Bott-Samelson Variety Research Articles

TWISTED ACTIONS ON COHOMOLOGIES AND BIMODULES

AbstractFor closed subgroups L and R of a compact Lie group G, a left L-space X, and an L-equivariant continuous map $A:X\to G/R$ , we introduce the twisted action of the equivariant cohomology $H_R^{\bullet }(\mathrm {pt},\Bbbk )$ on the equivariant cohomology $H_L^{\bullet }(X,\Bbbk )$ . Considering this action as a right action, $H_L^{\bullet }(X,\Bbbk )$ becomes a bimodule together with the canonical left action of $H_L^{\bullet }(\mathrm {pt},\Bbbk )$ . Using this bimodule structure, we prove an equivariant version of the Künneth isomorphism. We apply this result to the computation of the equivariant cohomologies of Bott–Samelson varieties and to a geometric construction of the bimodule morphisms between them.

K‐theory Soergel bimodules

AbstractWe initiate the study of ‐theory Soergel bimodules, a global and ‐theoretic version of Soergel bimodules. We show that morphisms of ‐theory Soergel bimodules can be described geometrically in terms of equivariant ‐theoretic correspondences between Bott–Samelson varieties. We thereby obtain a natural categorification of ‐theory Soergel bimodules in terms of equivariant coherent sheaves. We introduce a formalism of stratified equivariant ‐motives on varieties with an affine stratification, which is a ‐theoretic analog of the equivariant derived category of Bernstein–Lunts. We show that Bruhat‐stratified torus‐equivariant ‐motives on flag varieties can be described in terms of chain complexes of ‐theory Soergel bimodules. Moreover, we propose conjectures regarding an equivariant/monodromic Koszul duality for flag varieties and the quantum ‐theoretic Satake.

Cluster algebra structures on Poisson nilpotent algebras

Various coordinate rings of varieties appearing in the theory of Poisson Lie groups and Poisson homogeneous spaces belong to the large, axiomatically defined class of symmetric Poisson nilpotent algebras, e.g. coordinate rings of Schubert cells for symmetrizable Kac–Moody groups, affine charts of Bott-Samelson varieties, coordinate rings of double Bruhat cells (in the last case after a localization). We prove that every symmetric Poisson nilpotent algebra satisfying a mild condition on certain scalars is canonically isomorphic to a cluster algebra which coincides with the corresponding upper cluster algebra, without additional localizations by frozen variables. The constructed cluster structure is compatible with the Poisson structure in the sense of Gekhtman, Shapiro and Vainshtein. All Poisson nilpotent algebras are proved to be equivariant Poisson Unique Factorization Domains. Their seeds are constructed from sequences of Poisson-prime elements for chains of Poisson UFDs; mutation matrices are effectively determined from linear systems in terms of the underlying Poisson structure. Uniqueness, existence, mutation, and other properties are established for these sequences of Poisson-prime elements.

K-theory of flag Bott manifolds

Abstract The aim of this paper is to describe the topological K-ring in terms of generators and relations of a flag Bott manifold. We apply our results to give a presentation for the topological K-ring, and hence the Grothendieck ring of algebraic vector bundles over flag Bott–Samelson varieties.

K-orbit closures and Barbasch–Evens–Magyar varieties

We define the Barbasch-Evens-Magyar varieties. We show they are isomorphic to the smooth varieties defined in [D.~Barbasch-S.~Evens '94] that map generically finitely to symmetric orbit closures, thereby giving resolutions of singularities in certain cases. Our definition parallels [P.~Magyar '98]'s construction of the Bott-Samelson varieties [H.~C.~Hansen '73, M.~Demazure '74]. From this alternative viewpoint, one deduces a graphical description in type $A$, stratification into closed subvarieties of the same kind, and determination of the torus-fixed points. Moreover, we explain how these manifolds inherit a natural symplectic structure with Hamiltonian torus action. We then express the moment polytope in terms of the moment polytope of a Bott-Samelson variety.

The Gromov Width of Bott-Samelson Varieties

We prove that the Gromov width of any Bott-Samelson variety associated to a reduced expression and equipped with a rational Kähler form equals the symplectic area of a minimal curve. From this, we derive an estimate for the Seshadri constants of ample line bundles on Bott-Samelson varieties.

Push–Pull Operators on Convex Polytopes

Abstract A classical result of Schubert calculus is an inductive description of Schubert cycles using divided difference (or push–pull) operators in Chow rings. We define convex geometric analogs of push–pull operators and describe their applications to the theory of Newton–Okounkov convex bodies. Convex geometric push–pull operators yield an inductive construction of Newton–Okounkov polytopes of Bott–Samelson varieties. In particular, we construct a Minkowski sum of Feigin–Fourier–Littelmann–Vinberg polytopes using convex geometric push–pull operators in type $A$.

