Desingularizations of Schubert varieties in double Grassmannians

Let X = Gr(k, V) × Gr(l, V) be the direct product of two Grassmann varieties of k-and l-planes in a finite-dimensional vector space V, and let B ? GL(V) be the isotropy group of a complete flag in V. We consider B-orbits in X, which are an analog of Schubert cells in Grassmannians. We describe this set of orbits combinatorially and construct desingularizations for the closures of these orbits, similar to the Bott-Samelson desingularizations for Schubert varieties.     

On quiver varieties and affine Grassmannians of type A

We construct Nakajima's quiver varieties of type A in terms of affine Grassmannians of type A. This gives a compactification of quiver varieties and a decomposition of affine Grassmannians into a disjoint union of quiver varieties. Consequently, singularities of quiver varieties, nilpotent orbits and affine Grassmannians are the same in type A. The construction also provides a geometric framework for skew (GL(m),GL(n)) duality and identifies the natural basis of weight spaces in Nakajima's construction with the natural basis of multiplicity spaces in tensor products which arises from affine Grassmannians. To cite this article: I. Mirkovi?, M. Vybornov, C. R. Acad. Sci. Paris, Ser. I 336 (2003).     

Smooth affine varieties and complete intersections

This paper consists of two independent parts. First I give a Chern class condition that is sufficient for a smooth surface in affinen-space to be a set-theoretic complete intersection. In the second part I show the existence of a smooth affine fourfold over C which is not a complete intersection in anyA ⁿ although its canonical bundle is trivial.     

Initial ideals of tangent cones to Schubert varieties in orthogonal Grassmannians

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 define 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.     

Rigidity of Schubert classes in orthogonal Grassmannians

A Schubert class σ in the cohomology of a homogeneous variety X is called rigid if the only projective subvarieties of X representing σ are Schubert varieties. A Schubert class σ is called multi rigid if the only projective subvarieties representing positive integral multiples of σ are unions of Schubert varieties. In this paper, we discuss the rigidity and multi rigidity of Schubert classes in orthogonal Grassmannians. For a large set of non-rigid classes, we provide explicit deformations of Schubert varieties using combinatorially defined varieties called restriction varieties. We characterize rigid and multi rigid Schubert classes of Grassmannian and quadric type. We also characterize all the rigid classes in OG(2, n) if n > 8.     

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.     

Radon transforms on affine Grassmannians

We develop an analytic approach to the Radon transform , where is a function on the affine Grassmann manifold of -dimensional planes in , and is a -dimensional plane in the similar manifold k$">. For , we prove that this transform is finite almost everywhere on if and only if , and obtain explicit inversion formulas. We establish correspondence between Radon transforms on affine Grassmann manifolds and similar transforms on standard Grassmann manifolds of linear subspaces of . It is proved that the dual Radon transform can be explicitly inverted for , and interpreted as a direct, ``quasi-orthogonal" Radon transform for another pair of affine Grassmannians. As a consequence we obtain that the Radon transform and the dual Radon transform are injective simultaneously if and only if . The investigation is carried out for locally integrable and continuous functions satisfying natural weak conditions at infinity.

     

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. Received: November 27, 1997     

Schubert varieties and short braidedness

LetW be a finite Weyl group. We give a characterization of those elements ofW whose reduced expressions avoid substrings of the formsts wheres andt are noncommuting generators. We give as an application a family of singular Schubert varieties.Supported in part by a NSF postdoctoral fellowship     

Schubert unions in Grassmann varieties

We study subsets of Grassmann varieties G(l,m) over a field F, such that these subsets are unions of Schubert cycles, with respect to a fixed flag. We study unions of Schubert cycles of Grassmann varieties G(l,m) over a field F. We compute their linear span and, in positive characteristic, their number of F_q-rational points. Moreover, we study a geometric duality of such unions, and give a combinatorial interpretation of this duality. We discuss the maximum number of F_q-rational points for Schubert unions of a given spanning dimension, and as an application to coding theory, we study the parameters and support weights of the well-known Grassmann codes. Moreover, we determine the maximum Krull dimension of components in the intersection of G(l,m) and a linear space of given dimension in the Plücker space.     

Schubert polynomials for the affine Grassmannian

Confirming a conjecture of Mark Shimozono, we identify polynomial representatives for the Schubert classes of the affine Grassmannian as the -Schur functions in homology and affine Schur functions in cohomology. The results are obtained by connecting earlier combinatorial work of ours to certain subalgebras of Kostant and Kumar's nilHecke ring and to work of Peterson on the homology of based loops on a compact group. As combinatorial corollaries, we settle a number of positivity conjectures concerning -Schur functions, affine Stanley symmetric functions and cylindric Schur functions.

     

Combinatorics of the K-theory of affine Grassmannians

We introduce a family of tableaux that simultaneously generalizes the tableaux used to characterize Grothendieck polynomials and k-Schur functions. We prove that the polynomials drawn from these tableaux are the affine Grothendieck polynomials and k-K-Schur functions – Schubert representatives for the K-theory of affine Grassmannians and their dual in the nil Hecke ring. We prove a number of combinatorial properties including Pieri rules.     

Degenerations of flag and Schubert varieties to toric varieties

In this paper, we prove the degenerations of Schubert varieties in a minusculeG/P, as well as the class of Kempf varieties in the flag varietySL(n)/B, to (normal) toric varieties. As a consequence, we obtain that determinantal varietes degenerate to (normal) toric varieties. Both of the authors are partially supported by NSF Grant DMS 9502942.