Quantum Double Schubert Polynomials

• Quantum double Schubert polynomials are explicit representatives of equivariant quantum Schubert classes, blending classical, double, and quantum deformations.
• They are constructed via equivariant divided difference operators and a stable quantization map that incorporates quantum parameters and Chevalley–Monk rules.
• They extend to parabolic analogues for partial flag varieties and connect with affine Schubert calculus through Toda lattice transformations, ensuring structural unification.

Quantum double Schubert polynomials are a canonical family of explicit polynomials that represent equivariant quantum Schubert classes in the torus-equivariant quantum cohomology rings of complete and partial flag varieties. These polynomials provide a bridge connecting combinatorial constructions (notably generalizations of divided difference operators and pipe dream models) with the algebraic and geometric structure of quantum cohomology. Their definition refines and quantum-deforms classical and equivariant double Schubert polynomials by incorporating quantum parameters, and their algebraic properties mirror and extend those in classical Schubert calculus, notably through quantum analogues of the Chevalley–Monk and Giambelli formulas. Quantum double Schubert polynomials additionally admit parabolic analogues that describe Schubert classes in partial flag variety cohomology and are interconnected with affine Schubert calculus through rational transformations mediated by solutions to the Toda lattice.

1. Classical, Double, and Quantum Double Schubert Polynomials

Schubert polynomials $\mathcal{S}_w(x)$, indexed by permutations $w \in S_n$, represent the cohomology classes $[X_w]$ of Schubert varieties in $H^*(\mathrm{Fl})$. The double Schubert polynomials $\mathcal{S}_w(x; a)$, introduced by Lascoux and Schützenberger, refine these to represent equivariant classes $[X_w]_T \in H_T^*(\mathrm{Fl})$, with the parameters $a_i$ encoding the action of the torus $T$.

Quantum deformations arise in the context of quantum cohomology, specifically $QH^*(\mathrm{Fl})$, which incorporates counts of rational curves in the flag variety via quantum parameters $q = (q_1, \ldots, q_{n-1})$. The quantum Schubert polynomials of Fomin–Gelfand–Postnikov give explicit representatives for Schubert classes in the quantum ring. The “quantum double Schubert polynomials” $\widetilde{\mathcal{S}}_w(x; a)$, as constructed in (Lam et al., 2011), combine equivariant and quantum deformations:

• They are obtained from double Schubert polynomials by a stable quantization map $\varphi$, i.e., $$\varphi(\mathcal{S}_w(x; a)) = \widetilde{\mathcal{S}}_w(x; a)$$.
• In Kim's presentation, the quantum equivariant cohomology ring is realized as $QH_T^*(\mathrm{Fl}) \cong S[x; q]/J_{q, a}$, with $S = \mathbb{Z}[a]$ and $J_{q, a}$ a determinantal ideal, where $q$ tracks degrees of rational curves.
• Under this identification, the quantum double Schubert polynomials provide explicit polynomial representatives for the quantum Schubert basis:

$$\sigma_w \leftrightarrow \widetilde{\mathcal{S}}_w(x; a),\quad \text{in}\ QH_T^*(\mathrm{Fl}).$$

For a simple reflection $s_i$, $$\widetilde{\mathcal{S}}_{s_i}(x; a) = \mathcal{S}_{s_i}(x; a) = (x_1 + \dots + x_i) - (a_1 + \dots + a_i)$$.

2. Algebraic and Combinatorial Definition

Quantum double Schubert polynomials are defined recursively via equivariant divided difference operators, with quantum deformation governed by the stable quantization map. The defining properties include:

• Divided differences: The recursion on permutations via divided difference operators in the $x$ and $a$ variables, commuting with the quantization map.
• Quantization: The stable quantization map modifies the relations satisfied by classical double Schubert polynomials, introducing quantum parameters into the recursion and the product structure.

The quantum double Schubert polynomials satisfy a quantum, equivariant Chevalley–Monk rule (see [(Lam et al., 2011), Proposition 13]). For $w \in S_n$ and simple reflection $s_i$,

$$\widetilde{\mathcal{S}}_{s_i}(x; a)\,\widetilde{\mathcal{S}}_{w}(x; a) = \left(-w\cdot w_i(a) + w\cdot w_i(a)\right) \widetilde{\mathcal{S}}_{w}(x; a) + \sum_{\alpha \in A_w} \langle \alpha^\vee, w_i \rangle\, \widetilde{\mathcal{S}}_{ws_\alpha}(x; a) + (\text{quantum terms}),$$

where $w_i(a) = a_1 + \cdots + a_i$, and $A_w$ is a set of positive roots parametrizing covers in the Bruhat order. The quantum terms are polynomials in the $q_i$ and encode Gromov–Witten invariants.

