Schubert Varieties In Grassmannians Research Articles

The isomorphism problem for cominuscule Schubert varieties

Cominuscule flag varieties generalize Grassmannians to other Lie types. Schubert varieties in cominuscule flag varieties are indexed by posets of roots labeled long/short. These labeled posets generalize Young diagrams. We prove that Schubert varieties in potentially different cominuscule flag varieties are isomorphic as varieties if and only if their corresponding labeled posets are isomorphic, generalizing the classification of Grassmannian Schubert varieties using Young diagrams by the last two authors. Our proof is type-independent.

The isomorphism problem for Grassmannian Schubert varieties

We prove that Schubert varieties in potentially different Grassmannians are isomorphic as varieties if and only if their corresponding Young diagrams are identical up to a transposition. We also discuss a generalization of this result to Grassmannian Richardson varieties. In particular, we prove that Richardson varieties in potentially different Grassmannians are isomorphic as varieties if their corresponding skew diagrams are semi-isomorphic as posets, and we conjecture the converse. Here, two posets are said to be semi-isomorphic if there is a bijection between their sets of connected components such that the corresponding components are either isomorphic or opposite.

Castelnuovo-Mumford regularity of ladder determinantal varieties and patches of Grassmannian Schubert varieties

We give degree formulas for Grothendieck polynomials indexed by vexillary permutations and 1432-avoiding permutations via tableau combinatorics. These formulas generalize a formula for degrees of symmetric Grothendieck polynomials which appeared in previous joint work of the authors with Y. Ren and A. St. Dizier.We apply our formulas to compute Castelnuovo-Mumford regularity of classes of generalized determinantal ideals. In particular, we give combinatorial formulas for the regularities of all one-sided mixed ladder determinantal ideals. We also derive formulas for the regularities of certain Kazhdan-Lusztig ideals, including those coming from open patches of Schubert varieties in Grassmannians. This provides a correction to a conjecture of Kummini-Lakshmibai-Sastry-Seshadri (2015).

Bases of the intersection cohomology of Grassmannian Schubert varieties

The parabolic Kazhdan–Lusztig polynomials for Grassmannians can be computed by counting Dyck partitions. We “lift” this combinatorial formula to the corresponding category of singular Soergel bimodules to obtain bases of the Hom spaces between indecomposable objects. In particular, we describe bases of intersection cohomology of Schubert varieties in Grassmannians parametrized by Dyck partitions which extend (after dualizing) the classical Schubert basis of the ordinary cohomology.

Degrees of symmetric Grothendieck polynomials and Castelnuovo-Mumford regularity

We give an explicit formula for the degree of the Grothendieck polynomial of a Grassmannian permutation and a closely related formula for the Castelnuovo-Mumford regularity of the Schubert determinantal ideal of a Grassmannian permutation. We then provide a counterexample to a conjecture of Kummini-Lakshmibai-Sastry-Seshadri on a formula for regularities of standard open patches of particular Grassmannian Schubert varieties and show that our work gives rise to an alternate explicit formula in these cases. We end with a new conjecture on the regularities of standard open patches of arbitrary Grassmannian Schubert varieties.

The Nash blow-up of a cominuscule Schubert variety

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.

Cyclic Demazure Modules and Positroid Varieties

A positroid variety is an intersection of cyclically rotated Grassmannian Schubert varieties. Each graded piece of the homogeneous coordinate ring of a positroid variety is the intersection of cyclically rotated (rectangular) Demazure modules, which we call the cyclic Demazure module. In this note, we show that the cyclic Demazure module has a canonical basis, and define the cyclic Demazure crystal.

Pattern avoidance and fiber bundle structures on Schubert varieties

We give a permutation pattern avoidance criteria for determining when the projection map from the flag variety to a Grassmannian induces a fiber bundle structure on a Schubert variety. In particular, we introduce the notion of a split pattern and show that a Schubert variety has such a fiber bundle structure if and only if the corresponding permutation avoids the split patterns 3|12 and 23|1. Continuing, we show that a Schubert variety is an iterated fiber bundle of Grassmannian Schubert varieties if and only if the corresponding permutation avoids (non-split) patterns 3412, 52341, and 635241. This extends a combined result of Lakshmibai–Sandhya, Ryan, and Wolper who prove that Schubert varieties whose permutation avoids the “smooth” patterns 3412 and 4231 are iterated fiber bundles of smooth Grassmannian Schubert varieties.

Minimum distance and the minimum weight codewords of Schubert codes

We consider linear codes associated to Schubert varieties in Grassmannians. A formula for the minimum distance of these codes was conjectured in 2000 and after having been established in various special cases, it was proved in 2008 by Xiang. We give an alternative proof of this formula. Further, we propose a characterization of the minimum weight codewords of Schubert codes by introducing the notion of Schubert decomposable elements of certain exterior powers. It is shown that codewords corresponding to Schubert decomposable elements are of minimum weight and also that the converse is true in many cases. A lower bound, and in some cases, an exact formula, for the number of minimum weight codewords of Schubert codes is also given. From a geometric point of view, these results correspond to determining the maximum number of Fq-rational points that can lie on a hyperplane section of a Schubert variety in a Grassmannian with its nondegenerate embedding in a projective subspace of the Plücker projective space, and also the number of hyperplanes for which the maximum is attained.

