Schubert Varieties Research Articles

A lifted square formulation for certifiable Schubert calculus

Formulating a Schubert problem as the solutions to a system of equations in either Plücker space or in the local coordinates of a Schubert cell usually involves more equations than variables. Using reduction to the diagonal, we previously gave a primal–dual formulation for Schubert problems that involved the same number of variables as equations (a square formulation). Here, we give a different square formulation by lifting incidence conditions which typically involves fewer equations and variables. Our motivation is certification of numerical computation using Smale's α-theory.

Schubert Quiver Grassmannians

Quiver Grassmannians are projective varieties parametrizing subrepresentations of given dimension in a quiver representation. We define a class of quiver Grassmannians generalizing those which realize degenerate flag varieties. We show that each irreducible component of the quiver Grassmannians in question is isomorphic to a Schubert variety. We give an explicit description of the set of irreducible components, identify all the Schubert varieties arising, and compute the Poincare polynomials of these quiver Grassmannians.

Restriction formula for stable basis of the Springer resolution

We give restriction formula for stable basis of the Springer resolution, and generalize it to cotangent bundles of homogeneous spaces. By a limiting process, we get the restriction formula of Schubert varieties.

Cellular homology of real maximal isotropic grassmannians

Abstract In this paper we obtain a formula for the boundary map of the cellular ℤ-homology of the real maximal isotropic Grassmannians of type B, C and D which are obtained as minimal flag manifolds of split real forms of semi-simple Lie algebras of the respective type. We use the model of shifted Young diagrams presented in [9] to describe the Schubert varieties and the cellular decompositions of such flag manifolds. The results are applied to determine the orientability of the real maximal isotropic Grassmannians.

Newton–Okounkov convex bodies of Schubert varieties and polyhedral realizations of crystal bases

A Newton-Okounkov convex body is a convex body constructed from a projective variety with a valuation on its homogeneous coordinate ring; this is deeply connected with representation theory. For instance, the Littelmann string polytopes and the Feigin-Fourier-Littelmann-Vinberg polytopes are examples of Newton-Okounkov convex bodies. In this paper, we prove that the Newton-Okounkov convex body of a Schubert variety with respect to a specific valuation is identical to the Nakashima-Zelevinsky polyhedral realization of a Demazure crystal. As an application of this result, we show that Kashiwara's involution (*-operation) corresponds to a change of valuations on the rational function field.

Toric matrix Schubert varieties and their polytopes

Given a matrix Schubert variety $\overline{X_\pi}$, it can be written as $\overline{X_\pi}=Y_\pi\times \mathbb{C}^q$ (where $q$ is maximal possible). We characterize when $Y_{\pi}$ is toric (with respect to a $(\mathbb{C}^*)^{2n-1}$-action) and study the associated polytope $\Phi(\mathbb{P}(Y_\pi))$ of its projectivization. We construct regular triangulations of $\Phi(\mathbb{P}(Y_\pi))$ which we show are geometric realizations of a family of subword complexes. Subword complexes were introduced by Knutson and Miller in 2004, who also showed that they are homeomorphic to balls or spheres and raised the question of their polytopal realizations.

Cluster structures on strata of flag varieties

We introduce some new Frobenius subcategories of the module category of a preprojective algebra of Dynkin type, and we show that they have a cluster structure in the sense of Buan–Iyama–Reiten–Scott. These categorical cluster structures yield cluster algebra structures in the coordinate rings of intersections of opposite Schubert cells.

On the Automorphism Group of a Smooth Schubert Variety

Let G be a simple algebraic group of adjoint type over the field \(\mathbb {C}\) of complex numbers. Let B be a Borel subgroup of G containing a maximal torus T of G. Let w be an element of the Weyl group W and let X(w) be the Schubert variety in G/B corresponding to w. Let α0 denote the highest root of G with respect to T and B. Let P be the stabiliser of X(w) in G. In this paper, we prove that if G is simply laced and X(w) is smooth, then the connected component of the automorphism group of X(w) containing the identity automorphism equals P if and only if w−1(α0) is a negative root (see Theorem 4.2). We prove a partial result in the non simply laced case (see Theorem 6.6).

