In this paper, we consider the GIT quotients of Schubert varieties for the action of a maximal torus. We describe the minuscule Schubert varieties for which the semistable locus is contained in the smooth locus. As a consequence, we study the smoothness of torus quotients of Schubert varieties in the Grassmannian. We also prove that the torus quotient of any Schubert variety in the homogeneous space $SL(n, \mathbb C)/P$ is projectively normal with respect to the line bundle $\mathcal L_{α_0}$ and the quotient space is a projective space, where the line bundle $\mathcal L_{α_0}$ and the parabolic subgroup $P$ of $SL (n, \mathbb C)$ are associated to the highest root $α_0$.
Let $G=SL(n, \mathbb{C}),$ and $T$ be a maximal torus of $G,$ where $n$ is a positive even integer. In this article, we study the GIT quotients of the Schubert varieties in the Grassmannian $G_{2,n}.$ We prove that the GIT quotients of the Richardson varieties in the minimal dimensional Schubert variety admitting stable points in $G_{2,n}$ are projective spaces. Further, we prove that the GIT quotients of certain Richardson varieties in $G_{2,n}$ are projective toric varieties. Also, we prove that the GIT quotients of the Schubert varieties in $G_{2,n}$ have at most finite set of singular poin…
We compute the Nash blow-up of a cominuscule Schubert variety. In particular, we show that the Nash blow-up is algebraically isomorphic to another Schubert variety of the same Lie type. As a consequence, we give a new characterization of the smooth locus and, for Grassmannian Schubert varieties, determine when the Nash blow-up is a resolution of singularities. We also study the induced torus action on the Nash blow-up and give a bijection between its torus fixed points and Peterson translates on the Schubert variety.
Let $r < n$ be positive integers and further suppose $r$ and $n$ are coprime. We study the GIT quotient of Schubert varieties $X(w)$ in the Grassmannian $G_{r,n}$, admitting semistable points for the action of $T$ with respect to the $T$-linearized line bundle ${\cal L}(nω_r)$. We give necessary and sufficient combinatorial conditions for the GIT quotient $T\backslash\mkern-6mu\backslash X(w)^{ss}_{T}({\cal L}(nω_r))$ to be smooth.
In this paper, for any simple, simply connected algebraic group $G$ of type $B_n,C_n$ or $D_n$ and for any maximal parabolic subgroup $P$ of $G$, we describe all minimal dimensional Schubert varieties in $G/P$ admitting semistable points for the action of a maximal torus $T$ with respect to an ample line bundle on $G/P$. In this paper, we also describe, for any semi-simple simply connected algebraic group $G$ and for any Borel subgroup $B$ of $G$, all Coxeter elements $τ$ for which the Schubert variety $X(τ)$ admits a semistable point for the action of the torus $T$ with respect to a non-trivi…
The closure of a generic torus orbit in the flag variety $G/B$ of type $A$ is known to be a permutohedral variety and its Poincare polynomial agrees with the Eulerian polynomial. In this paper, we study the Poincare polynomial of a generic torus orbit closure in a Schubert variety in $G/B$. When the generic torus orbit closure in a Schubert variety is smooth, its Poincare polynomial is known to agree with a certain generalization of the Eulerian polynomial. We extend this result to an arbitrary generic torus orbit closure which is not necessarily smooth.
Let $G=SO(8n+4,\mathbb{C})$ ($n\ge 1$). Let $B$ be a Borel subgroup of $G$ containing a maximal torus $T$ of $G.$ Let $P (\supset B)$ denote the maximal parabolic subgroup of $G$ corresponding to the simple root $α_{4n+2}$. In this article, we prove projective normality of the GIT quotients of certain Schubert varieties in the orthogonal Grassmannian $G/P$ with respect to the descent of a suitable $T$-linearized very ample line bundle.
Let $G$ be a simple, simply connected algebraic group over the field of complex numbers. We give a necessary and a sufficient condition for a Schubert variety $X(τ)$ for which all the higher cohomologies $H^{i}(X(τ), E)$ vanish for the restriction $E$ of the tangent bundle of $G/B$ to X(τ)$. We further show that the global sections $H^{0}(X(τ), E)$ is the adjoint representation of $G$ when $G$ is simply laced.
We give a new proof for the fact that the number of torus fixed points for the degenerated flag variety is equal to the normalized median Genocchi number, using the identification with a certain Schubert variety. We further study the torus fixed points for the symplectic degenerated flag variety and develop a combinatorial model, symplectic Dellac configurations, so parametrize them. The number of these symplectic fixed points is conjectured to be the median Euler number.
