RSK Correspondence: Combinatorics and Geometry

• RSK correspondence is a bijection that maps words, matrices, or permutations to pairs of Young tableaux, serving as a bridge between combinatorics and geometry.
• Modified insertion algorithms, including adjusted bumping rules and standardization processes, extend RSK to partial flag varieties using semistandard tableaux.
• These adaptations unify classical and modern perspectives, enhancing our understanding of flag varieties, nilpotent orbits, and the interplay of algebra, combinatorics, and geometry.

The Robinson-Schensted-Knuth (RSK) correspondence is a foundational combinatorial bijection that associates words, matrices, or permutations to pairs of Young tableaux of the same shape, thereby establishing a deep connection between the representation theory of symmetric groups, algebraic geometry, and combinatorics. The classical RSK correspondence and its modern extensions not only organize and enumerate combinatorial structures but also encode geometric and algebraic data via insertion algorithms and tableau parametrizations. Recent developments have seen the RSK correspondence generalized to the geometry of partial flag varieties, where the construction and its consequences reveal new links between the geometry of nilpotent orbits and explicit combinatorial data in the form of semistandard tableaux, as well as modifications of the insertion process to accommodate richer geometric settings.

1. Classical RSK Correspondence and Complete Flag Varieties

In its classical version, the RSK correspondence provides a bijection between permutations in the symmetric group $S_d$ (or more generally, between words, biwords, and nonnegative integer matrices) and pairs of standard (or semistandard) Young tableaux of the same shape. This map is tightly interwoven with the geometry of the full flag variety $\mathcal{F}$ of a $d$-dimensional complex vector space $V$: $$\mathcal{F} = \{ 0 = F_0 \subset F_1 \subset \dots \subset F_d = V \}.$$ When considered in conjunction with a nilpotent operator $x \in \operatorname{End}(V)$ of Jordan type $\lambda$, one studies the subvariety

$$\mathcal{F}_x = \{ F \in \mathcal{F} \mid x(F_i) \subset F_{i-1} \}$$

of $x$-stable flags. The irreducible components of $\mathcal{F}_x$ are indexed by standard Young tableaux of shape $\lambda$. For two generic flags $F,G$ in components indexed by tableaux $T$, $S$, the relative position $w(F,G)$—viewed as a permutation—is exactly the permutation associated to $(T,S)$ under the classical RSK bijection (Rosso, 2010, Rosso, 2010). Thus, the RSK correspondence serves as a combinatorial dictionary encoding how pairs of irreducible components intersect in the flag variety.

2. Generalization to Partial Flag Varieties

The RSK correspondence admits a nontrivial refinement in the setting of partial flag varieties. For a composition $p = (p_1, \dots, p_n)$ with $\sum p_i = d$, the corresponding partial flag variety is

$$\mathcal{F}^p = \{ 0 = F_0 \subset F_1 \subset \dots \subset F_n = V \mid \dim(F_i/F_{i-1}) = p_i \}.$$

Given a nilpotent operator $x$, the subvariety of $x$-stable partial flags analogously is

$$\mathcal{F}_x^p = \{ F \in \mathcal{F}^p \mid x(F_i) \subset F_{i-1} \}.$$

The parametrization of irreducible components now involves semistandard tableaux (specifically, strict in rows and weak in columns—and transposed relative to the usual convention), constructed by "adding boxes in chunks", i.e., adding several boxes (with at most one per row) at each step and filling each new box with the same integer (see Definitions 2.4 and 2.5 of (Rosso, 2010)). This reflects the partial flag type $p$ in the content of the tableau.

For two generic partial flags $F, G$ in components indexed by semistandard tableaux $T, S$ of shape $\lambda$ and content $p$, the relative position becomes a nonnegative integer $n \times n$ matrix: $$M(F,G)_{ij} = \dim \left( \frac{F_i \cap G_j}{F_i \cap G_{j-1} + F_{i-1} \cap G_j} \right).$$ This matrix replaces the permutation (which is a special case when $p = (1,1,\dots,1)$), capturing the refined intersection data appropriate for the partial flag context (Rosso, 2010, Rosso, 2010).

