Richardson Varieties in Algebraic Geometry

• Richardson varieties are subvarieties formed by intersecting Schubert and opposite Schubert varieties in generalized flag varieties, linking algebraic geometry, combinatorics, and representation theory.
• They exhibit core geometric properties such as normality, Cohen–Macaulayness, and rational singularities, with smoothness characterized by precise cohomological criteria.
• Their generalizations—including intersection, projection, and rank varieties—serve as building blocks for advances in total positivity, Poisson geometry, and combinatorial studies in algebraic groups.

A Richardson variety is a subvariety of a generalized flag variety formed as the (schematic) intersection of a Schubert variety and an opposite Schubert variety. Introduced in the context of the geometric interpretation of the Littlewood–Richardson rule, these varieties serve as a natural geometric object linking the realms of algebraic geometry, combinatorics, and representation theory. Their geometric structure encodes central results in intersection theory on flag varieties, contributes to cohomological calculations, and provides building blocks for more general geometric constructs in the theory of algebraic groups.

1. Definition and Fundamental Properties

Let $G$ be a connected reductive (or symmetrizable Kac–Moody) group over an algebraically closed field, $B$ and $B_-$ be opposite Borel subgroups, $T=B\cap B_-$ a maximal torus, $W$ the Weyl group, and $G/P$ a generalized flag variety for a parabolic $P\subset G$. For elements $u,v \in W^P$ (minimal representatives in $W/W_P$), define:

• The Schubert variety $X(u) = \overline{B\cdot uP}/P$,
• The opposite Schubert variety $X^v = \overline{B_-\cdot vP}/P$.

The Richardson variety is the scheme-theoretic intersection: $R(u,v) := X(u)\cap X^v \subset G/P.$ In the case $G/B$, this can be realized as the intersection of two opposite cells or their closures. The main properties include:

• $R(u,v)$ is reduced, irreducible, and equidimensional when nonempty; it is nonempty if and only if $u \leq v$ in Bruhat order.
• In type $A$ and more generally, $R(u,v)$ is normal, Cohen–Macaulay, with rational singularities and well-understood cohomological invariants (Billey et al., 2010, Knutson et al., 2010).
• The open Richardson $R^\circ(u,v) = X^\circ(u)\cap X^{v\circ}$ is a locally closed subvariety (an open subset in both Schubert varieties) and is smooth if and only if a precise cohomological criterion is satisfied (see Section 4).

2. Generalizations: Intersection, Projection, and Rank Varieties

Richardson varieties serve as the prototypical case for broader families of subvarieties in flag varieties (Billey et al., 2010):

• Intersection varieties: Intersections of more than two (essentially transverse) translated Schubert varieties $g_1X(u_1)\cap\ldots\cap g_rX(u_r)$ in $G/P$. For $r=2$, one recovers Richardson varieties. When $r>2$, these intersection varieties have connected components that are normal and have rational singularities, as controlled by Kleiman's transversality theorem.
• Projection varieties: Given a Richardson variety in $G/P$ and a natural projection map $T_Q: G/P\rightarrow G/Q$ (for $Q\supset P$), the image is a projection variety. They form a stable class under further projections, are normal, have rational singularities, and are critical in geometric inductive constructions (e.g., Bott–Samelson resolutions, total positivity, Poisson geometry).
• Rank varieties (Grassmannian projection varieties): In $\operatorname{Gr}(k,n)$, a rank variety is defined by a collection of rank conditions encoded by a set of subspaces spanned by consecutive basis vectors. Theorem 4.5 in (Billey et al., 2010) implies that a subvariety is a projection variety if and only if it is a rank variety. These varieties lead to combinatorial and geometric descriptions that generalize Richardson varieties.

This hierarchy elevates projection and intersection varieties as the "fundamental blocks" for the geometry and combinatorics of generalized flag varieties.