Equivariant Hirzebruch class for singular varieties

We develop an equivariant version of the Hirzebruch class for singular varieties. When the group acting is a torus we apply localization theorem of Atiyah–Bott and Berline–Vergne. The localized Hirzebruch class is an invariant of a singularity germ. The singularities of toric varieties and Schubert varieties are of special interest. We prove certain positivity results for simplicial toric varieties. The positivity for Schubert varieties is illustrated by many examples, but it remains mysterious.

Accessible proof of standard monomial basis for coordinatization of Schubert sets of flags

The main results of this paper are accessible with only basic linear algebra. Given an increasing sequence of dimensions, a flag in a vector space is an increasing sequence of subspaces with those dimensions. The set of all such flags (the flag manifold) can be projectively coordinatized using products of minors of a matrix. These products are indexed by tableaux on a Young diagram. A basis of “standard monomials” for the vector space generated by such projective coordinates over the entire flag manifold has long been known. A Schubert variety is a subset of flags specified by a permutation. Lakshmibai, Musili, and Seshadri gave a standard monomial basis for the smaller vector space generated by the projective coordinates restricted to a Schubert variety. Reiner and Shimozono made this theory more explicit by giving a straightening algorithm for the products of the minors in terms of the right key of a Young tableau. Since then, Willis introduced scanning tableaux as a more direct way to obtain right keys. This paper uses scanning tableaux to give more-direct proofs of the spanning and the linear independence of the standard monomials. In the appendix it is noted that this basis is a weight basis for the dual of a Demazure module for a Borel subgroup of GLn. This paper contains a complete proof that the characters of these modules (the key polynomials) can be expressed as the sums of the weights for the tableaux used to index the standard monomial bases.

Schubert’s Winter Journey: Anatomy of an Obsession by Ian Bostridge (review)

VOICES Schubert's Winter Journey: Anatomy of an Obsession. By Ian Bostridge. New York: Alfred A. Knopf, 2015. [xxi, 502 p. ISBN 9780307961631 (hardcover), $29; ISBN 9780307961648 (e-book), $14.99.] Music examples, illustrations, bibliography.The Winter Journey in the title under review is, of course, the English translation of Franz Schubert's second and last song cycle, Winterreise, composed in 1827, one year before his death at age thirty-one. It remains unclear whose obsession noted in the title is anatomized, but readers expecting to learn more about Schubert's Winter Journey will soon learn that one can write a 500-page small but surprisingly heavy book about the twenty-four songs Schubert set to the poetry of Wilhelm Muller and yet rarely address the relationship of music and text. While often stylish and engaging, Winter Journey seems designed for readers who appreciate the singer-author's distinctive personality and agile, digressive mind. Readers who wish to know more about such matters as the genesis and sources, texts, music, and musically informed discussions about Schubert's cycle, might save themselves considerable frustration by revisiting Susan Youens's authoritative Retracing a Winter's Journey: Schubert's Winterreise (Ithaca: Cornell University Press, 1991) or the worthwhile nontechnical musical and textual discussion in Paul Robinson's Opera & Ideas: From Mozart to Strauss (Ithaca: Cornell University Press, 1986).Each of Ian Bostridge's twenty-four chapters, without a footnote or endnote in sight, corresponds to a song in Schubert's published ordering of the cycle, a serviceable structure the author credits to his spouse. In many of these chapters Bostridge, a tenor who has sung this cycle numerous times professionally, addresses what the song means-or rather what Muller's text means-and also occasionally offers pertinent remarks on the performance of these songs. Longer chapters provide a forum for Bostridge to launch into discussions of topics that usually go far beyond the song's immediate context. Thus the chapter on Die Wetterfahne (no. 2 in the cycle) soon leads to a discussion of marriage laws in Schubert's Vienna and an extended tangent on Schubert's sexuality that manages to obfuscate the issues and facts and to belittle and dismiss rather than engage the work of Maynard Solomon. In Gefrorne Tranen (no. 3) readers learn about the disastrous winter journeys of Napoleon and Hitler, while the chapter on Der Lindenbaum (no. 5) offers a discourse on the Linden tree in literature and some thoughtful commentary on how Thomas Mann famously used this song in his novel The Magic Mountain.The chapter on Auf dem Flusse (no. 7) informs readers about weather fluctuations in Europe between 1350 and 1850. The charcoal burner's hut mentioned in Rast (no. 10) leads to a detailed discourse on energy sources in England and Wales between 1560 and 1860 and an unsubstantiated theory on how Muller's text conceals ideas that would have merited censorship in the repressive post-Congress of Vienna Age of Metternich. In Die Post (no. 13) we learn about modes and speeds of nineteenth-century transportation, in Die Krahe (no. 15) about crows in paintings, literature, and science-although not, for my taste, nearly enough about the compelling recent research on how this intelligent bird manages to communicate information to its associates and its highly developed ability to recognize individual human faces. In Tauschung (no. 19), we can savor an illustration of the nine dance positions of the waltz, and in Der Wegweiser (no. 20), discover Schubert's passion for the Leatherstocking tales of James Fenimore Cooper during the composer's final illness. In Mut (no. 22), Schubert, God, and Nietzsche coexist, in Die Nebensonnen (no. 23), sun dogs and parhelia, and finally in Der Leiermann (no. 24), the hurdy-gurdy and its place in German culture gives way to ruminations on possible parallels between the jinglejangle of Bob Dylan's tambourine man and Schubert's hurdy-gurdy man. …