3. Parabolic Analogues for Partial Flag Varieties

For a partial flag variety $\mathrm{SL}_n/P$ defined by a parabolic subgroup $P$ (corresponding to a composition $(n_1, \dots, n_k)$), the quantum product involves additional corrections not present in the classical “functoriality.” The authors of (Lam et al., 2011) define parabolic quantum double Schubert polynomials $P^P_w(x; a)$, indexed by minimal length coset representatives $w \in W^P$, by a determinantal formula with block structure reflecting the parabolic subgroup: $P^P_w(x; a) = (-1)^? \det(D_j - a_? I_?),$ where entries of the matrix $D$ encode the roles of $x$ and $q$-variables across the blocks. These polynomials constitute explicit representatives for the Torus-equivariant quantum Schubert classes in $QH_T^*(\mathrm{SL}_n/P)$ under the natural quotient isomorphism,

$QH_T^*(\mathrm{SL}_n/P) \cong S[x; q]/J^P_{q, a},$

with $J^P_{q, a}$ the block-structured generalization of the ideal in the complete flag case.

4. Structural Properties and Product Rules

Quantum double Schubert polynomials inherit and extend several key properties from their classical counterparts:

• Basis Property: The set $\{\widetilde{\mathcal{S}}_w(x; a): w \in S_n\}$ (or $\{P^P_w(x; a): w \in W^P\}$ for a parabolic) forms a $Z[q, a]$-basis of $S[x; q]/J_{q, a}$.
• Chevalley-Monk Rules: The polynomials satisfy product rules mirroring equivariant and quantum Chevalley-Monk formulas, with explicit combinatorial corrections for quantum parameters.
• Unification via Universal Polynomials: In the equivariant quantum Giambelli formula for partial flag varieties, the representatives are specializations of the universal double Schubert polynomials of Fulton (Anderson et al., 2011).
• Positivity: The structure constants in the quantum cohomology ring—given as coefficients in the expansions of products of quantum double Schubert polynomials—exhibit Graham-positivity: all coefficients are non-negative polynomials (in negative roots) reflecting enumerative geometry (Anderson et al., 2011).
• Functorial Properties: For smooth Schubert divisors $X \subset \mathrm{Fl}_n$, the quantum Schubert polynomials representing classes in $QH^*(X)$ are identical to those for the complete flag variety, despite nontrivial changes to the quantum product and ring presentation (Li et al., 22 Sep 2025).

5. Connections with Affine Schubert Calculus and the Toda Lattice

Quantum double Schubert polynomials are in explicit rational correspondence with $k$-double Schur functions ($k = n-1$), which represent torus-equivariant Schubert classes in the homology of the affine Grassmannian $\mathrm{Gr}_{\mathrm{SL}_n}$ (Lam et al., 2011). Via Kostant’s explicit solution to the (generalized) Toda lattice, variables $(x, q)$ in the quantum double Schubert polynomials become certain rational functions in symmetric function parameters $(y; a)$:

• For instance, the change of variables is given by

$$x_1 + \cdots + x_i \mapsto a_1 + \cdots + a_i + \frac{S_{R_i^t}(y\parallel a)}{S_{R_{i-1}^t}(y\parallel a)},$$

$$q_i \mapsto \frac{S_{R_{i+1}^t}(y\parallel a)\, S_{R_{i-1}^t}(y\parallel a)}{S_{R_i^t}(y\parallel a)^2},$$

where $S_{R_i^t}(y \parallel a)$ are dual Schur functions for rectangular partitions. This correspondence is tightly coupled to the isomorphism between the quantum cohomology of the full flag and the homology of the affine Grassmannian mediated by the geometry of the Toda lattice.

6. Approaches and Comparative Methods

There are two principal constructions for the (equivariant) quantum Schubert class representatives:

• Algebraic–Combinatorial Approach: The polynomials are constructed via explicit formulas using divided difference operators, stable quantization maps, and block determinantal forms (Lam et al., 2011). This approach allows for detailed combinatorial and algebraic formulae and recursive characterizations.
• Geometric Approach: Independent constructions, notably by Anderson and Chen, employ the geometry of Quot schemes (parameter spaces for stable maps and compactifications of moduli spaces) to produce polynomial representatives for quantum Schubert classes by geometric means. These constructions are logically distinct from, but ultimately convergent with, the algebraic-combinatorial presentations (Lam et al., 2011).

Both methods confirm that the quantum double Schubert polynomials, as constructed combinatorially, represent the correct quantum Schubert classes by verification of Chevalley–Monk and Cauchy-type product formulas, and by graded dimension counts in the relevant rings.

7. Extensions, Unification, and Future Directions

Quantum double Schubert polynomials unify and extend the theory of Schubert polynomials to encompass quantum and equivariant deformation:

• Via specializations and further generalizations, they connect with the factorial $P$- and $Q$-Schur functions (for types $B$, $C$, $D$) and to “master” universal double Schubert polynomials (Ikeda et al., 2014, Anderson et al., 2011).
• They admit positivity properties, both in the sense of Newton polytope saturation (every exponent in the Newton polytope appears) (Castillo et al., 2021) and in terms of the Graham-positivity of structure constants (Anderson et al., 2011).
• Their combinatorial realizations are being actively developed in terms of (quantum) bumpless pipe dreams (Le et al., 24 Mar 2024), and their algebraic structure is closely tied to cluster algebra and quantum group frameworks (Jakobsen, 2015).

This robust algebraic-combinatorial-geometric synthesis advances the understanding of quantum deformation phenomena in Schubert calculus and opens further avenues in the paper of quantum cohomology rings, quantum cluster algebras, and integrable systems.