It is well-known that the $T$-fixed points of a Schubert variety in the flag variety $GL_n(\mathbb{C})/B$ can be characterized purely combinatorially in terms of Bruhat order on the symmetric group $\mathfrak{S}_n$. In a recent preprint, Cho, Hong, and Lee give a combinatorial description of the $T$-fixed points of Hessenberg analogues of Schubert varieties (which we call Hessenberg Schubert varieties) in a regular semisimple Hessenberg variety. This note gives an interpretation of their result in terms of Bruhat order by making use of a partition of the symmetric group defined using so-called…
We describe the torus-equivariant cohomology ring of isotropic Grassmannians by using a localization map to the torus fixed points. We present two types of formulas for equivariant Schubert classes of these homogeneous spaces. The first formula involves combinatorial objects which we call ``excited Young diagrams'' and the second one is written in terms of factorial Schur $Q$- or $P$-functions. As an application, we give a Giambelli-type formula for the equivariant Schubert classes. We also give combinatorial and Pfaffian formulas for the multiplicity of a singular point in a Schubert variety.
The closure of a generic torus orbit in the flag variety $G/B$ of type $A_{n-1}$ is known to be a permutohedral variety and well studied. In this paper we introduce the notion of a generic torus orbit in the Schubert variety $X_w$ $(w\in \mathfrak{S}_n)$ and study its closure $Y_w$. We identify the maximal cone in the fan of $Y_w$ corresponding to a fixed point $uB$ $(u\le w)$, associate a graph $Γ_w(u)$ to each $u\le w$, and show that $Y_w$ is smooth at $uB$ if and only if $Γ_w(u)$ is a forest. We also introduce a polynomial $A_w(t)$ for each $w$, which agrees with the Eulerian polynomial whe…
This paper aims to focus on Richardson varieties on symplectic groups, especially their combinatorial characterization and defining equations. Schubert varieties and opposite Schubert varieties have profound significance in the study of generalized flag varieties which are not only research objects in algebraic geometry but also ones in representation theory. A more general research object is Richardson variety, which is obtained by the intersection of a Schubert variety and an opposite Schubert variety. The structure of Richardson variety on Grassmannian and its combinatorial characterization…
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.
We develop a method of finding a Cox ring of a crepant resolution of a quotient singularity with a torus action and apply it to examples of symplectic quotient singularities in dimension 4. In addition we obtain a bound on the degrees of homogeneous generators of the Cox ring in a more general setup, based on the multigraded Castelnuovo-Mumford regularity and Kawamata-Viehweg vanishing.
We introduce the notion of a cominuscule point in a Schubert variety in a generalized flag variety for a semisimple group. We derive formulas expressing the Hilbert series and multiplicity of a Schubert variety at a cominuscule point in terms of the restrictions of classes in torus-equivariant K-theory and cohomology to that point, generalizing previously known formulas for flag varieties of cominuscule type. Thus, we can calculate Hilbert series and multiplicities in cases where these were previously unknown. The formulas for Schubert varieties are special cases of more general formulas valid…
We consider torus actions on Mori dream spaces and ask whether the associated Chow quotient is again a Mori dream space and, if so, what does its Cox ring look like. We provide general tools for the study of these problems and give solutions for k*-actions on smooth quadrics.
This article is motivated by the following local-to-global question: is every variety with tame quotient singularities globally the quotient of a smooth variety by a finite group? We show that this question has a positive answer for all quasi-projective varieties which are expressible as a quotient of a smooth variety by a split torus (e.g. simplicial toric varieties). Although simplicial toric varieties are rarely toric quotients of smooth varieties by finite groups, we give an explicit procedure for constructing the quotient structure using toric techniques. This result follow from a charact…
Let $G$ be a compact and $1$--connected Lie group with a maximal torus $T$. Based on Schubert calculus on the flag manifold $G/T$ [15] we construct the integral cohomology ring $H^{\ast}(G)$ uniformly for all $G$.
We study a variation of Mumford's quotient for the action of a maximal torus $T$ on a flag variety $G/B$ depending on projective embedding $G/B \hookrightarrow \Bbb P(V(χ))$, where $T$-linearization is induced by the standard $G$-linearization. We describe the linear spans of the supports of semistable orbits, that allow us to calculate the rank of the Picard group of quotient $(G/B)^{ss}//T$, when $G$ does not contain simple factors of type $A_n$.