Schubert Varieties and Distances between Subspaces of Different Dimensions

We resolve a basic problem on subspace distances that often arises in applications: How can the usual Grassmann distance between equidimensional subspaces be extended to subspaces of different dimensions? We show that a natural solution is given by the distance of a point to a Schubert variety within the Grassmannian. This distance reduces to the Grassmann distance when the subspaces are equidimensional and does not depend on any embedding into a larger ambient space. Furthermore, it has a concrete expression involving principal angles and is efficiently computable in numerically stable ways. Our results are largely independent of the Grassmann distance---if desired, it may be substituted by any other common distances between subspaces. Our approach depends on a concrete algebraic geometric view of the Grassmannian that parallels the differential geometric perspective that is well established in applied and computational mathematics.

Accessible proof of standard monomial basis for coordinatization of Schubert sets of flags

The main results of this paper are accessible with only basic linear algebra. Given an increasing sequence of dimensions, a flag in a vector space is an increasing sequence of subspaces with those dimensions. The set of all such flags (the flag manifold) can be projectively coordinatized using products of minors of a matrix. These products are indexed by tableaux on a Young diagram. A basis of "standard monomials" for the vector space generated by such projective coordinates over the entire flag manifold has long been known. A Schubert variety is a subset of flags specified by a permutation. Lakshmibai, Musili, and Seshadri gave a standard monomial basis for the smaller vector space generated by the projective coordinates restricted to a Schubert variety. Reiner and Shimozono made this theory more explicit by giving a straightening algorithm for the products of the minors in terms of the right key of a Young tableau. Since then, Willis introduced scanning tableaux as a more direct way to obtain right keys. This paper uses scanning tableaux to give more-direct proofs of the spanning and the linear independence of the standard monomials. In the appendix it is noted that this basis is a weight basis for the dual of a Demazure module for a Borel subgroup of GL(n). This paper contains a complete proof that the characters of these modules (the key polynomials) can be expressed as the sums of the weights for the tableaux used to index the standard monomial bases.

Giambelli and Degeneracy Locus Formulas for Classical G/P spaces

Let G be a classical complex Lie group, P any parabolic subgroup of G, and X = G/P the corresponding homogeneous space, which parametrizes (isotropic) partial flags of subspaces of a fixed vector space. In the mid 1990s, Fulton, Pragacz, and Ratajski asked for global formulas which express the cohomology classes of the universal Schubert varieties in flag bundles -- when the space X varies in an algebraic family -- in terms of the Chern classes of the vector bundles involved in their definition. This has applications to the theory of degeneracy loci of vector bundles and is closely related to the Giambelli problem for the torus-equivariant cohomology ring of X. In this article, we explain the answer to these questions which was obtained in arXiv:0908.3628, in terms of combinatorial data coming from the Weyl group.

