Bott-Samelson Variety Research Articles

Chern class of Schubert cells in the flag manifold and related algebras

We discuss a relationship between Chern–Schwartz–MacPherson classes for Schubert cells in flag manifolds, the Fomin–Kirillov algebra, and the generalized nil-Hecke algebra. We show that the nonnegativity conjecture in the Fomin–Kirillov algebra implies the nonnegativity of the Chern–Schwartz–MacPherson classes for Schubert cells in flag manifolds for type A. Motivated by this connection, we also prove that the (equivariant) Chern–Schwartz–MacPherson classes for Schubert cells in flag manifolds are certain summations of the structure constants of the equivariant cohomology of Bott–Samelson varieties. We also discuss refined positivity conjectures of the Chern–Schwartz–MacPherson classes for Schubert cells motivated by the nonnegativity conjecture in the Fomin–Kirillov algebra.

The symplectic plactic monoid, crystals, and MV cycles

We study cells in generalised Bott-Samelson varieties for type C. These cells are parametrised by certain galleries in the affine building. We define a set of readable galleries - we show that the closure in the affine Grassmannian associated to a gallery in this set is an MV cycle. This then defines a map from the set of readable galeries to the set of MV cycles, which we show to be a morphism of crystals. We further compute the fibres of this map in terms of the Littelmann path model.

Newton-Okounkov bodies of Bott-Samelson varieties and Grossberg-Karshon twisted cubes

We describe, under certain conditions, the Newton-Okounkov body of a Bott-Samelson variety as a lattice polytope defined by an explicit list of inequalities. The valuation that we use to define the Newton-Okounkov body is different from that used previously in the literature. The polytope that arises is a special case of the Grossberg-Karshon twisted cubes studied by Grossberg and Karshon in connection to character formulae for irreducible $G$-representations and also studied previously by the authors in relation to certain toric varieties associated to Bott-Samelson varieties.

Brick Manifolds and Toric Varieties of Brick Polytopes

Bott-Samelson varieties are a twisted product of $\mathbb{C}\mathbb{P}^1$'s with a map into $G/B$. These varieties are mostly studied in the case in which the map into $G/B$ is birational to the image; however in this paper we study a fiber of this map when it is not birational. We prove that in some cases the general fiber, which we christen a brick manifold, is a toric variety. In order to do so we use the moment map of a Bott-Samelson variety to translate this problem into one in terms of the "subword complexes" of Knutson and Miller. Pilaud and Stump realized certain subword complexes as the dual of the boundary of a polytope which generalizes the brick polytope defined by Pilaud and Santos. For a nice family of words, the brick polytope is the generalized associahedron realized by Hohlweg, Lange and Thomas. These stories connect in a nice way: we show that the moment polytope of the brick manifold is the brick polytope. In particular, we give a nice description of the toric variety of the associahedron. We give each brick manifold a stratification dual to the subword complex. In addition, we relate brick manifolds to Brion's resolutions of Richardon varieties.

Crystal bases and Newton–Okounkov bodies

Let G be a connected reductive algebraic group. We prove that the string parametrization of a crystal basis for a finite dimensional irreducible representation of G extends to a natural valuation on the field of rational functions on the flag variety G/B, which is a highest term valuation corresponding to a coordinate system on a Bott-Samelson variety. This shows that the string polytopes associated to irreducible representations, can be realized as Newton-Okounkov bodies for the flag variety. This is closely related to an earlier result of A. Okounkov for the Gelfand-Cetlin polytopes of the symplectic group. As a corollary we recover a multiplicativity property of the canonical basis due to P. Caldero. We generalize the results to spherical varieties. From these the existence of SAGBI bases for the homogeneous coordinate rings of flag and spherical varieties, as well as their toric degenerations follow, recovering previous results of Caldero, Alexeev-Brion and the author.

Littlewood–Richardson coefficients for reflection groups

In this paper we explicitly compute all Littlewood–Richardson coefficients for semisimple and Kac–Moody groups G, that is, the structure constants (also known as the Schubert structure constants) of the cohomology algebra H⁎(G/P,C), where P is a parabolic subgroup of G. These coefficients are of importance in enumerative geometry, algebraic combinatorics and representation theory. Our formula for the Littlewood–Richardson coefficients is purely combinatorial and is given in terms of the Cartan matrix and the Weyl group of G. However, if some off-diagonal entries of the Cartan matrix are 0 or −1, the formula may contain negative summands. On the other hand, if the Cartan matrix satisfies aijaji≥4 for all i, j, then each summand in our formula is nonnegative that implies nonnegativity of all Littlewood–Richardson coefficients. We extend this and other results to the structure coefficients of the T-equivariant cohomology of flag varieties G/P and Bott–Samelson varieties Γi(G).