For any simple, simply connected algebraic group $G$ of type $B,C$ and $D$ and for any maximal parabolic subgroup $P$ of $G$, we provide a criterion for a Richardson variety in $G/P$ to admit semistable points for the action of a maximal torus $T$ with respect to an ample line bundle on $G/P$.
We study the GIT quotient of the minimal Schubert variety in the Grassmannian admitting semistable points for the action of maximal torus $T$, with respect to the $T$-linearized line bundle ${\cal L}(n ω_r)$ and show that this is smooth when $gcd(r,n)=1$. When $n=7$ and $r=3$ we study the GIT quotients of all Richardson varieties in the minimal Schubert variety. This builds on previous work by Kumar \cite{kumar2008descent}, Kannan and Sardar \cite{kannan2009torusA}, Kannan and Pattanayak \cite{kannan2009torusB}, and recent work of Kannan et al \cite{kannan2018torus}. It is known that the GIT q…
Let $G=Spin(8n, \mathbb{C})(n\ge 1)$ and $T_{G}$ be a maximal torus of $G.$ Let $P^{α_{4n}}(\supset T_{G})$ be the maximal parabolic subgroup of $G$ corresponding to the simple root $α_{4n}.$ Let $X$ be a Schubert variety in $G/P^{α_{4n}}$ admitting semi-stable point with respect to the $T$-linearized very ample line bundle $\mathcal{L}(2ω_{4n}).$ Let $R=\bigoplus_{k \in \mathbb{Z}_{\geq 0}}R_k,$ where $R_k=H^{0}(X, \mathcal{L}^{\otimes k}(2ω_{4n}))^{T_{G}}.$ In this article, we prove that for $n=1$ and $X=G/P^{α_4},$ the graded $\mathbb{C}$-algebra $R$ is generated by $R_1.$ As a consequence,…
Let $G/P$ be a complex cominuscule flag manifold. We prove a type independent formula for the torus equivariant Mather class of a Schubert variety in $G/P$, and for a Schubert variety pulled back via the natural projection $G/Q \to G/P$. We apply this to find formulae for the local Euler obstructions of Schubert varieties, and for the torus equivariant localizations of the conormal spaces of these Schubert varieties. We conjecture positivity properties for the local Euler obstructions and for the Schubert expansion of Mather classes. We check the conjectures in many cases, by utilizing results…
Tangent spaces to Schubert varieties of type A were characterized by Lakshmibai and Seshadri. This result was extended to the other classical types by Lakshmibai. We give a uniform characterization of tangent spaces to Schubert varieties in cominuscule G/P. Our results extend beyond cominuscule G/P; they describe the tangent space to any Schubert variety in G/B at a point xB, where x is a cominuscule Weyl group element in the sense of Peterson. Our results also give partial information about the tangent space to any Schubert variety at any point. Our method is to describe the tangent spaces of…
The aim of this paper is to study the representation theory of quantum Schubert cells. Let $\g$ be a simple complex Lie algebra. To each element $w$ of the Weyl group $W$ of $\g$, De Concini, Kac and Procesi have attached a subalgebra $U_q[w]$ of the quantised enveloping algebra $U_q(\g)$. Recently, Yakimov showed that these algebras can be interpreted as the quantum Schubert cells on quantum flag manifolds. In this paper, we study the primitive ideals of $U_q[w]$. More precisely, it follows from the Stratification Theorem of Goodearl and Letzter that the primitive spectrum of $U_q[w]$ admits …
In this paper, we will look at actions on complex flag varieties $G/P$ of the torus $\hat{T}=\bigcap\limits_{α\inΔ\setminusΔ_P}\ker(α)$, and under reasonable assumptions, we will give a description of the set $X^{us}$ of unstable points for $\hat{T}$-linearized invertible sheaves. We will investigate the case where $P$ is a maximal parabolic subgroup, and show that $X^{us}$ can be written as a disjoint union of a Schubert variety and an opposite Schubert variety, and we deduce the vanishing of cohomology groups $H^i(Y,{\cal M})$ for invertible sheaves ${\cal M}$ on the quotient variety $Y$ for…
Let $B$ be a Borel subgroup of $\mathrm{GL}_n(\mathbb{C})$ and $\mathbb{T}$ a maximal torus contained in $B$. Then $\mathbb{T}$ acts on $\mathrm{GL}_{n}(\mathbb{C})/B$ and every Schubert variety is $\mathbb{T}$-invariant. We say that a Schubert variety is of complexity $k$ if a maximal $\mathbb{T}$-orbit in $X_w$ has codimension $k$. In this paper, we discuss topology, geometry, and combinatorics related to Schubert varieties of complexity one.
