Projections of Richardson varieties

Abstract

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, Cohen–Macaulay, have rational resolutions, 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, Kumar–Mehta) 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 EL-shellable ball (Björner–Wachs). We prove that the projection of such a complex into the order complex of the Bruhat order on W / W P $W/W_P$ is again a shellable ball. This requires extensive analysis of “P-Bruhat order”, a generalization of the k-Bruhat order of Bergeron–Sottile. In the case that G/P is minuscule (e.g. a Grassmannian), we show that its Gröbner degeneration takes each projected Richardson variety to the Stanley–Reisner scheme of its corresponding ball.