Quantum Schubert Polynomials

• Quantum Schubert polynomials are polynomial representatives for quantum cohomology classes, incorporating quantum parameters that account for Gromov–Witten invariants.
• They are constructed using explicit algebraic methods such as transition equations and quantum divided difference operators, with combinatorial models like pipe dreams.
• Their structure bridges integrable systems, notably the Toda lattice, with algebraic geometry, enabling efficient computation of quantum products and intersection numbers.

Quantum Schubert polynomials arise at the intersection of algebraic combinatorics, integrable systems, and quantum cohomology, serving as polynomial representatives for quantum Schubert classes in the (quantum) cohomology rings of flag varieties and related homogeneous spaces. They generalize classical Schubert polynomials by incorporating quantum parameters reflecting Gromov–Witten invariants (counts of rational curves), and, through deep geometric and combinatorial constructions, connect with objects such as $k$-Schur functions, the Toda lattice, and representations of quantum groups. Quantum Schubert polynomials admit explicit algebraic, combinatorial, and geometric presentations and possess structural properties unifying several branches of modern mathematics.

1. Algebraic Definitions and Explicit Presentations

Quantum Schubert polynomials, denoted $\mathfrak{S}_w^q$, are deformations of classical Schubert polynomials indexed by permutations $w \in S_n$, depending on variables $x_1,\dots,x_n$ and quantum parameters $q_1,\dots,q_{n-1}$. In the non-equivariant case for the complete flag variety $\mathrm{Fl}_n$, these polynomials are characterized by the following:

• They are polynomial representatives for the quantum Schubert class $\sigma^w$ in the quantum cohomology ring $QH^*(\mathrm{Fl}_n)$.
• The ring $QH^*(\mathrm{Fl}_n)$ admits a presentation:

$$QH^*(\mathrm{Fl}_n) \cong \mathbb{Z}[x_1,\dots,x_n, q_1,\dots,q_{n-1}]/(E_1^n,\ldots,E_n^n)$$

where the $E_i^n$ are quantum elementary symmetric functions defined as the coefficients in

$$\det(I_n + \lambda\, M_{\mathrm{Fl}_n}) = \sum_{i=0}^n E_i^n \lambda^i$$

with $M_{\mathrm{Fl}_n}$ the tridiagonal matrix with diagonal $x_i$, $-1$ on the superdiagonal, and $q_i$ on the subdiagonal (Li et al., 22 Sep 2025).

• The quantum Schubert polynomials are determined recursively (transition equations) or via quantum divided difference operators, and geometric representatives via degeneration loci formulas.

The equivariant and double (i.e., in additional variable sets) versions, known as quantum double Schubert polynomials $\mathcal{G}_w(x; a)$, further depend on equivariant parameters and retain polynomiality in all variables (Lam et al., 2011).

2. Rational Substitution and the Affine–Quantum Dictionary

A central structural result is the identification of a rational algebraic map $\Phi$ that connects quantum Schubert polynomials to $k$-Schur functions (representatives for affine Schubert classes) (Lam et al., 2010, Lam et al., 2011): $$\Phi:\ \mathbb{Q}[x_1,\ldots,x_n, q_1, \ldots, q_{n-1}] \to \Lambda_{(n)}[s_{R_1}^{-1}, \ldots, s_{R_{n-1}}^{-1}]$$ with

$$x_1+\cdots+x_i \longmapsto \frac{s_{R_i'}}{s_{R_i}},\qquad q_i \longmapsto \frac{s_{R_{i-1}} s_{R_{i+1}}}{s_{R_i}^2}$$

where $s_{R_i}$ denotes the Schur function indexed by the rectangular partition $R_i$ and $R_i'$ by removing the outer corner. The map realizes quantum Schubert classes as rational expressions in symmetric functions, with the numerator the $k$-Schur function $s_{\lambda(w)}^{(k)}$ associated to $w$ and denominator a product over $s_{R_i}$ indexed by the descent set of $w$: $$\Phi(\mathfrak{S}_w^q) = \frac{s_{\lambda(w)}^{(k)}}{\prod_{i \in \operatorname{Des}(w)} s_{R_i}}$$ This bijective correspondence is fundamentally underpinned by Peterson's isomorphism between localized quantum cohomology and (localized) homology of the affine Grassmannian, and by explicit action–angle coordinates originating in Kostant's solution to the nilpotent Toda lattice (Lam et al., 2010, Lam et al., 2011).

3. Integrable Systems, the Toda Lattice, and the Geometry of Flag Varieties

Kostant's solution provides an explicit isomorphism between an open subset of solutions to the nilpotent Toda lattice (considered as a Hamiltonian system with Lax matrix $L(x,q)$) and a space $X^\circ$ in the centralizer of a principal nilpotent element in the Lie algebra of $\mathrm{PGL}_n$.

• The variables $x_i$ and $q_i$ in the quantum Schubert polynomial are understood as action–angle variables for the Toda lattice, with the associated Hamiltonians $H_k = \mathrm{tr}\, L^{k+1}/(k+1)$ vanishing on the relevant leaf.