Standard Monomial Theory of RR Varieties

We construct the RR varieties as the fiber products of Bott-Samelson varieties over Richardson varieties. We study their homogeneous coordinate rings and standard monomial theory.

Cell Decompositions of Double Bott–Samelson Varieties

Let $G$ be a connected complex semisimple Lie group. Webster and Yakimov have constructed partitions of the double flag variety $G/B \times G/B_-$ , where $(B, B_-)$ is a pair of opposite Borel subgroups of $G$ , generalizing the Deodhar decompositions of $G/B$ . We show that these partitions can be better understood by constructing cell decompositions of a product of two Bott–Samelson varieties $Z_{ {\bf u}, {\bf v} }$ , where $ {\bf u}$ and $ {\bf v}$ are sequences of simple reflections. We construct coordinates on each cell of the decompositions and in the case of a positive subexpression, we relate these coordinates to regular functions on a particular open subset of $Z_{ {\bf u}, {\bf v} }$ . Our motivation for constructing cell decompositions of $Z_{ {\bf u}, {\bf v} }$ was to study a certain natural Poisson structure on $Z_{ {\bf u}, {\bf v} }$ .

ROW CONVEX TABLEAUX AND BOTT–SAMELSON VARIETIES

AbstractBy using row convex tableaux, we study the section rings of Bott–Samelson varieties of type A. We obtain flat deformations and standard monomial type bases of the section rings. In a separate section, we investigate a three-dimensional Bott–Samelson variety in detail and compute its Hilbert polynomial and toric degenerations.

GROSSBERG-KARSHON TWISTED CUBES AND BASEPOINT-FREE DIVISORS

Let $G$ be a complex semisimple simply connected linear algebraic group. The main result of this note is to give several equivalent criteria for the untwistedness of the twisted cubes introduced by Grossberg and Karshon. In certain cases arising from representation theory, Grossberg and Karshon obtained a Demazure-type character formula for irreducible $G$-representations as a sum over lattice points (counted with sign according to a density function) of these twisted cubes. A twisted cube is untwisted when it is a "true" (i.e. closed, convex) polytope; in this case, Grossberg and Karshon's character formula becomes a purely positive formula with no multiplicities, i.e. each lattice point appears precisely once in the formula, with coefficient $+1$. One of our equivalent conditions for untwistedness is that a certain divisor on the special fiber of a toric degeneration of a Bott-Samelson variety, as constructed by Pasquier, is basepoint-free. We also show that the strict positivity of some of the defining constants for the twisted cube, together with convexity (of its support), is enough to guarantee untwistedness. Finally, in the special case when the twisted cube arises from the representation-theoretic data of $\lambda$ an integral weight and $\underline{w}$ a choice of word decomposition of a Weyl group element, we give two simple necessary conditions for untwistedness which is stated in terms of $\lambda$ and $\underline{w}$.

Bott-Samelson Varieties, Subword Complexes and Brick Polytopes

Bott-Samelson varieties factor the flag variety $G/B$ into a product of $\mathbb{C}\mathbb{P}^1$'s with a map into $G/B$. These varieties are mostly studied in the case in which the map into $G/B$ is birational; however in this paper we study fibers of this map when it is not birational. We will see that in some cases this fiber is a toric variety. In order to do so we use the moment map of a Bott-Samelson variety to translate this problem into a purely combinatorial one in terms of a subword complex. These simplicial complexes, defined by Knutson and Miller, encode a lot of information about reduced words in a Coxeter system. Pilaud and Stump realized certain subword complexes as the dual to the boundary of a polytope which generalizes the brick polytope defined by Pilaud and Santos. For a nice family of words, the brick polytope is the generalized associahedron realized by Hohlweg and Lange. These stories connect in a nice way: the moment polytope of a fiber of the Bott-Samelson map is the Brick polytope. In particular, we give a nice description of the toric variety of the associahedron.

Standard monomial theory for desingularized Richardson varieties in the flag variety GL(n)/B

We consider a desingularization Γ of a Richardson variety in the variety of complete flags, obtained as a fibre of a projection from a certain Bott-Samelson variety Z. For any projective embedding of Z via a complete linear system, we construct a basis of the homogeneous coordinate ring of Γ inside Z, indexed by combinatorial objects which we call w 0 -standard tableaux.

