am trying to prove that for type $A$ , rational smoothness of Schubert varieties implies smoothness. So suppose we are in Type $A_{n-1}$, so let $G=Sl(n,\mathbb C)$, $B=$ the group of upper ...
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