On the Mori theory and Newton–Okounkov bodies of Bott–Samelson varieties

We prove that on a Bott–Samelson variety X every movable divisor is nef. This enables us to consider Zariski decompositions of effective divisors, which in turn yields a description of the Mori chamber decomposition of the effective cone. This amounts to information on all possible birational morphisms from X. Applying this result, we prove the rational polyhedrality of the global Newton–Okounkov cone of a Bott–Samelson variety with respect to the so called ‘horizontal’ flag. In fact, we prove the stronger property of the finite generation of the corresponding global value semigroup.

Minimal rational curves on generalized Bott–Samelson varieties

We investigate families of minimal rational curves on Schubert varieties, their Bott–Samelson desingularizations, and their generalizations constructed by Nicolas Perrin in the minuscule case. In particular, we describe the minimal families on small resolutions of minuscule Schubert varieties.

Algebraic and geometric properties of flag Bott–Samelson varieties and applications to representations

We introduce the notion of flag Bott-Samelson variety as a generalization of Bott-Samelson variety and flag variety. Using a birational morphism from an appropriate Bott-Samelson variety to a flag Bott-Samelson variety, we compute Newton-Okounkov bodies of flag Bott-Samelson varieties as generalized string polytopes, which are applied to give polyhedral expressions for irreducible decompositions of tensor products of $G$-modules. Furthermore, we show that flag Bott-Samelson varieties are degenerated into flag Bott manifolds with higher rank torus actions, and find the Duistermaat-Heckman measures of the moment map images of flag Bott-Samelson varieties with the torus action together with invariant closed $2$-forms.

Deformation of Bott–Samelson varieties and variations of isotropy structures

In the framework of the problem of characterizing complete flag manifolds by their contractions, the complete flags of type F4 and G2 satisfy the property that any possible tower of Bott–Samelson varieties dominating them birationally deforms in a nontrivial moduli. In this paper we illustrate the fact that, at least in some cases, these deformations can be explained in terms of automorphisms of Schubert varieties, providing variations of certain isotropy structures on them. As a corollary, we provide a unified and completely algebraic proof of the characterization of complete flag manifolds in terms of their contractions.

Bott–Samelson Varieties and Poisson Ore Extensions

Abstract We show that associated with any $n$-dimensional Bott–Samelson variety of a complex semi-simple Lie group $G$, one has $2^n$ Poisson brackets on the polynomial algebra $A={\mathbb{C}}[z_1, \ldots , z_n]$, each an iterated Poisson Ore extension and one of them a symmetric Poisson Cauchon–Goodearl–Letzter (CGL) extension in the sense of Goodearl–Yakimov. We express the Poisson brackets in terms of root strings and structure constants of the Lie algebra of $G$. It follows that the coordinate rings of all generalized Bruhat cells have presentations as symmetric Poisson CGL extensions. The paper establishes the foundation on generalized Bruhat cells and sets the stage for their applications to integrable systems, cluster algebras, total positivity, and toric degenerations of Poisson varieties, some of which are discussed in the Introduction.

Categories of Bott-Samelson Varieties

We consider all Bott-Samelson varieties BS(s) for a fixed connected semisimple complex algebraic group with maximal torus T as the class of objects of some category. The class of morphisms of this category is an extension of the class of canonical (inserting the neutral element) morphisms BS(s)↪BS(s′), where s is a subsequence of s′. Every morphism of the new category induces a map between the T-fixed points but not necessarily between the whole varieties. We construct a contravariant functor from this new category to the category of graded \(\phantom {\dot {i}\!}H^{\bullet }_{T}(\text {pt})\)-modules coinciding on the objects with the usual functor \(\phantom {\dot {i}\!}H_{T}^{\bullet }\) of taking T-equivariant cohomologies. We also discuss the problem how to define a functor to the category of T-spaces from a smaller subcategory. The exact answer is obtained for groups whose root systems have simply laced irreducible components by explicitly constructing morphisms between Bott-Samelson varieties (different from the canonical ones).

Polyhedra and parameter spaces for matroids over valuation rings

In this paper we address two of the major foundational questions in the theory of matroids over rings. First, we provide a cryptomorphic axiomatisation, by introducing an analogue of the base polytope for matroids. Second, we describe a parameter space for matroids over a valuation ring, which turns out to be a tropical version of the Bott–Samelson varieties for the full flag variety. Thus a matroid over a valuation ring is a sequence of flags of tropical linear spaces a.k.a. valuated matroids.

