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     

Toric degenerations of Schubert varieties

LetG be a simply connected semisimple complex algebraic group. We prove that every Schubert variety ofG has a flat degeneration into a toric variety. This provides a generalization of results of [9], [7], [6]. Our basic tool is Lusztig's canonical basis and the string parametrization of this basis.Supported in part by the EC TMR network Algebraic Lie Representations, contract No. ERB FMTX-CT97-0100.     

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.     

A note on CIP varieties

It is proved that every variety satisfying the Congruence Intersection Property (CIP) is Abelian. In addition, a CM Abelian variety has the CIP if and only if it has a constant term operation. Finally, a CM variety is Abelian if and only if it has the weak CIP. Received October 8, 1998; accepted in final form January 5, 1999.     

A note on Veronese varieties

We show that for every primep, there is a class of Veronese varieties which are set-theoretic complete intersections if and only if the ground field has characteristicp. Partially supported by PRIN Algebra Commutativa e Computazionale, Italian Ministry of Education, University and Research.     

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.     

Generic singularities of certain Schubert varieties

Let G be a connected semisimple algebraic group, B a Borel subgroup, T a maximal torus in B with Weyl group W, and Q a subgroup containing B. For , let denote the Schubert variety . For such that , one knows that ByQ / Q admits a T-stable transversal in , which we denote by . We prove that, under certain hypotheses, is isomorphic to the orbit closure of a highest weight vector in a certain Weyl module. We also obtain a generalisation of this result under slightly weaker hypotheses. Further, we prove that our hypotheses are satisfied when Q is a maximal parabolic subgroup corresponding to a minuscule or cominuscule fundamental weight, and is an irreducible component of the boundary of (that is, the complement of the open orbit of the stabiliser in G of ). As a consequence, we describe the singularity of along ByQ / Q and obtain that the boundary of equals its singular locus. Received October 9, 1997; in final form February 19, 1998     

Homology operations inG/TOP

Inventiones mathematicae -     

Quadratic degenerations of odd-orthogonal Schubert varieties

This paper is the second in a series leading to a type B_n geometric Littlewood-Richardson rule. The rule will give an interpretation of the B_n Littlewood-Richardson numbers as an intersection of two odd-orthogonal Schubert varieties and will consider a sequence of linear and quadratic deformations of the intersection into a union of odd-orthogonal Schubert varieties. This paper describes the setup for the rule and specifically addresses results for quadratic deformations, including a proof that at each quadratic degeneration, the results occur with multiplicity one. This work is strongly influenced by Vakil’s [14].     

A note on quantum products of Schubert classes in a Grassmannian

Given two Schubert classes σ_λ and σ_μ in the quantum cohomology of a Grassmannian, we construct a partition ν, depending on λ and μ, such that σ_ν appears with coefficient 1 in the lowest (or highest) degree part of the quantum product σ_λ⋆σ_μ. To do this, we show that for any two partitions λ and μ, contained in a k × (n − k) rectangle and such that the 180^∘-rotation of one does not overlap the other, there is a third partition ν, also contained in the rectangle, such that the Littlewood-Richardson number c _λμ ^ν is 1.