Richardson Tableaux in Combinatorics and Geometry

• Richardson tableaux are special standard Young tableaux that satisfy explicit combinatorial conditions to encode irreducible components of Springer fibers.
• They bridge geometry and combinatorics by representing intersections of Schubert and opposite Schubert varieties in flag manifolds and totally nonnegative cells.
• Their enumeration connects to Motzkin numbers and refined q-analogs, underpinning advances in Schubert calculus, smoothness criteria, and toric degenerations.

A Richardson tableau is a special class of standard Young tableau characterized by explicit combinatorial and geometric conditions, central to the paper of the geometry of flag varieties, Springer fibers, and their totally nonnegative analogues. These tableaux encode the precise set of irreducible components of Springer fibers that are simultaneously obtained as Richardson varieties—that is, as intersections of Schubert and opposite Schubert varieties in the flag manifold GLₙ/B. Recent developments have provided both combinatorial criteria for their identification and elegant enumerative results, establishing connections to Motzkin paths and refined q-enumerations. Richardson tableaux also play a fundamental role in Schubert calculus, smoothness criteria, and toric degenerations. This article synthesizes major results and formalism related to Richardson tableaux, with emphasis on their definition, algebraic and geometric properties, and enumerative structures (Karp et al., 25 Jun 2025, Guo, 19 Nov 2025, Spink et al., 14 Oct 2025).

1. Combinatorial Definition and Characterizations

Given a partition $\lambda=(\lambda_1,\ldots,\lambda_\ell)\vdash n$, let $\sigma\in \mathrm{SYT}(\lambda)$ be a standard Young tableau of shape $\lambda$. The tableau $\sigma$ is called a Richardson tableau if for every $j$ with $r_\sigma(j) > 1$ ($r_\sigma(j)$ being the row of $j$ in $\sigma$), the last entry of $\sigma[j-1]$ in row $r_\sigma(j)-1$ exceeds every entry in rows $\geq r_\sigma(j)$. Here, $\sigma[j-1]$ denotes the subtableau formed by removing entries $\geq j$.

Equivalent formulations include:

• In terms of the Schützenberger evacuation involution, $\sigma$ is Richardson if and only if $\sigma^{\vee}$ is Richardson, and all evacuation slides are $L$-shaped paths—meaning each empty box in evacuation proceeds straight down the first column, then right to an outer corner.
• Recursively, a tableau is Richardson if after deleting the first row and renumbering, the resulting tableau is itself Richardson, and every entry $j$ in the second row has $j-1$ in the first row.
• The insertion tableaux produced by the Robinson–Schensted correspondence applied to all noncrossing involutions on $n$ letters are precisely the set of Richardson tableaux of size $n$ (Guo, 19 Nov 2025).

2. Geometric Interpretation and the Springer Fiber–Richardson Correspondence

In the geometric context, let $N$ be a nilpotent matrix of Jordan type $\lambda$, and $\mathcal{B}_N$ the Springer fiber over $N$ (the variety of Borel subgroups stabilized by $N$). The components of $\mathcal{B}_N$ are indexed by $\mathrm{SYT}(\lambda)$; the component corresponding to $\sigma$ is denoted $B_\sigma$.

Crucially, $B_\sigma$ is a Richardson variety—meaning $$B_\sigma = \overline{X_{v_\sigma} \cap X^{w_\sigma}}$$ for suitable $v_\sigma, w_\sigma \in S_n$—if and only if $\sigma$ is a Richardson tableau. This equivalence links the combinatorial conditions on the tableau to a geometric property: precisely those components of the Springer fiber that are Richardson varieties are the ones indexed by Richardson tableaux.

Moreover, in the totally nonnegative flag variety (Lusztig’s $Fl_n^{\geq 0}$), the top-dimensional cells of the totally nonnegative Springer fiber are in bijection with Richardson tableaux, and these cells correspond to cells of the form $\overline{R^{>0}_{v_\sigma, w_\sigma}}$; see (Karp et al., 25 Jun 2025, Guo, 19 Nov 2025).