MV-polytopes via affine buildings

We give a construction of MV-polytopes of a complex semisimple algebraic group G in terms of the geometry of the Bott-Samelson variety and the affine building. This is done by using the construction of dense subsets of MV-cycles by Gaussent and Littelmann. They used LS-gallery to define subsets in the Bott-Samelson variety that map to subsets of the affine Grassmannian, whose closure are MV-cycles. Since points in the Bott-Samelson variety correspond to galleries in the affine building one can look at the image of a point in such a special subset under all retractions at infinity. We prove that these images can be used to construct the corresponding MV-polytope in an explicit way, by using the GGMS strata. Furthermore we give a combinatorial construction for these images by using the crystal structure of LS-galleries and the action of the ordinary Weyl group on the coweight lattice.

Vanishing theorem for the cohomology of line bundles on Bott–Samelson varieties

We use the Grossberg–Karshon's degeneration of Bott–Samelson varieties to toric varieties and the description of cohomology of line bundles on toric varieties to deduce vanishing results for the cohomology of lines bundles on Bott–Samelson varieties.

K -THEORY OF QUASI-TORIC MANIFOLDS

In this note we shall give a description of the K -ring of a quasi-toric manifolds in terms of generators and relations. We apply our results to describe the K -ring of Bott-Samelson varieties.

K-théorie équivariante des tours de Bott. Application à la structure multiplicative de la K-théorie équivariante des variétés de drapeaux

(Equivariant K-theory of Bott towers. Application to the multiplicative structure of the equivariant K-theory of flag varieties) We construct a basis of the equivariant K-theory of Bott towers, and we describe precisely the multiplicative structure of these algebras. We deduce similar results for Bott-Samelson varieties. Thanks to the link between flag varieties and Bott-Samelson varieties, we give a method to compute the structure constants of the equivariant K-theory of flag varieties in the basis constructed by Kostant and Kumar in [18]

Corrections and new results on the fibre of the Bott-Samelson resolution

In these notes, we present some counter-examples of two propositions concerning cellular decompositions occuring in a Bott-Samelson variety [G, Proposition 4 and Proposition 7]. Then, we prove that the description of the fibre of a Bott-Samelson resolution over a point of a Schubert variety is still valid in the Kac-Moody setting.

Standard Monomial Theory for Bott–Samelson Varieties

Bott–Samelson varieties are an important tool in geometric representation theory [1, 3, 10, 25]. They were originally defined as desingularizations of Schubert varieties and share many of the properties of Schubert varieties. They have an action of a Borel subgroup, and the projective coordinate ring of a Bott–Samelson variety splits into certain generalized Demazure modules (which also appear in other contexts [22, 23]). Standard Monomial Theory, developed by Seshadri and the first author [15, 16], and recently completed by the second author [20], gives explicit bases for the Demazure modules associated to Schubert varieties. In this paper, we extend the techniques of [20] to give explicit bases for the generalized Demazure modules associated to Bott–Samelson varieties, thus proving a strengthened form of the results announced by the first and third authors in [12] (see also [13]). We also obtain more elementary proofs of the cohomology vanishing theorems of Kumar [10] and Mathieu [25]; of the projective normality of Bott–Samelson varieties; and of the Demazure character formula.

Schubert polynomials and Bott-Samelson varieties

Schubert polynomials generalize Schur polynomials, but it is not clear how to generalize several classical formulas: the Weyl character formula, the Demazure character formula, and the generating series of semistandard tableaux. We produce these missing formulas and obtain several surprising expressions for Schubert polynomials. The above results arise naturally from a new geometric model of Schubert polynomials in terms of Bott-Samelson varieties. Our analysis includes a new, explicit construction for a Bott-Samelson variety Z as the closure of a B-orbit in a product of flag varieties. This construction works for an arbitrary reductive group G, and for G = GL(n) it realizes Z as the representations of a certain partially ordered set. This poset unifies several well-known combinatorial structures: generalized Young diagrams with their associated Schur modules; reduced decompositions of permutations; and the chamber sets of Berenstein-Fomin-Zelevinsky, which are crucial in the combinatorics of canonical bases and matrix factorizations. On the other hand, our embedding of Z gives an elementary construction of its coordinate ring, and allows us to specify a basis indexed by tableaux.

Standard Monomial Theory for Bott–Samelson Varieties of $\textit{GL}(n)$

We construct an explicit basis for the coordinate ring of the Bott–Samelson variety Z_i associated to G= \textit{GL}(n) and an arbitrary sequence of simple reflections \mathbf i . Our basis is parametrized by certain standard tableaux and generalizes the Standard Monomial basis for Schubert varieties.