V. Lakshmibai

Criterion for smoothness of Schubert varieties in Sl(n)/B

LetG=Sl(n) andB, the Borel subgroup ofG consisting of upper triangular matrices. Letw∈Sn andX(w)=BwB(modB), the associated Schubert variety inG/B. In this paper, we give a geometric criterion for the smoothness ofX(w). This criterion admits a neat combinatorial description in terms of the permutationw.

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.

Singular locus of a Schubert variety

In this note we give a characterization of the singular locus of a Schubert variety in the flag variety G/B, where G is a classical group and B is a Borel subgroup of G, or, more generally, for a Schubert variety in G/(J, where Q is a parabolic subgroup of G of classical type (cf. [4, 5]) of a semisimple algebraic group G. This turns out to be a rather easy consequence of the standard monomial theory, developed in Geometry ofG/P, I-V (cf. [9, 6, 3, 4, 7]). A corollary of this theory is the determination of the ideal defining X in G/B, and our result follows from the Jacobian criterion for smoothness. When G = SL(n) this characterization takes a particularly nice form (cf. Theorem 1 and Remark 1). To the best of our knowledge, our result is new even for a Schubert variety in a Grassmannian. Let G be a semisimple algebraic group, which we assume, for simplicity, to be defined over an algebraically closed field of characteristic zero (the following discussion is valid in any characteristic, in fact even over Z). Let T be a maximal torus, B a Borel subgroup containing T, and W the Weyl group of G. Let L be a line bundle on G/B associated to a dominant character \ of T (or B). Let V be the G-module H°(G/B, L). Recall that V can be identified with the set of regular functions ƒ on G such that

Standard Monomial Theory and applications

In these notes, we explain how one can construct Standard Monomial Theory for reductive algebraic groups by using the path models of their representations and quantum groups at a root of unity. As applications, we obtain a combinatorial proof of the Demazure character formula and representation theoretic proofs of geometrical properties of Schubert varieties, such as normality, vanishing theorems, ideal theory and so on. Further applications of Standard Monomial Theory are made to prove geometrical properties of certain ladder determinantal varieties and certain quiver varieties. We sketch at the end an extension of the theory to Bott-Samelson varieties and configuration varieties.

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.

Richardson varieties and equivariant K-theory

We generalize Standard Monomial Theory (SMT) to intersections of Schubert varieties and opposite Schubert varieties; such varieties are called Richardson varieties. The aim of this article is to get closer to a geometric interpretation of the standard monomial theory as constructed in (P. Littelmann, J. Amer. Math. Soc. 11 (1998) 551–567). In fact, the construction given here is very close to the ideas in (P. Lakshmibai, C.S. Seshadri, J. Algebra 100 (1986) 462–557). Our methods show that in order to develop a SMT for a certain class of subvarieties in G/B (which includes G/B), it suffices to have the following three ingredients, a basis for H0(G/B,Lλ), compatibility of such a basis with the varieties in the class, certain quadratic relations in the monomials in the basis elements. An important tool (as in (P. Lakshmibai, C.S. Seshadri, J. Algebra 100 (1986) 462–557)) will be the construction of nice filtrations of the vanishing ideal of the boundary of the varieties above. This provides a direct connection to the equivariant K-theory (products of classes of structure sheaves with classes of line bundles), where the combinatorially defined notion of standardness gets a geometric interpretation.

Multiplicities of Singular Points in Schubert Varieties of Grassmannians

We give a closed-form formula for the Hilbert function of the tangent cone at the identity of a Schubert variety X in the Grassmannian in both group theoretic and combinatorial terms. We also give a formula for the multiplicity of X at the identity, and a Grobner basis for the ideal defining X(w) ∩ O − as a closed subvariety of O −, where O − is the opposite cell in the Grassmannian. We give conjectures for the Hilbert function and multiplicity at points other than the identity.

Singular loci of Schubert varieties for classical groups

In this note, we give an explicit description of the singular locus of a Schubert variety in the flag variety G/B, where G is a classical group, and B a Borel subgroup of G. Let G be a classical group, and T a maximal torus in G. Let W be the Weyl group, and R the system of roots, of G relative to T. Let B be a Borel subgroup of G, where B D T. Let S (resp. i?) be the set of simple (resp. positive) roots of R relative to B. For a G Ü , let 8a be the reflection with respect to a, and Xa the element in the Chevalley basis for the Lie algebra of G, associated to a. For w € W, let e(w) denote the point in G/B corresponding to w. The Schubert variety X(w)y where w G W, is by definition the Zariski closure of B e(w) in G/B. {X(w) is understood to be endowed with the canonical reduced structure.) Let >: denote the Bruhat order in W. It is well known that for w\,W2 £ W,

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.

Wahl’s conjecture holds in odd characteristics for symplectic and orthogonal Grassmannians

It is shown that the proof by Mehta and Parameswaran of Wahl’s conjecture for Grassmannians in positive odd characteristics also works for symplectic and orthogonal Grassmannians.