Let X be an affine T-variety. We study two different quotients for the action of T on X: the toric Chow quotient X/_CT and the toric Hilbert scheme H. We introduce a notion of the main component H_0 of H which parameterizes general T-orbit closures in X and their flat limits. The main component U_0 of the universal family U over H is a preimage of H_0. We define an analogue of a universal family W_X over the main component of the X/_CT. We show that the toric Chow morphism restricted on the main components lifts to a birational projective morphism from U_0 to W_X. The variety W_X also provides…
Let L be a homogeneous ample line bundle on any flag variety G/P and let T be a maximal torus of G. We prove a general necessary and sufficient condition for L to descend as a line bundle on the GIT quotient of G/P by T. We use this result to explicitly determine exactly which L descend to the GIT quotient for any simple complex algebraic group G and any parabolic subgroup P.
We compute the initial ideals, with respect to certain conveniently chosen term orders, of ideals of tangent cones at torus fixed points to Schubert varieties in orthogonal Grassmannians. The initial ideals turn out to be square-free monomial ideals and therefore Stanley-Reisner face rings of simplicial complexes. We describe these complexes. The maximal faces of these complexes encode certain sets of non-intersecting lattice paths.
We investigate the class of Kazhdan-Lusztig varieties, and its subclass of matrix Schubert varieties, endowed with a naturally defined torus action. Writing a matrix Schubert variety $\overline{X_w}$ as $\overline{X_w}=Y_w\times \mathbb{C}^d$ (where $d$ is maximal possible), we show that $Y_w$ can be of complexity-$k$ exactly when $k\neq 1$. Also, we give a combinatorial description of the extremal rays of the weight cone of a Kazhdan-Lusztig variety, which in particular turns out to be the edge cone of an acyclic directed graph. As a consequence we show that given permutations $v$ and $w$, th…
Let $G$ be a complex quasi-simple algebraic group and $G/P$ be a partial flag variety. The projections of Richardson varieties from the full flag variety form a stratification of $G/P$. We show that the closure partial order of projected Richardson varieties agrees with that of a subset of Schubert varieties in the affine flag variety of $G$. Furthermore, we compare the torus-equivariant cohomology and $K$-theory classes of these two stratifications by pushing or pulling these classes to the affine Grassmannian. Our work generalizes results of Knutson, Lam, and Speyer for the Grassmannian of t…
An embedded variety is said to be well-poised when the associated initial ideal degenerations coming from points of the tropical variety are reduced and irreducible. Varieties with a well-poised embedding admit a large collection of explicitly constructible Newton-Okounkov bodies. This paper aims to study the well-poised property under torus quotients. Our first result states that GIT quotients of normal well-poised varieties by quasi-tori also have well-poised embeddings. As an application, we show that several Hassett spaces, $\overline{M}_{0,β}$, are well-poised under Alexeev's embedding. C…
We give a stratification of the GIT quotient of the Grassmannian $G_{2,n}$ modulo the normaliser of a maximal torus of $SL_{n}(k)$ with respect to the ample generator of the Picard group of $G_{2,n}$. We also prove that the flag variety $GL_{n}(k)/B_{n}$ can be obtained as a GIT quotient of $GL_{n+1}(k)/B_{n+1}$ modulo a maximal torus of $SL_{n+1}(k)$ for a suitable choice of an ample line bundle on $GL_{n+1}(k)/B_{n+1}$.
