Olivier Dudas
We show that for an algebraic reductive group $G$, the partition of a double Schubert cell in the flag variety $G/B$ defined by Deodhar, and coming from a Bialynicki-Birula decomposition, is not a stratification in general. We give a counterexample for a group of type B$_n$, where the closure of some specific cell of dimension $2n$ has a non-trivial intersection with a cell of dimension $3n-3$
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A footnote to a paper of Deodhar
本文证明了Schubert簇的Bott-Samelson分解定理的有效版本，提出了一种无需极小集即可从Deodhar多项式中提取Kazhdan-Lusztig多项式的递归方法，并展示了如何在Hecke代数中定义新基及计算转移矩阵。
We describe a stratification on the double flag variety $G/B^+\times G/B^-$ of a complex semisimple algebraic group $G$ analogous to the Deodhar stratification on the flag variety $G/B^+$, which is a refinement of the stratification into orbits both for $B^+\times B^-$ and for the diagonal action of $G$, just as Deodhar's stratification refines the orbits of $B^+$ and $B^-$. We give a coordinate system on each stratum, and show that all strata are coisotropic subvarieties. Also, we discuss possible connections to the positive and cluster geometry of $G/B^+\times G/B^-$, which would generalize …
Richardson varieties, projected Richardson varieties and positroid varieties
本文综述了Richardson簇与positroid簇的组合学、坐标环、参数化及分解等已知成果，并提出了开Richardson簇坐标环是UFD等原创结果。
Using Deodhar's decomposition of a double Schubert cell, we study the regular representations of finite groups of Lie type arising in the cohomology of Deligne-Lusztig varieties associated to tori. We deduce that the Deligne-Lusztig restriction of a Gelfand-Graev module is a shifted Gelfand-Graev module.
In this work we extend some previously known results on the automorphism group of Schubert varieties. We consider the Schubert conditions which define a Schubert variety. An automorphism of the Grassmannian fixes a Schubert variety contained in it if and only if it fixes certain elements of the flag used to define the Schubert conditions.
Given two Schubert classes $σ_λ$ and $σ_μ$ in the quantum cohomology of a Grassmannian, we construct a partition $ν$, depending on $λ$ and $μ$, such that $σ_ν$ appears with coefficient 1 in the lowest (or highest) degree part of the quantum product $σ_λ\starσ_μ$. To do this, we show that for any two partitions $λ$ and $μ$, contained in a $k$-by-$(n-k)$ rectangle and such that the 180-degree rotation of one does not overlap the other, there is a third partition $ν$, also contained in the rectangle, such that the Littlewood-Richardson number $c_{λμ}^ν$ is 1.
In this note, we discuss the flexibility of Schubert classes in homogeneous varieties. We give several constructions for representing multiples of a Schubert class by irreducible subvarieties. We sharpen [R, Theorem 3.1] by proving that every positive multiple of an obstructed class in a cominuscule homogeneous variety can be represented by an irreducible subvariety.
Present notes can be viewed as an attempt to extend the notion of Schubert/Grothendieck polynomial to the context of an arbitrary algebraic oriented cohomology theory and, hence, of a commutative one-dimensional formal group law.
Elliptic Schubert Classes and the Poincare Duality
本文通过 Kostant-Kumar 方法，证明了 Schubert 簇的椭圆类满足庞加莱对偶性。
This note constructs the flat toric degeneration of the manifold FL_n of flags in C^n from [Gonciulea-Lakshmibai 96] as an explicit GIT quotient of the Gr"obner degeneration in [Knutson-Miller 03]. This implies that Schubert varieties degenerate to reduced unions of toric varieties, associated to faces indexed by rc-graphs (reduced pipe dreams) in the Gel'fand-Cetlin polytope. Our explicit description of the toric degeneration of FL_n provides a simple explanation of how Gel'fand-Cetlin decompositions for irreducible polynomial representations of GL_n arise via geometric quantization.
The purpose of this note is to give a refinement of the product formula proved in [1] for the Poincare polynomial of a smooth Schubert variety in the flag variety of an algebraic group G over C. This yields a factorization of the number of elements in a Bruhat interval [e,w] in the Weyl group W of G provided the Schubert variety associated to w is smooth. This gives an elementary necessary condition for a Schubert variety in the flag variety to be smooth.