EFFECTIVE DIVISORS ON BOTT–SAMELSON VARIETIES

We compute the cone of effective divisors on a Bott–Samelson variety corresponding to an arbitrary sequence of simple roots. The main tool is a general result concerning effective cones of certain B-equivariant ℙ1 bundles. As an application, we compute the cone of effective codimension-two cycles on Bott–Samelson varieties corresponding to reduced words. We also obtain an auxiliary result giving criteria for a Bott–Samelson variety to contain a dense B-orbit, and we construct desingularizations of intersections of Schubert varieties. An appendix exhibits an explicit divisor showing that any Bott–Samelson variety is log Fano.

Newton–Okounkov bodies for Bott–Samelson varieties and string polytopes for generalized Demazure modules

A Newton–Okounkov body is a convex body constructed from a projective variety with a valuation on its homogeneous coordinate ring; this generalizes a Newton polytope for a toric variety. This convex body inherits information about algebraic, geometric, and combinatorial properties of the original projective variety; for instance, Kaveh showed that string polytopes in the representation theory of algebraic groups are examples of Newton–Okounkov bodies for Schubert varieties. In this paper, we extend the notion of string polytopes for Demazure modules to generalized Demazure modules, and prove that the resulting generalized string polytopes are identical to the Newton–Okounkov bodies for Bott–Samelson varieties with respect to a specific valuation. As an application of this result, we show that these are indeed polytopes.

Singular string polytopes and functorial resolutions from Newton–Okounkov bodies

The main result of this paper is that the toric degenerations of flag and Schubert varieties associated to string polytopes and certain Bott–Samelson resolutions of flag and Schubert varieties fit into a commutative diagram which gives a resolution of singularities of singular toric varieties corresponding to string polytopes. Our main tool is a result of Anderson which shows that the toric degenerations arising from Newton–Okounkov bodies are functorial in an appropriate sense. We also use results of Fujita which show that Newton–Okounkov bodies of Bott–Samelson varieties with respect to a certain valuation $\nu_{\mathrm{max}}$ coincide with generalized string polytopes, as well as previous results by the authors which explicitly describe the Newton–Okounkov bodies of Bott–Samelson varieties with respect to a different valuation $\nu_{\mathrm{min}}$ in terms of Grossberg–Karshon twisted cubes. A key step in our argument is that, under a technical condition, these Newton–Okounkov bodies coincide.

Rhombic tilings and Bott–Samelson varieties

S. Elnitsky (1997) gave an elegant bijection between rhombic tilings of 2 n 2n -gons and commutation classes of reduced words in the symmetric group on n n letters. P. Magyar (1998) found an important construction of the Bott–Samelson varieties introduced by H. C. Hansen (1973) and M. Demazure (1974). We explain a natural connection between S. Elnitsky’s and P. Magyar’s results. This suggests using tilings to encapsulate Bott–Samelson data (in type A A ). It also indicates a geometric perspective on S. Elnitsky’s bijection. We also extend this construction by assigning desingularizations of Schubert varieties to the zonotopal tilings considered by B. Tenner (2006).

Global Okounkov bodies for Bott–Samelson varieties

We use the theory of Mori dream spaces to prove that the global Okounkov body of a Bott–Samelson variety with respect to a natural flag of subvarieties is rational polyhedral. As a corollary, Okounkov bodies of effective line bundles over Schubert varieties are shown to be rational polyhedral. In particular, it follows that the global Okounkov body of a flag variety G/B is rational polyhedral. As an application we show that the asymptotic behaviour of dimensions of weight spaces in section spaces of line bundles is given by the volume of polytopes.

A comparison of Newton–Okounkov polytopes of Schubert varieties

AbstractA Newton–Okounkov body is a convex body constructed from a polarized variety with a valuation on its function field. Kaveh (respectively, the first author and Naito) proved that the Newton–Okounkov body of a Schubert variety associated with a specific valuation is identical to the Littelmann string polytope (respectively, the Nakashima–Zelevinsky polyhedral realization) of a Demazure crystal. These specific valuations are defined algebraically to be the highest term valuations with respect to certain local coordinate systems on a Bott–Samelson variety. Another class of valuations, which is geometrically natural, arises from some sequence of subvarieties of a polarized variety. In this paper, we show that the highest term valuation used by Kaveh (respectively, by the first author and Naito) and the valuation coming from a sequence of specific subvarieties of the Schubert variety are identical on a perfect basis with some positivity properties. The existence of such a perfect basis follows from a categorification of the negative part of the quantized enveloping algebra. As a corollary, we prove that the associated Newton–Okounkov bodies coincide through an explicit affine transformation.