Projections of Richardson Varieties
Abstract
While the projections of Schubert varieties in a full generalized flag manifold G/B to a partial flag manifold $G/P$ are again Schubert varieties, the projections of Richardson varieties (intersections of Schubert varieties with opposite Schubert varieties) are not always Richardson varieties. The stratification of G/P by projections of Richardson varieties arises in the theory of total positivity and also from Poisson and noncommutative geometry. In this paper we show that many of the geometric properties of Richardson varieties hold more generally for projected Richardson varieties; they are normal, CohenMacaulay, have rational singularities, and are compatibly Frobenius split with respect to the standard splitting. Indeed, we show that the projected Richardson varieties are the only compatibly split subvarieties, providing an example of the recent theorem [Schwede, KumarMehta] that a Frobenius split scheme has only finitely many compatibly split subvarieties. (The G/B case was treated by [Hague], whose proof we simplify somewhat.) One combinatorial analogue of a Richardson variety is the order complex of the corresponding Bruhat interval in W; this complex is known to be an ELshellable ball [BjornerWachs '82]. We prove that the projection of such a complex into the order complex of the Bruhat order on W/W_P is again a shellable ball. This requires extensive analysis of "PBruhat order", a generalization of the kBruhat order of [BergeronSottile '98]. In the case that G/P is minuscule (e.g. a Grassmannian), we show that its Grobner degeneration takes each projected Richardson variety to the StanleyReisner scheme of its corresponding ball.
 Publication:

arXiv eprints
 Pub Date:
 August 2010
 arXiv:
 arXiv:1008.3939
 Bibcode:
 2010arXiv1008.3939K
 Keywords:

 Mathematics  Algebraic Geometry;
 Mathematics  Combinatorics
 EPrint:
 New appendix, proving Richardon varieties have rational singularities in all characterisitcs. Many other minor edits, suggested by referees. Contains, and greatly improves on, material from arXiv:0903.3694