ar X iv : m at h / 98 07 07 9 v 1 [ m at h . C O ] 1 5 Ju l 1 99 8 RECOGNIZING SCHUBERT CELLS

1.1. Formulations of the main problems. Let G be a simply connected complex semisimple Lie group of rank r with a fixed Borel subgroup B and a maximal torus H ⊂ B. Let W = NormG(H)/H be the Weyl group of G. The generalized flag manifold G/B can be decomposed into the disjoint union of Schubert cells X w = (BwB)/B, for w ∈ W . To any weight γ that is W -conjugate to some fundamental weight of G, one can associate a generalized Plücker coordinate pγ on G/B (see [9] or Section 3 below). In the case of type An−1 (i.e., G = SLn), the pγ are the usual Plücker coordinates on the flag manifold. The closure of a Schubert cell X w is the Schubert variety Xw , an irreducible projective subvariety of G/B that can be described as the set of common zeroes of some collection of generalized Plücker coordinates pγ . It is also known (see, e.g., Proposition 4.1 below) that every Schubert cell X w can be defined by specifying vanishing and/or non-vanishing of some collection of Plücker coordinates. The main two problems studied in this paper are the following.