Tableaux Skew Schubert Polynomials

• Tableaux skew Schubert polynomials are generating functions defined using tableau and inversion diagram models that encode skew elements of Weyl groups.
• They offer positive combinatorial expansions connecting double Schubert formulas with excited diagrams, standard Young tableaux, and classical symmetric functions.
• Recent developments extend Grassmannian cases to non-symmetric settings by employing inversion tableaux that capture Schubert structure constants.

Searching arXiv for the cited papers and directly related context. arxiv_search query: (Morales et al., 2017) Hook formulas for skew shapes III multivariate and product formulas arxiv_search query: (Tamvakis, 2020) Tableau formulas for skew Schubert polynomials arxiv_search query: (Axelrod-Freed, 15 Jul 2025) Inversions Tableaux Tableaux skew Schubert polynomials form a combinatorial framework in which Schubert polynomials attached to skew data are realized as generating functions over tableaux or tableau-like fillings. In the classical Lie types, skew elements of Weyl groups determine skew Young diagrams or their $k$-strict analogues, and the corresponding double Schubert polynomials admit positive expansions over bitableaux, tritableaux, and typed tritableaux (Tamvakis, 2020). Related type-$A$ product formulas arise when principal evaluations are expressed through excited diagrams and standard Young tableaux of skew shape (Morales et al., 2017). A distinct recent development defines tableaux-skew Schubert polynomials $\mathfrak S^t_{w/u}$ from skew inversion tableaux on staircase inversion diagrams, recovering skew Schur polynomials in the Grassmannian case while remaining generally non-symmetric (Axelrod-Freed, 15 Jul 2025).

1. Weyl-group indexing and skew diagrams

The basic input is a notion of a skew element in a Weyl group. In type $A$, a permutation $w\in S_\infty$ is called skew if there exist $m$ and $m$-Grassmannian permutations $u$ and $v$, corresponding to partitions $\lambda\supset\mu$ of length $A$0, such that $A$1 is a reduced factorization. The associated skew Young diagram is then $A$2. Here $A$3-Grassmannian means that $A$4 has exactly one descent at position $A$5 (Tamvakis, 2020).

In type $A$6, one fixes $A$7. A signed permutation is $A$8-Grassmannian if $A$9 for all $\mathfrak S^t_{w/u}$0, and such elements correspond to $\mathfrak S^t_{w/u}$1-strict partitions, meaning that no part $\mathfrak S^t_{w/u}$2 can repeat. If $\mathfrak S^t_{w/u}$3 are $\mathfrak S^t_{w/u}$4-strict partitions and $\mathfrak S^t_{w/u}$5 is reduced, then $\mathfrak S^t_{w/u}$6 is a compatible pair, $\mathfrak S^t_{w/u}$7 is a $\mathfrak S^t_{w/u}$8-horizontal strip, and the resulting element is skew of type $\mathfrak S^t_{w/u}$9 (Tamvakis, 2020).

In type $A$0, the indexing data are typed $A$1-strict partitions, where the type is $A$2, $A$3, or $A$4 according to how many times the part $A$5 appears. Compatible factorizations in the even-signed Weyl group $A$6 define typed $A$7-horizontal strips $A$8 and hence skew elements of type $A$9 (Tamvakis, 2020).

The ambient polynomial objects are defined in nilCoxeter algebras. In type $w\in S_\infty$0, the double Schubert polynomial $w\in S_\infty$1 is obtained as a coefficient in the product of $w\in S_\infty$2 and $w\in S_\infty$3; in type $w\in S_\infty$4, $w\in S_\infty$5 inserts a central $w\in S_\infty$6 factor; in type $w\in S_\infty$7, $w\in S_\infty$8 uses $w\in S_\infty$9. Type $m$0 is obtained from type $m$1 by

$m$2

where $m$3 is the number of negative entries of $m$4. All these families are stable under the natural inclusions $m$5, so they define formal power series in infinitely many $m$6 (Tamvakis, 2020).

These definitions organize skew Schubert theory around reduced factorizations rather than only around partitions. A plausible implication is that the skew diagram is not merely a shape parameter: it is a record of a factorization pattern in the Weyl group.

2. Tableau formulas in the classical types

Tamvakis gives tableau formulas for double skew Schubert polynomials in all four classical types. The proofs follow a common blueprint: expand the double Schubert polynomial in the nilCoxeter algebra, isolate factorizations into increasing and decreasing pieces together with a middle Stanley-type factor, and interpret the resulting reduced factorizations as fillings of the relevant skew diagram (Tamvakis, 2020).

For type $m$7, the tableaux are $m$8-bitableaux. Their entries lie in the ordered alphabet

$m$9

with primed letters marked and unprimed letters unmarked. Rows and columns are weakly increasing; marked letters are strictly increasing down columns; unmarked letters are strictly increasing across rows; and row $m$0 satisfies an interval condition determined by $m$1, $m$2, and $m$3. If

$m$4

then

$m$5

where the sum is over all $m$6-bitableaux $m$7 of shape $m$8 (Tamvakis, 2020).