3. Modified Insertion Algorithms and Standardization

The combinatorial algorithm required to recover the pair of tableaux from the relative position matrix is a variant of RSK. The essential modification is in the row-bumping rule during insertion: instead of bumping the leftmost entry strictly greater than a new value $z$ (as in classical RSK), the algorithm now bumps the leftmost entry greater than or equal to $z$. This alteration is crucial: it ensures that the output tableaux have strictly increasing rows and weakly increasing columns, matching the tableau conditions required to parameterize irreducible components of $\mathcal{F}_x^p$ (Rosso, 2010, Rosso, 2010).

An explicit standardization procedure maps semistandard tableaux (and two-rowed arrays) to standard ones. A key lemma (Lemma 3.8 in (Rosso, 2010)) establishes that standardization commutes with the RSK: one may "lift" to the complete flag (standard tableau) case, perform calculations there, and "descend" back to the partial flag situation. This reduction greatly simplifies both the combinatorial and geometric aspects of the analysis.

4. Geometric and Combinatorial Implications

With these modifications, the main theorem (Theorem 4.1 in (Rosso, 2010), Theorem 4.1 in (Rosso, 2010)) asserts that for generic partial flags $F \in C_T$, $G \in C_S$ (with $C_T, C_S$ irreducible components indexed by semistandard tableaux $T, S$), the relative position matrix satisfies

$M(F,G) = M(T,S),$

where $M(T,S)$ is the matrix output by the modified RSK applied to $(T,S)$. This result generalizes the classical parametrization (in terms of permutation matrices for complete flags and standard tableaux) to the much broader context of partial flags and semistandard tableaux.

Some significant consequences and new perspectives include:

• The geometry of partial flag varieties supports an RSK-type correspondence provided the insertion algorithm is suitably adapted.
• The bumping rule modification is essential: it reflects the correct combinatorial data for components of $\mathcal{F}_x^p$.
• Standardization enables reduction to the complete flag case, providing a uniform framework for both classical and partial flag settings.
• This unifies geometric results due to Spaltenstein, Steinberg, and Haines in a combinatorial language, opening new avenues in geometric representation theory and combinatorics (Rosso, 2010, Rosso, 2010).

5. Mirabolic RSK and Further Generalizations

An additional development elaborated in (Rosso, 2010) is the mirabolic RSK, extending these ideas to parametrizations involving pairs of partial flags and a line. Here, the action is on $\mathcal{F}^p \times \mathcal{F}^{p'} \times V$, and orbits are classified by a pair $(M, A)$, with $M$ as above and $A$ ("decorated" data) encoding the position of a line with respect to the flags.

The mirabolic RSK correspondence assigns "decorated arrays" to orbits and realizes an explicit combinatorial algorithm (again, a variant of classical RSK with distinct insertion and bumping procedures) that produces tableau data signifying geometric invariants—aligning the geometry of conormal bundles to orbits with combinatorial structures.

Such generalizations enable refined parametrizations in contexts relevant to:

• Convolution algebras of equivariant functions on spaces of partial flags.
• Identification and description of central elements and structural features in these algebras.

6. Summary Table: RSK in Flag Varieties

Flag Type	Parametrization of Irreducible Components	Relative Position Data	Bumping Rule in Insertion
Complete	Standard Young tableau of shape $\lambda$	Permutation matrix	$> \;$ (strict bumping)
Partial	Semistandard tableau ("transpose" type)	Nonnegative integer matrix	$\geq \;$ (weak bumping)

All statements in the table refer to (Rosso, 2010, Rosso, 2010).

7. Unified Framework and Future Directions

The resilience of the RSK framework—its capacity to be suitably modified for partial flag varieties, and its compatibility with powerful techniques such as standardization—demonstrates the ubiquity of combinatorial bijections in geometric representation theory. These connections deepen understanding of orbit parametrizations, decompositions of varieties, and intersections in invariant theory.

The adoption of modified insertion rules is not a mere technicality, but rather a fundamental adaptation required to synchronize the geometric data (e.g., stabilizing subspaces and their intersections) with the combinatorial objects parametrizing irreducible components. These geometric-combinatorial correspondences also suggest new strategies for investigating orbit closures, characteristic cycles, and intersection patterns in more general geometric representation-theoretic contexts.

This synthesis of geometric and combinatorial ideas highlights the robustness of the RSK framework and its adaptability to various geometric contexts beyond the classical case (Rosso, 2010).