3. Enumerative Structure and Motzkin Numbers

Richardson tableaux admit succinct and elegant enumerative formulas:

• Total number: The number of Richardson tableaux of size $n$ is the $n$th Motzkin number $M_n$, due to a bijection with noncrossing partial matchings (or Motzkin paths of $n$ steps). That is, $|\mathrm{RT}(n)| = M_n$ (Karp et al., 25 Jun 2025, Guo, 19 Nov 2025).
• Fixed shape: For a fixed partition shape $\lambda$, the count is given by a product of binomial coefficients:

$$|R_\lambda| = \prod_{i=1}^{\ell-1} \binom{\lambda_i + \lambda_{i+2} + \cdots + \lambda_\ell}{\lambda_{i+1} + \lambda_{i+2} + \cdots + \lambda_\ell}.$$

• q-Analogs and Refined Counts: Refined enumerations express the major index generating function over Richardson tableaux (for fixed shape or even/odd column constraints) in terms of $q$-Catalan and $q$-Narayana numbers, and conjectural expressions relate the $q$-enumeration by the number of odd columns to explicit $q$-binomial formulas (Guo, 19 Nov 2025).

4. Schubert Calculus and Cohomology Classes

The homology class of a Richardson variety, and hence that of the corresponding Springer fiber component, can be expressed via Schubert polynomials:

$$[\overline{R^{v_\sigma}_{w_\sigma}}] = \mathfrak{S}_{v_\sigma} \cdot \mathfrak{S}_{w_0 w_\sigma}$$

where $w_0$ is the long word in $S_n$ and $\mathfrak{S}_\ast$ denotes the Schubert polynomial. For each Richardson tableau $\sigma$, $[B_\sigma] = [\overline{R^{v_\sigma}_{w_\sigma}}]$ (Karp et al., 25 Jun 2025, Spink et al., 14 Oct 2025). In particular, the interval $[v_\sigma, w_\sigma]$ is "very well-aligned" in the Bruhat order, ensuring positivity and combinatorial tractability for Schubert expansions.

Explicit combinatorial rules for these structure constants in the homology of the flag manifold have been worked out in terms of column-strip decompositions of the tableau (Spink et al., 14 Oct 2025).

5. Structural Properties: Smoothness, Evacuation, and Algorithms

• Smoothness: Every Richardson variety $\overline{R^{v_\sigma}_{w_\sigma}}$ associated to a Richardson tableau is smooth, as attested by combinatorial checks on Bruhat graphs aligning with Deodhar’s and Billey–Coskun’s smoothness criteria (Karp et al., 25 Jun 2025, Spink et al., 14 Oct 2025).
• Evacuation-Invariance: The set of Richardson tableaux is stable under Schützenberger’s evacuation involution. This is mirrored at the geometric level by the Springer correspondence and, combinatorially, by the invariance of noncrossing involutions under conjugation by the longest element in $S_n$ (Guo, 19 Nov 2025).
• Recursive Structure: Richardson tableaux can be characterized recursively by cropping the first row and requiring each entry in the second row to have its predecessor in the previous row; prime Richardson tableaux (those not obtainable by row-concatenation) correspond to prime noncrossing involutions.
• Algorithmic and Representation-Theoretic Connections: Invariant subspace varieties of nilpotent operators are parametrized by "one-entry" Littlewood–Richardson tableaux; Richardson tableaux with all entries equal to 1 yield a monoid structure reflecting generic extension and degeneration order in the invariant subspace context (Kaniecki et al., 2018).

6. Connections to Toric Degenerations and Standard Monomial Theory

In the context of standard monomial theory for desingularizations of Richardson varieties, a basis of the homogeneous coordinate ring is provided by "w₀-standard" tableaux—combinatorial objects akin to Richardson tableaux, defined by liftings containing reduced expressions for the long element. These tableaux parameterize explicit toric degenerations in the Grassmannian case, and a bijection with appropriate semi-standard Young tableaux is established via matching field combinatorics (Balan, 2011, Bonala et al., 2021).

7. Illustrative Examples and Special Shapes

Key examples and shapes include:

Shape $\lambda$	Richardson Tableaux Count	Corresponding Motzkin Path Structure
$(2,2)$	$1$	Unique; cell structure via noncrossing match.
$(3,1)$	$3$	Binomial: $\binom{3}{1}=3$
size $n=4$	$9 = M_4$	All Motzkin paths of length 4
$(k,1^{n-k})$ (hook)	all SYT of shape are Richardson; cohomology via G\"uemes’ rule	-

For each, the unique geometric and Schubert-theoretic features reflect the combinatorics of their defining conditions (Karp et al., 25 Jun 2025, Spink et al., 14 Oct 2025).

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References: Primary sources (arXiv ids) for this exposition are (Karp et al., 25 Jun 2025, Guo, 19 Nov 2025, Spink et al., 14 Oct 2025, Balan, 2011, Bonala et al., 2021), and (Kaniecki et al., 2018). See these for detailed proofs, full bijections, and further generalizations.