For type $m$9, the tableaux are $u$0-tritableaux. Their alphabet has single-primed $u$1-letters, unmarked middle $u$2-letters, and double-primed $u$3-letters. Each fixed letter fills a $u$4-horizontal strip; the single-primed letters encode $u$5-strips, the double-primed letters encode $u$6-strips, and the unmarked letters form a $u$7-tableau $u$8. With

$u$9

the formula is

$v$0

The middle factor is the type-$v$1 Stanley function $v$2, expanded by a known type-$v$3 tableau rule (Tamvakis, 2020).

For type $v$4, the tableaux are typed $v$5-tritableaux. Their alphabet enlarges the type-$v$6 one by including unprimed and “$v$7” letters to encode the middle $v$8-strips together with a type $v$9 or $\lambda\supset\mu$0 choice. The fixed-letter condition is now that each letter occupies a typed $\lambda\supset\mu$1-horizontal strip, with extremality conditions for certain marked letters. If

$\lambda\supset\mu$2

then

$\lambda\supset\mu$3

Type $\lambda\supset\mu$4 inherits the same $\lambda\supset\mu$5-tritableaux model up to the global factor $\lambda\supset\mu$6 (Tamvakis, 2020).

Type	Tableau model	Generating formula
$\lambda\supset\mu$7	$\lambda\supset\mu$8-bitableaux	$\lambda\supset\mu$9
$A$00	$A$01-tritableaux	$A$02
$A$03	typed $A$04-tritableaux	$A$05

The significance of these formulas is twofold. First, they provide positive tableau expansions for skew Schubert representatives in all classical types. Second, they place type $A$06 alongside the symplectic and orthogonal cases in a single nilCoxeter-algebraic framework.

3. Single specializations and Grassmannian limits

The single skew Schubert polynomials are obtained by setting $A$07: $A$08 In the tableau formulas, this removes the $A$09-contributions. In type $A$10 one recovers the single bitableaux rule of Billey–Jockusch–Stanley, while in types $A$11 and $A$12 one recovers the single tritableaux formulas first proved by Tamvakis (Tamvakis, 2020).

The Grassmannian cases recover the standard symmetric-function families. If $A$13 has length $A$14 and $A$15 is $A$16-Grassmannian in $A$17, then

$A$18

the classical double Schur polynomial. In this specialization the type-$A$19 skew tableau rule reduces to the flagged bitableaux formula associated with Littlewood and Wachs (Tamvakis, 2020).

If $A$20 is $A$21-strict and $A$22 is $A$23-Grassmannian in type $A$24, then

$A$25

the Ikeda–Mihalcea–Naruse double theta polynomial. If $A$26 is typed $A$27-strict and $A$28 is $A$29-Grassmannian in type $A$30, then

$A$31

the double eta polynomial of Tamvakis (Tamvakis, 2020).

These Grassmannian specializations identify tableau skew Schubert formulas as genuine extensions of familiar tableau rules for Schur, theta, and eta polynomials. This suggests that the skew theory interpolates between ordinary Schubert combinatorics and classical symmetric-function theory.

4. Excited diagrams, principal evaluations, and product formulas

A complementary type-$A$32 theory connects skew Schubert polynomials to excited diagrams, hook formulas, and path enumerations. For a vexillary, equivalently $A$33-avoiding, permutation $A$34 of shape $A$35 and supershape $A$36, the Knutson–Miller–Yong formula gives

$A$37

and Macdonald’s identity implies

$A$38

Therefore, whenever the number of excited diagrams has a product formula, the principal evaluation of the Schubert polynomial has one as well (Morales et al., 2017).

For $A$39-avoiding permutations, the principal evaluation is explicitly related to standard Young tableaux of skew shape. If $A$40 is $A$41-avoiding of length $A$42 with reduced word $A$43 and corresponding skew shape $A$44, then Theorem 4.8 states

$A$45

Thus product formulas for $A$46 immediately induce product formulas for principal Schubert evaluations (Morales et al., 2017).

The proof mechanism uses two multivariate sums over excited diagrams,

$A$47

Lemma 3.3 identifies $A$48 with an evaluation of a factorial Schur function and hence shows symmetry in the $A$49-variables. Via Lindström–Gessel–Viennot, the sums become nonintersecting-path enumerators; the symmetry permits a boundary-path flip that yields the multivariate path identities of Theorems 3.6 and 3.7. After the specialization $A$50 and $A$51, one combines the Naruse hook-length formula with the MacMahon box formula to collapse the path sum to a closed product (Morales et al., 2017).

This branch of the subject is not a tableau rule in the direct semistandard sense, but it is closely allied. It replaces tableaux by excited diagrams and nonintersecting paths, while retaining skew-shape indexing and producing explicit product formulas for Schubert evaluations.

