A combinatorial approach to the singularities of Schubert varieties

Abstract

Schubert subvarieties of Grassmannians and, more generally, varieties G/P (with G a linear algebraic group and P a parabolic subgroup) have long been a rich source for algebraic geometers and topologists. Recent interest has focussed on their singularities, which are of interest due to the relationship between the intersection homological properties of a singularity of a Schubert variety in G/P and invariants of composition series of Verma modules [KL, GM]. This work presents a simple combinatorial algorithm for deciding whether a Schubert variety in G/P, where G = SL,, is singular. This leads to a geometric characterization of the nonsingular Schubert varieties, as sequences of Grassmannian bundles over Grassmannians. The combinatorial techniques are those of [P] (of whose existence the author was unaware). A more general result of this nature has been proved by Seshadri and Lakshmibai [LakS], who use Standard Monomial Theory to find the ideal of the singular locus of a Schubert variety in G/B, where G is an algebraic group with a Standard Monomial Theory, and B is a Bore1 subgroup of G. Ryan [R] has obtained results similar to ours, using general techniques for understanding determinantal varieties. The present work has the advantage of being quite elementary, while sharing aspects of both of the above. Or special interest here are the combinatorial techniques, which appear to be a basic connection between the singularities of Schubert Varieties, the geometry of the tibres of Springer’s resolution of the singularities of the nilpotent scheme (see [Wl ] ), and the Kazhdan-Lusztig polynomials. A related note [W2] discusses some applications of these techniques to the computation of the Kazhdan-Lusztig polynomials.