Relative Schubert Filtration in Algebraic Geometry

• Relative Schubert Filtration is a canonical, combinatorially-indexed filtration that organizes local cohomology and D-module structures to reveal the geometry of Schubert varieties.
• It is constructed using the Grothendieck–Cousin complex and mixed Hodge modules, with strict spectral sequences that align graded pieces with geometric and representation-theoretic data.
• The filtration interfaces with parabolic Verma modules and uses combinatorial tools like augmented Dyck patterns and LS‐tableaux to index simple modules and standard monomials.

A relative Schubert filtration is a deep structural feature of several algebraic and geometric objects arising in the representation theory of algebraic groups, Schubert calculus, D-module theory, local cohomology, and standard monomial theory. It provides a canonical, typically combinatorially indexed, filtration whose subquotients or graded pieces reflect the geometry of Schubert varieties, the representation-theoretic combinatorics of Weyl groups, and mixed Hodge-theoretic or D-module invariants. The “relative” aspect refers to the filtration’s dependence on an intrinsic poset or stratification data, such as the Schubert cell decomposition or LS-tableaux, that encodes the interaction between closed subvarieties and algebraic invariants.

1. Mixed Hodge Module Structures and Local Cohomology with Schubert Support

Let $G$ be a complex semisimple group, $P\supset B$ a parabolic subgroup, and $X = G/P$ a generalized flag variety. For each Schubert variety $X_w\subset X$, the local cohomology modules $H^q_{X_w}(O_X)$ carry mixed Hodge module (MHM) structures in the sense of Saito, encompassing two canonical filtrations: the Hodge filtration $F^\bullet$ and the weight filtration $W_\bullet$.

These local cohomology sheaves are holonomic $D_X$-modules of finite length, with $B$-equivariance and pure graded pieces given by intersection cohomology modules $\operatorname{IC}^H(X_v)$ for $v\leq w$. This structure enables the calculation and interpretation of D-module invariants and representation-theoretic multiplicities through the lens of algebraic geometry (Perlman, 2024).

2. The Grothendieck–Cousin Complex and Its Filtration Properties

The foundational device for constructing the relative Schubert filtration on cohomology modules is the Grothendieck–Cousin (GC) complex, which is built from the affine paving of $X$ by Schubert cells. This yields a filtered complex whose terms consist of local cohomology sheaves with support in the differences of Schubert cell closures. Each term in the GC complex is naturally a MHM, and the differentials are strict with respect to both Hodge and weight filtrations.

By strictness, the associated spectral sequences for passing to the associated-graded degenerate at $E_1$, producing canonical isomorphisms:

• $$\operatorname{gr}^F_p H^q_{X_w}(O_X) \cong H^q(\operatorname{gr}^F_p\,GC^\bullet)_w$$
• $$\operatorname{gr}^W_k H^q_{X_w}(O_X) \cong H^q(\operatorname{gr}^W_k\,GC^\bullet)_w$$

This means that the relative Schubert filtration—the filtration on $H^q_{X_w}(O_X)$ induced from the GC complex—simultaneously refines geometric, combinatorial, and representation-theoretic structures (Perlman, 2024).

3. Filtration in Representation Theory: Parabolic Verma Modules and Jantzen Filtration

Globally, the relative Schubert filtration reflects explicitly in representation theory as a filtration on (dual) parabolic Verma modules $N_I(v)$. In this context, the graded pieces of the weight filtration are semisimple and determined by Kazhdan–Lusztig theory, and in type A are controlled by combinatorial objects such as augmented Dyck patterns.

Notably, for each highest weight, the filtration coincides (rigidly) with the Jantzen filtration, the socle filtration, and the radical filtration, so that the associated graded decomposes as a sum of simple modules $L(u)$ indexed by $u\leq v$, with multiplicity either 0 or 1 (Perlman, 2024).

4. Combinatorial Formulas: Dyck Patterns, LS-tableaux, and the Indexed Leaves

For Grassmannians, the structure of the relative Schubert filtration is governed by augmented Dyck pattern combinatorics. The set of Dyck patterns and their augmentations label the steps and graded pieces of the weight and Hodge filtrations. The main combinatorial result states that in the Grothendieck group of $D_X$-modules, the class $[H^q_a]$ for local cohomology supported in a Schubert variety $X_a$ is given by: $$[H^q_a] = \sum_{D\in A(a;q)} [L(a \setminus \operatorname{supp}(D))]$$ where $A(a;q)$ is the set of admissible augmented Dyck patterns in $a$ fitting codimension and bullet constraints.

The graded pieces of the weight filtration are then: $$\operatorname{gr}^W_p H^q_a = \bigoplus_{D\in A_p(a;q)} \operatorname{IC}^H_{X_{a\setminus \operatorname{supp}(D)}} \left(\tfrac{|a\setminus \operatorname{supp}(D)| - p}{2}\right)$$ This description is type-uniform: in more general settings, the leaves of the relevant filtration in the coordinate rings or algebras are indexed by Lakshmibai–Seshadri (LS) tableaux, which provide a basis for the standard monomial theory and a canonical indexing set for the one-dimensional subquotients of the filtration (Perlman, 2024, Müller, 2024).

5. Coordinate Rings, Standard Monomial Filtrations, and Seshadri Stratification

Schematically, the relative Schubert filtration generalizes to multihomogeneous coordinate rings of Schubert varieties embedded multiprojectively. The construction goes via Seshadri stratifications, which induce a chain of closed subvarieties indexed by a poset $A$. The resulting quasi-valuation $\mathcal{V}$ on the coordinate ring yields a filtration whose one-dimensional leaves are indexed by standard monomials associated to LS-tableaux.

For each multi-degree, the standard monomial basis, parameterized by these combinatorial tableaux, coincides with the set of leaves of the filtration. In type $A$, this gives a multigraded (or, for $m=1$, a classical Demazure-type) Schubert filtration (Müller, 2024).

6. Worked Examples and Explicit Calculations

In low-rank settings (e.g., $Gr(2,4)$), explicit calculation of the relative Schubert filtration reveals the effect of Koszul-type subcomplexes inside the GC complex, and provides a direct connection between graded pieces and simple modules, intersection cohomology, and explicit combinatorics of Dyck patterns.

For example, local cohomology $H^3_{X_{(2,1)}}(O_X)$ on $Gr(2,4)$ has a single pure summand $L(2,1)$ in weight 7, while $H^4_{X_{(2,1)}}(O_X)$ has $L(1,0)$ in weight 8. The filtration structure stays compatible when restricting to the opposite big cell, recovering the determinantal variety picture and corresponding cohomological invariants (Perlman, 2024).

7. Summary and Interconnected Structures

The relative Schubert filtration encompasses and connects multiple algebraic structures:

• The cohomology of the Grothendieck–Cousin complex with MHM and D-module filtrations,
• The representation-theoretic Jantzen and Kazhdan–Lusztig filtrations on parabolic Verma modules,
• Combinatorial indexing by Dyck patterns and LS-tableaux, producing standard monomial bases,
• Filtration and stratification structures on coordinate rings, realized geometrically via Seshadri stratification.

This filtration thus encapsulates the interplay between geometry of Schubert and determinantal varieties, combinatorics of Weyl groups and tableaux, and deep representation-theoretic categories, providing a powerful and unifying framework for the analysis of cohomological, D-module, and algebraic invariants (Perlman, 2024, Müller, 2024).