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.

Hilbert coefficients of Schubert varieties in Grassmannians

In this paper, we give combinatorial formulas for the Hilbert coefficients, h-polynomial and the Cohen–Macaulay type of Schubert varieties in Grassmannians in terms of the posets associated with them. As a consequence, necessary conditions for a Schubert variety to be a complete intersection and combinatorial criteria are given for a Schubert variety to be Gorenstein and almost Gorenstein, respectively.

Some degenerations of Kazhdan–Lusztig ideals and multiplicities of Schubert varieties

We study Hilbert–Samuel multiplicity for points of Schubert varieties in the complete flag variety, by Gröbner degenerations of the Kazhdan–Lusztig ideal. In the covexillary case, we give a manifestly positive combinatorial rule for multiplicity by establishing (with a Gröbner basis) a reduced limit whose Stanley–Reisner simplicial complex is homeomorphic to a shellable ball or sphere. We show that multiplicity counts the number of facets of this complex. We also obtain a formula for the Hilbert series of the local ring. In particular, our work gives a multiplicity rule for Grassmannian Schubert varieties, providing alternative statements and proofs to formulae of Lakshmibai and Weyman (1990) [26], Rosenthal and Zelevinsky (2001) [37], Krattenthaler (2001) [22], Kodiyalam and Raghavan (2003) [21], Kreiman and Lakshmibai (2004) [24], Ikeda and Naruse (2009) [13] and Woo and Yong (2009) [40]. We suggest extensions of our methodology to the general case.

A combinatorial rule for (co)minuscule Schubert calculus

We prove a root system uniform, concise combinatorial rule for Schubert calculus of minuscule and cominuscule flag manifolds G / P (the latter are also known as compact Hermitian symmetric spaces). We connect this geometry to the poset combinatorics of Proctor, thereby giving a generalization of Schützenberger's jeu de taquin formulation of the Littlewood–Richardson rule that computes the intersection numbers of Grassmannian Schubert varieties. Our proof introduces cominuscule recursions, a general technique to relate the numbers for different Lie types.

Schubert varieties, linear codes and enumerative combinatorics

We consider linear error correcting codes associated to higher-dimensional projective varieties defined over a finite field. The problem of determining the basic parameters of such codes often leads to some interesting and difficult questions in combinatorics and algebraic geometry. This is illustrated by codes associated to Schubert varieties in Grassmannians, called Schubert codes, which have recently been studied. The basic parameters such as the length, dimension and minimum distance of these codes are known only in special cases. An upper bound for the minimum distance is known and it is conjectured that this bound is achieved. We give explicit formulae for the length and dimension of arbitrary Schubert codes and prove the minimum distance conjecture in the affirmative for codes associated to Schubert divisors.

Hilbert functions of points on Schubert varieties in Grassmannians

Hilbert functions of points on Schubert varieties in Grassmannians

Good filtrations of symmetric algebras and strong F-regularity of invariant subrings

In this paper, we prove that a linear action of a reductive group on a polynomial ring with good filtrations over a field of characteristic p>0 yields a strongly F-regular (in particular, Cohen-Macaulay) invariant subring. The strongly F-regular property of some known examples of invariant subrings, such as the coordinate rings of Schubert varieties in Grassmannians, are recovered. A similar result over a field of characteristic zero is also proved.

Multiplicities of Points on Schubert Varieties in Grassmannians

We obtain an explicit determinantal formula for the multiplicity of any point on a classical Schubert variety.

Young bitableaux, lattice paths and Hilbert functions

A recent result on the enumeration of p-tuples of nonintersecting lattice paths in an integral rectangle is used to deduce a formula of Abhyankar for standard Young bitableaux of certain type, which gives the Hilbert function of a class of determinantal ideals. The lattice path formula is also shown to yield the numerator of the Hilbert series of these determinantal ideals and the h-vectors of the associated simplicial complexes. As a consequence, the a-invariant of these determinantal ideals is obtained in some cases, extending an earlier result of Gräbe. Some problems concerning generalizations of these results to ‘higher dimensions’ are also discussed. In an appendix, the equivalence of Abhyankar's formula for unitableaux of a given shape and a formula of Hodge, obtained in connection with his determination of Hilbert functions of Schubert varieties in Grassmannians, is outlined.

Vanishing Theorems for Flag Manifolds

Introduction. In this paper, I prove some vanishing theorems for the cohomology groups of invertible sheaves on flag manifolds. Most of these vanishing theorems follow from Kodaira's vanishing theorem in characteristic zero. I prove these by induction. This induction involves certain subvarieties of the flag manifold, which I call Schubert manifolds. These vanishing theorems are used to deduce the Cohen-Macaulayness of Schubert varieties in Grassmannians and cones over them. They were recently proved to be Cohen-Macaulay by Eagon, Hochster and Laksov [3, 5, and 8] by more algebraic methods. I prove that the Schubert varieties have rational resolutions. This is a stronger property than Cohen-Macaulayness. The original idea for this proof was to generalize the results in [6]. The vanishing theorem has been applied by T. Svanes to show the rigidity of some Schubert varieties [12].