Geometric mitosis

We describe an elementary convex geometric algorithm for realizing Schubert cycles in complete flag varieties by unions of faces of polytopes. For GL_n and Gelfand--Zetlin polytopes, combinatorics of this algorithm coincides with that of the mitosis on pipe dreams introduced by Knutson and Miller. For Sp_4 and a Newton--Okounkov polytope of the symplectic flag variety, the algorithm yields a new combinatorial rule that extends to Sp_{2n}.

Free resolutions of some Schubert singularities in the Lagrangian Grassmannian

In this paper we construct free resolutions of certain class of closed subvarieties of affine space of symmetric matrices (of a given size). Our class covers the symmetric determinantal varieties (i.e., determinantal varieties in the space of symmetric matrices), whose resolutions were first constructed by Jozefiak-Pragacz-Weyman. Our approach follows the techniques developed in Free Resolutions of Some Schubert Singularities by Kummini-Lakshmibai-Sastry-Seshadri, and uses the geometry of Schubert varieties.

PBW-degenerated Demazure modules and Schubert varieties for triangular elements

We study certain faces of the normal polytope introduced by Feigin, Littelmann and the author whose lattice points parametrize a monomial basis of the PBW-degenerated of simple modules for sln+1. We show that lattice points in these faces parametrize monomial bases of PBW-degenerated Demazure modules associated to Weyl group elements satisfying a certain closure property, for example Kempf elements.These faces are again normal polytopes and their Minkowski sum is compatible with tensor products, which implies that we obtain flat degenerations of the corresponding Schubert varieties to PBW degenerated and toric varieties.

COTANGENT BUNDLE TO THE GRASSMANN VARIETY

We show that there is an affine Schubert variety in the infinite-dimensional partial Flag variety (associated to the two-step parabolic subgroup of the Kac–Moody group , corresponding to omitting α 0; α d ) which is a natural compactification of the cotangent bundle to the Grassmann variety.

A positive Grassmannian analogue of the permutohedron

The classical permutohedron Perm is the convex hull of the points (w(1),...,w(n)) in R^n where w ranges over all permutations in the symmetric group. This polytope has many beautiful properties -- for example it provides a way to visualize the weak Bruhat order: if we orient the permutohedron so that the longest permutation w_0 is at the top and the identity e is at the bottom, then the one-skeleton of Perm is the Hasse diagram of the weak Bruhat order. Equivalently, the paths from e to w_0 along the edges of Perm are in bijection with the reduced decompositions of w_0. Moreover, the two-dimensional faces of the permutohedron correspond to braid and commuting moves, which by the Tits Lemma, connect any two reduced expressions of w_0. In this note we introduce some polytopes Br(k,n) (which we call bridge polytopes) which provide a positive Grassmannian analogue of the permutohedron. In this setting, BCFW bridge decompositions of reduced plabic graphs play the role of reduced decompositions. We define Br(k,n) and explain how paths along its edges encode BCFW bridge decompositions of the longest element pi(k,n) in the circular Bruhat order. We also show that two-dimensional faces of Br(k,n) correspond to certain local moves for plabic graphs, which by a result of Postnikov [Pos06], connect any two reduced plabic graphs associated to pi(k,n). All of these results can be generalized to the positive parts of Schubert cells. A useful tool in our proofs is the fact that our polytopes are isomorphic to certain Bruhat interval polytopes. Conversely, our results on bridge polytopes allow us to deduce some corollaries about the structure of Bruhat interval polytopes.

Billey–Postnikov decompositions and the fibre bundle structure of Schubert varieties

A theorem of Ryan and Wolper states that a type A Schubert variety is smooth if and only if it is an iterated fibre bundle of Grassmannians. We extend this theorem to arbitrary finite type, showing that a Schubert variety in a generalized flag variety is rationally smooth if and only if it is an iterated fibre bundle of rationally smooth Grassmannian Schubert varieties. The proof depends on deep combinatorial results of Billey-Postnikov on Weyl groups. We determine all smooth and rationally smooth Grassmannian Schubert varieties, and give a new proof of Peterson's theorem that all simply-laced rationally smooth Schubert varieties are smooth. Taken together, our results give a fairly complete geometric description of smooth and rationally smooth Schubert varieties using primarily combinatorial methods.