3. Singularities and Smoothness Criteria

Richardson varieties and their generalizations have mild singularities:

• Normality and Rational Singularities: Both Richardson and projection varieties are normal and have rational singularities. This is deduced via vanishing higher cohomology for nef line bundles and the fact that projections from Richardson varieties preserve these properties [(Billey et al., 2010), Theorem 1.1].
• Describing Singular Loci: The singular locus of an intersection variety $g_1X(u_1)\cap\ldots\cap g_rX(u_r)$ is:

$$\bigcup_{i=1}^r \left((g_iX(u_i))_{\text{sing}} \cap \bigcap_{j\ne i}g_jX(u_j)\right)$$

(Proposition 2.8, (Billey et al., 2010)). For $r=2$, $R(u,v)$ is smooth if and only if $[X(u)_{\text{sing}}] \cup [X'(v)] = 0$ in cohomology (see Section 4).

• Rank and Projection Varieties: In the Type A Grassmannian, the singular locus of a projection/rank variety $X(M)$ is exactly the image of the singular locus of the minimal Richardson variety under the projection, together with the image of the exceptional locus. Exceptional loci have codimension at least $2$.
• Frobenius Splitting: Projection and Richardson varieties are compatibly Frobenius split (i.e., split with respect to the standard splitting), ensuring normality and extra vanishing theorems. This feature is crucial both in ordinary and positive characteristic (Knutson et al., 2010).

4. Cohomological and Intersection-Theoretic Aspects

Cohomology classes deeply control the geometry of Richardson varieties:

• Smoothness via Cup Products: The smoothness of $R(u,v) = X(u) \cap X'(v)$ is characterized by vanishing cup products:

$$R(u,v) \text{ smooth }\iff [X(u)_{\text{sing}}] \cup [X'(v)] = 0 \text{ in } H^*(G/P,\mathbb{Z})$$

This result generalizes to higher intersections (intersection varieties) via cup products of the singular loci classes. The singular locus is detected precisely by these cohomological intersections (Billey et al., 2010).

• Intersection Numbers and Structure Constants: Since Richardson varieties provide a geometric realization for Littlewood–Richardson and Schubert calculus structure constants, their combinatorics and singular loci have implications for explicit calculation of intersection numbers, positivity results, and multiplicity formulas. Methods from Kleiman's transversality theorem provide reliable criteria for nonvanishing and general-position intersections.
• Use of Kleiman’s Transversality Theorem: This theorem asserts that for suitable translates, intersections are transverse and their local tangent spaces intersect appropriately, enabling precise identification of singularity loci and their relation to Schubert varieties.

5. Combinatorial and Geometric Applications

Richardson varieties and their projections occupy a central role in several branches:

• Parametrization and Degenerations: In type A Grassmannians, rank conditions, standard monomial theory, and Schubert/Bruhat combinatorics yield explicit Plücker coordinate-based generators for Richardson and projection varieties (Billey et al., 2010). These descriptions facilitate explicit computation and decomposition, providing bases for homogeneous coordinate rings and explicit degenerations.
• Connections to Total Positivity and Poisson Geometry: Projection varieties stratify the totally nonnegative part of the flag variety, and compose the positroid stratification in the Grassmannian case, as developed by Lusztig, Postnikov, and others (Knutson et al., 2010). These stratifications have ramifications in cluster algebras, representation theory, and quantum groups.
• Further Research Directions: Open problems remain regarding explicit, type-independent descriptions of singular loci in arbitrary $G/P$, as well as the combinatorics of degenerations and $q$-enumerations (with connections to $q$-Stirling numbers). The stability of projection varieties under morphisms supports iterative geometric constructions (e.g., Bott–Samelson and toric degenerations).

6. Role in the Broader Theory and Further Implications

Richardson varieties unify and extend combinatorial and geometric structures in algebraic geometry:

• They provide the geometric realization for central combinatorial relations (Littlewood–Richardson rule) in the cohomology of flag varieties.
• Their singularity theory informs minimal model program investigations, vanishing theorems, and rationality questions.
• The interplay between geometry, cohomology, and combinatorics in their theory exemplifies the unity—and depth—of algebraic geometry and representation theory.
• Their projection and intersection generalizations suggest an even broader framework for interpreting intersection-theoretic and singularity-theoretic phenomena.

This integration of geometric, combinatorial, and cohomological methods gives a comprehensive understanding of the structure and role of Richardson varieties and their generalizations in modern algebraic geometry and representation theory.