A lookup conjecture for rational smoothness

The purpose of this paper is to present a new combinatorial criterion for rational smoothness of a point in a Schubert variety. The main result proves the criterion for Schubert varieties associated to partial flag varieties of the form G/P where G is semisimple of classical type and P is a maximal parabolic subgroup; in the case where G/P is of cominuscule type (not necessarily classical), we prove a stronger result. We conjecture that the criterion is valid for all Schubert varieties. The paper also contains some general results related to rational smoothness in Schubert varieties.