5. Inversion tableaux and tableaux-skew Schubert polynomials

A newer model defines Schubert polynomials directly on inversion diagrams. For $A$52, the inversion set is

$A$53

viewed inside the staircase of matrix-indexed boxes $A$54. An inversion tableau of shape $A$55 fills the shaded boxes of $A$56 by positive integers subject to three rules: the Rectangle Rule or weak-balance condition (IT1), column-strictness (IT2), and the row-bound condition (IT3) requiring the first-diagonal box $A$57, when shaded, to carry an entry $A$58. If $A$59 is the number of entries equal to $A$60, then $A$61 and $A$62. The generating theorem is

$A$63

This gives a tableau realization of ordinary Schubert polynomials that directly specializes to semistandard Young tableaux in the Grassmannian case (Axelrod-Freed, 15 Jul 2025).

If $A$64 in left weak Bruhat order, then $A$65, so one may form the skew inversion diagram $A$66. A skew inversion tableau of shape $A$67 fills exactly these boxes, with the boxes of $A$68 treated as empty, and satisfies the same three rules (IT1)–(IT3). The resulting tableaux-skew Schubert polynomial is

$A$69

This definition is explicitly distinct from the skew-element construction in Weyl groups: the indexing object is now a skew staircase inversion diagram rather than a skew Young diagram $A$70 (Axelrod-Freed, 15 Jul 2025).

Several structural properties are established. The family is stable under adding fixed points at the beginning of $A$71 and $A$72; the stable limit, analogous to the Stanley symmetric-function limit, is obtained by dropping (IT3). If both $A$73 and $A$74 are $A$75-Grassmannian, then

$A$76

and therefore

$A$77

Outside the Grassmannian case, the polynomials are generally not symmetric in the $A$78-variables (Axelrod-Freed, 15 Jul 2025).

The positivity theory is modeled on Schubert structure constants. If

$A$79

with $A$80, then equivalently

$A$81

The lexicographically largest monomial of $A$82 is $A$83, with coefficient $A$84, and the lexicographically minimal monomial is described via the column-Lehmer code. For dominant permutations, where the Lehmer code is weakly decreasing, the polynomial collapses to a single monomial, and in the skew-dominant case there is a unique skew inversion tableau. The paper also states that no closed-form determinant or Pfaffian is known in general beyond the Schur-Grassmannian case (Axelrod-Freed, 15 Jul 2025).

6. Comparison, applications, and scope

The subject contains several related but non-identical notions of a skew Schubert polynomial. One usage refers to double Schubert polynomials indexed by skew elements of Weyl groups and expressed by tableau formulas on skew Young diagrams or their $A$85-strict analogues (Tamvakis, 2020). A second usage concerns principal evaluations of certain type-$A$86 Schubert polynomials indexed by skew-shape data and controlled by excited diagrams, standard Young tableaux, and hook-type product formulas (Morales et al., 2017). A third defines the tableaux-skew Schubert polynomials $A$87 on skew inversion diagrams in the staircase (Axelrod-Freed, 15 Jul 2025).

These theories intersect most clearly in the Grassmannian regime. In type $A$88, Grassmannian skew data recover skew Schur polynomials, whether through flagged bitableaux on Young diagrams or through reverse semistandard tableaux arising from inversion tableaux (Tamvakis, 2020, Axelrod-Freed, 15 Jul 2025). Outside that regime, the distinctions become pronounced: the Weyl-group tableau formulas remain positive combinatorial expansions for double Schubert representatives, while $A$89 is generally non-symmetric and is designed to model Schubert structure constants directly (Axelrod-Freed, 15 Jul 2025).

The broader enumerative significance is reinforced by tiling models. The bijection between excited diagrams and lozenge tilings of a region with base $A$90 identifies

$A$91

so factorial-Schur symmetry yields Jacobi–Trudi-type determinantal formulas for weighted tilings. In the boxed case $A$92, one obtains a determinantal formula for weighted plane partitions in an $A$93 box, and Theorem 6.6 gives a determinantal expression for the probability of any fixed nonintersecting path in a random hook-weighted tiling of the hexagon (Morales et al., 2017). This suggests that skew Schubert combinatorics sits at an interface among Schubert calculus, tableau theory, symmetric functions, and exactly solvable tiling models.

A common misconception is that “skew Schubert polynomial” denotes a single canonical polynomial family. The literature represented here does not support that identification. Instead, it exhibits multiple constructions, each tailored to a different combinatorial or geometric problem: Weyl-group skew elements and classical-type tableaux, excited-diagram evaluations and product formulas, and inversion-diagram skew polynomials with Schubert-positive expansions. Their overlap in the Grassmannian case is substantial, but their general domains and structural properties are not the same (Tamvakis, 2020, Morales et al., 2017, Axelrod-Freed, 15 Jul 2025).