Let $G$ be a complex reductive group, $T$ be a maximal torus of $G$, $B$ be a Borel subgroup of $G$ containing $T$, $W$ be the Weyl group of $G$ with respect to $T$. To each element $w$ of $W$ one can associate the Schubert subvariety $X_w$ of the flag variety $G/B$, the tangent cone to $X_w$ at the identity point $p$ considered as a subcheme of the tangent space $T_p(G/B)$, and the reduced tangent cone to $X_w$ at $p$ considered as a subvariety of $T_p(G/B)$. Let $w_1$, $w_2$ be distinct involutions in $W$. We prove that if $G$ is of type $B_n$ or $C_n$, then the tangent cones corresponding t…
For a semisimple adjoint algebraic group $G$ and a Borel subgroup $B$, consider the double classes $BwB$ in $G$ and their closures in the canonical compactification of $G$: we call these closures large Schubert varieties. We show that these varieties are normal and Cohen-Macaulay; we describe their Picard group and the spaces of sections of their line bundles. As an application, we construct geometrically van der Kallen's filtration of the algebra of regular functions on $B$. We also construct a degeneration of the flag variety $G/B$ embedded diagonally in $G/B\times G/B$, into a union of Schu…
Let $G$ be a simple algebraic group of adjoint type over the field $\mathbb{C}$ of complex numbers, $B$ be a Borel subgroup of $G$ containing a maximal torus $T$ of $G.$ In this article, we show that $α$ is a co-minuscule root if and only if for any parabolic subgroup $Q$ containing $B$ properly, there is no Schubert variety $X_{Q}(w)$ in $G/Q$ such that the minimal parabolic subgroup $P_α$ of $G$ is the connected component, containing the identity automorphism of the group of all algebraic automorphisms of $X_{Q}(w).$
We construct all possible Hamiltonian torus actions for which all the non-empty reduced spaces are two dimensional (and not single points) and the manifold is connected and compact, or, more generally, the moment map is proper as a map to a convex set. This construction completes the classification of tall complexity one spaces.
Let $G$ be a simple simply-laced algebraic group of adjoint type over the field $\mathbb{C}$ of complex numbers, $B$ be a Borel subgroup of $G$ containing a maximal torus $T$ of $G.$ In this article, we show that $ω_α$ is a minuscule fundamental weight if and only if for any parabolic subgroup $Q$ containing $B$ properly, there is no Schubert variety $X_{Q}(w)$ in $G/Q$ such that the minimal parabolic subgroup $P_α$ of $G$ is the connected component, containing the identity automorphism of the group of all algebraic automorphisms of $X_{Q}(w).$
Consider an algebraic action of a connected complex reductive algebraic group on a complex polarized projective variety. In this paper, we first introduce the nilpotent quotient, the quotient of the polarized projective variety by a maximal unipotent subgroup. Then, we introduce and investigate three induced actions: one by the reductive group, one by a Borel subgroup, and one by a maximal torus, respectively. Our main result is that there are natural correspondences among quotients of these three actions. In the end, we mention a possible application to the moduli spaces of parabolic bundles …
Let $G$ be a complex reductive algebraic group, $T$ a maximal torus in $G$, $B$ a Borel subgroup of $G$ containing $T$, $W$ the Weyl group of $G$ with respect to $T$. Let $w$ be an element of $W$. Denote by $X_w$ the Schubert subvariety of the flag variety $G/B$ corresponding to $w$. Let $C_w$ be the tangent cone to $X_w$ at the point $p=eB$ (we consider $C_w$ as a subscheme of the tangent space to $G/B$ at $p$). In 2011, D.Yu. Eliseev and A.N. Panov computed all tangent cones for $G=SL(n)$, $n<6$. Using their computations, A.N. Panov formulated the following Conjecture: if $w$, $w'$ are disti…
For a complex toric variety $X$ the logarithmic absolute value induces a natural retraction of $X$ onto the set of its non-negative points and this retraction can be identified with a quotient of $X(\mathbb{C})$ by its big real torus. We prove an analogous result in the non-Archimedean world: The Kajiwara-Payne tropicalization map is a non-Archimedean analytic stack quotient of $X^{an}$ by its big affinoid torus. Along the way, we provide foundations for a geometric theory of non-Archimedean analytic stacks, particularly focussing on analytic groupoids and their quotients, the process of analy…
Schubert varieties in finite dimensional flag manifolds G/P are a well-studied family of projective varieties indexed by elements of the corresponding Weyl group W. In particular, there are many tests for smoothness and rational smoothness of these varieties. One key result due to Lakshmibai-Sandhya is that in type A the smooth Schubert varieties are precisely those that are indexed by permutations that avoid the patterns 4231 and 3412. Recently, there has been a flurry of research related to the infinite dimensional analogs of flag manifolds corresponding with G being a Kac-Moody group and W …
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 semi-simple algebraic group over the complex numbers, B a Borel subgroup of G, T a maximal torus in B and P a parabolic in G containing B. This paper deals with singularities of T-stable subvarieties of G/P. It turns out that under the restriction that G doesn't contain any G_2-factors, the key geometric invariant determining the singular T-fixed points of X is the linear span of the reduced tangent cone to X at a T-fixed point x provided the singularity is isolated. The goal of this paper is to describe this invariant at the maximal singular T-fixed points when X is a Schubert vari…