A formula for the structure constants of the multiplication of Schubert classes is obtained in arXiv:1909.05283. In this note, we prove analogous formulae for the Chern--Schwartz--MacPherson (CSM) classes and Segre--Schwartz--MacPherson (SSM) classes of Schubert cells in the flag variety. By the equivalence between the CSM classes and the stable basis elements for the cotangent bundle of the flag variety, a formula for the structure constants for the latter is also deduced.
These notes are the written version of my lectures at the Banach Center mini-school "Schubert Varieties" in Warsaw, May 18-22, 2003. Their aim is to give a self-contained exposition of some geometric aspects of Schubert calculus.
The polynomial ring $B$ in infinitely many indeterminates $(x_1,x_2,\ldots)$, with rational coefficients, has a vector space basis of Schur polynomials, parametrized by partitions. The goal of this note is to provide an explanation of the following fact. If $\blamb$ is a partition of weight $d$, then the partial derivative of order $d$ with respect to $x_1$ of the Schur polynomial $S_\blamb(\bfx)$ coincides with the Plücker degree of the Schubert variety of dimension $d$ associated to $\blamb$, equal to the number of standard Young tableaux of shape $\blamb$. The generating function encoding a…
Nil-Hecke rings and the Schubert calculus
本文系统阐述Nil-Hecke代数技术在旗簇等变Schubert演算局部化方法中的应用，并将其推广至非晶体根系与旗簇（连通）K理论研究。
These are extended notes of a talk given at Maurice Auslander Distinguished Lectures and International Conference (Woods Hole, MA) in April 2013. Their aim is to give an introduction into Schubert calculus on Grassmannians and flag varieties. We discuss various aspects of Schubert calculus, such as applications to enumerative geometry, structure of the cohomology rings of Grassmannians and flag varieties, Schur and Schubert polynomials. We conclude with a survey of results of V.Kiritchenko, V.Timorin and the author on a new approach to Schubert calculus on full flag varieties via combinatorics…
Let $Q$ be a quiver, $M$ a representation of $Q$ with an ordered basis $\cB$ and $\ue$ a dimension vector for $Q$. In this note we extend the methods of \cite{L12} to establish Schubert decompositions of quiver Grassmannians $\Gr_\ue(M)$ into affine spaces to the ramified case, i.e.\ the canonical morphism $F:T\to Q$ from the coefficient quiver $T$ of $M$ w.r.t.\ $\cB$ is not necessarily unramified. In particular, we determine the Euler characteristic of $\Gr_\ue(M)$ as the number of \emph{extremal successor closed subsets of $T_0$}, which extends the results of Cerulli Irelli (\cite{Cerulli11…
Let G be a complex semi-simple linear algebraic group without G_2 factors, B a Borel subgroup of G and T a maximal torus in B. The flag variety G/B is a projective G-homogeneous variety whose tangent space at the identity coset is isomorphic, as a B-module, to Lie(G)/Lie(B). Recall that if w is an element of the Weyl group W of the pair (G,T), the Schubert variety X(w) in G/B is by definition the closure of the Bruhat cell BwB. In this note we prove that X(w) is non-singular iff the following two conditions hold: 1) its Poincaré polynomial is palindromic and 2) the tangent space TE(X(w)) to th…
Let G be a semisimple algebraic group over an algebraically closed field of positive characteristic. In this note, we show that an irreducible closed subvariety of the flag variety of G is compatibly split by the unique canonical Frobenius splitting if and only if it is a Richardson variety, i.e. an intersection of a Schubert and an opposite Schubert variety.
In this short note we prove a lemma about the dimension of certain algebraic sets of matrices. This result is needed in our paper arXiv:1201.1672. The result presented here has also applications in other situations and so it should appear as part of a larger work. The statement of the lemma goes as follows: Suppose $X$ is a nonempty algebraically closed subset of the affine space of $n \times m$ complex matrices. Suppose that $X$ is column-invariant (i.e., belongingness to $X$ depends only on the column space of the matrix). Suppose $E$ is a vector subspace of $\mathbb{C}^n$ that is not contai…
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