The realization of quantum Schubert polynomials as pullbacks under $\Psi^*$ (the dual of Kostant's isomorphism) connects the Hamiltonian description of integrable systems with cohomological calculations. Symmetric functions such as $h_i$ and the special Schur functions $s_{R_i}$ naturally arise in the transition from Toda lattice variables to the field of algebraic combinatorics (Lam et al., 2010, Lam et al., 2011).

4. Quantum Schubert Polynomials in Partial Flags, Parabolic Cases, and Pfaffian Formulas

Quantum Schubert polynomials generalize to partial flag varieties $G/P$ via "parabolic" quantum double Schubert polynomials, defined relative to the block structure of the parabolic subgroup:

• For each minimal length coset representative $w$ in $W^P$, the parabolic quantum double Schubert polynomial $G^P_w(x;a)$ is given by a product of determinants of block-structured matrices $D_j$ that encode both the $x$ variables and the quantum parameters $q$ as prescribed by the parabolic type (Lam et al., 2011).
• For maximal isotropic Grassmannians (types B, C, D), quantum Schubert classes are represented by factorial $P$- and $Q$-Schur functions, with explicit Pfaffian expressions. The quantum and equivariant Giambelli formula is

$$[X_\lambda] = P_\lambda(x|t) = \operatorname{Pf} \left(P_{\lambda_i,\lambda_j}(x|t)\right)_{1 \leq i < j \leq r}$$

where $P_\lambda$ is the factorial Schur $P$-function corresponding to a strict partition $\lambda$, and the Pfaffian embodies quadratic relations among the generators reflecting Chern class identities of tautological bundles (Ikeda et al., 2014).

5. Representation Theory, Plactic Algebras, and Combinatorial Models

Quantum Schubert polynomials admit interpretations and explicit expansions in terms of plactic algebras, noncommutative Cauchy kernels, and combinatorial objects such as pipe dreams:

• The reduced rectangular plactic algebra provides a basis for Schubert-type polynomials via noncommutative Cauchy kernel decompositions, which can be $q$-deformed to yield quantum analogues (Kirillov, 2015).
• Quantum Schubert polynomials in this framework emerge as sums of quantum key polynomials indexed by (quantum) tableaux, corresponding to words in the $q$-deformed plactic algebra (Kirillov, 2015).
• Quantum bumpless pipe dreams constitute a combinatorial model for quantum double Schubert polynomials, yielding positive combinatorial formulas as binomial-weighted sums, and obeying transition equations aligning with the algebraic definition (Le et al., 24 Mar 2024).

The slide polynomial and Stanley symmetric function frameworks give further expansions (and are candidates for "quantum" extensions) with structural positivity and refined Littlewood–Richardson rules (Assaf et al., 2016).

6. Quantum, Equivariant, and $K$-theoretic Extensions

Quantum (double) Schubert polynomials represent Schubert classes not only in (quantum) cohomology, but also in equivariant and quantum $K$-theory settings:

• In the equivariant setting, specialization of universal double Schubert polynomials yields representatives for equivariant quantum Schubert classes, with an explicit Giambelli-type determinantal formula and Graham-positivity of structure constants (Anderson et al., 2011).
• In the torus-equivariant quantum $K$-theory of $\mathrm{Fl}_{n+1}$, quantum double Grothendieck polynomials, recursively generated by quantum Demazure operators, represent the (opposite) Schubert classes. The main explicit formula for the class corresponding to the longest element $w_0$ is

$$\mathfrak{G}^\mathrm{QG}(x,y) = \prod_{k=1}^{n} \left[ \sum_{\ell=0}^{k} (-1)^\ell (1-y_{n+1-k})^\ell F_k(x) \right]$$

where $F_k(x)$ encodes the quantum correction terms (Maeno et al., 2023).

The co-transition recurrence formulas for classical Schubert and Grothendieck polynomials in cohomology and $K$-theory have projectively dual analogues, and suggest pathways for recursion in quantum settings (Knutson, 2019).

7. Structural and Computational Consequences

The identification $$\Phi(\mathfrak{S}_w^q) = s_{\lambda(w)}^{(k)}/\prod s_{R_i}$$ provides a dictionary between quantum and affine Schubert calculus, enabling:

• Transfer of factorization rules, positivity, and combinatorial properties from $k$-Schur to quantum Schubert polynomials.
• New computational techniques for quantum products, Gromov–Witten invariants, and intersection numbers in quantum cohomology.
• Positive combinatorial rules for quantum structure constants via quantum Littlewood–Richardson and Monk formulas, e.g., through the Fomin–Kirillov algebra and Dunkl elements (Meszaros et al., 2012, Dalal et al., 2014).
• The "easy" computation of certain quantum Schubert polynomials corresponds to permutations avoiding classical patterns (such as 312 and 1432); in these cases, the quantum Schubert polynomial is simply obtained from a standard elementary monomial basis (Woodruff, 5 Mar 2025).

The underlying connection to integrable systems (nilpotent Toda lattice), and the use of combinatorial and representation-theoretic frameworks (Fock spaces, Heisenberg algebras, quantum cluster algebras) highlight the rich interplay between algebraic geometry, combinatorics, and mathematical physics in the paper of quantum Schubert polynomials.