Projections of Richardson Varieties

By Allen Knutson, Thomas Lam, David E. Speyer. 2010

96 citations recorded by the source corpus.

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 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, 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 [Bjorner-Wachs '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 "P-Bruhat order", a generalization of the k-Bruhat order of [Bergeron-Sottile '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 Stanley-Reisner scheme of its corresponding ball.

Subfields: Discrete Mathematics and Combinatorics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/77cec46e3c2c9f5de78bb76a46c5cc2f0a8fc1b2

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Citations in the corpus

Singularities of Generalized Richardson Varieties (2010, methodology): “1Shortly after this paper was posted, Knutson, Lam and Speyer posted a preprint [31] independently proving Theorem 1.”
Cluster structures on strata of flag varieties (2014, result): “nd 1 only if v ≤w in the Bruhat order of W, and it is a smooth irreducible locally closed subset of Cw of dimension ℓ(w)−ℓ(v). More recently, Rv,w has sometimes been called an open Richardson variety [KLS1], because its closure in X is known as a Richardson variety [Rich]. Intersecting the decomposi”
Regularity theorem for totally nonnegative flag varieties (2019, background): “The map φ̄u, which we call an affine Bruhat atlas, sends the projected Richardson stratification [21] of G/P to the affine Richardson stratification of its image inside G/B.”
Positive geometries and canonical forms (2017, background): “was essentially established in [19], but in somewhat different language.”