Elliptic Schubert Calculus: Dualities & Combinatorics

• Elliptic Schubert Calculus is the study of intersection theory and algebraic structures of Schubert varieties using equivariant elliptic cohomology.
• It employs twisted group algebras, elliptic Demazure–Lusztig operators, and theta functions to capture refined geometric and representation-theoretic phenomena.
• The theory extends classical Schubert calculus by unifying combinatorial models, duality theorems, and singularity insights with applications in mirror symmetry and quantum groups.

Elliptic Schubert calculus is the paper of intersection theory and algebraic/combinatorial structures associated to Schubert varieties in the context of equivariant elliptic cohomology. This theory generalizes classical Schubert calculus and its quantum, K-theoretic, and connective extensions by incorporating modular/elliptic parameters, thereby encoding refined geometric and representation-theoretic phenomena. Elliptic Schubert calculus unifies and extends several strands of modern geometry and combinatorics, making essential use of oriented cohomology theories, Demazure and Hecke algebras, theta functions, and deep ideas from mirror symmetry, representation theory, and singularity theory.

1. Algebraic Framework: Twisted Group Algebras and Demazure-Lusztig Operators

Elliptic Schubert calculus is formulated through the algebraic machinery of twisted group algebras and elliptic analogues of Demazure–Lusztig operators. The central object is the elliptic twisted group algebra $Q_{W \times W}$, built to accommodate two commuting Weyl group actions—one geometric, one "dynamical"—with structure constants given in terms of Jacobi theta functions. A typical operator takes the form

$T_w = \sum_{v \leq w} a_v^{(w)} \cdot \_v,$

with $$a_v^{(w)} = \prod_{\alpha \in \Sigma_w} \frac{\theta(\hbar - z_\alpha) \theta(_\alpha^\vee)}{\theta(z_\alpha)\theta(\hbar - _\alpha^\vee)}$$. Here, the parameters $\hbar$ (Planck parameter) and $z_\alpha$ (torus weights) encode the elliptic deformation.

The elliptic Demazure–Lusztig operators $T_i^\pm$ are defined as endomorphisms acting on suitable modules:

• $(T_i)^2 = 1$ and the braid relations are satisfied.
• These operators generalize the classical divided difference operators, making explicit use of theta functions.

Elliptic Schubert classes are then constructed as

$\mathbb{E}_w = T_{w^{-1}} \bullet f_e,$

where $f_e$ is the identity element and $\bullet$ denotes the twisted action. Opposite elliptic Schubert classes are similarly defined using $T^-_{w^{-1}w_0}$.

2. Poincaré Duality for Elliptic Schubert Classes

A central result is the establishment of a Poincaré duality pairing between elliptic Schubert classes and their opposites. This pairing is defined using a sum over Weyl group elements and theta-function-weighted terms: $$Y_\Pi := \sum_{v, w \in W} {}_v {}_w \prod_{\alpha > 0} \theta(z_\alpha)^{-1} \theta(\hbar - z_\alpha),$$ which can be compactly written as $Y_\Pi = (\sum_{v \in W} {}_v ) \cdot Y'_\Pi$ with $Y'_\Pi$ acting via the twisted action.

The duality theorem states: $$Y_\Pi \bullet (\mathbb{E}_w \cdot \mathbb{E}_v^-) = \delta_{w,v} \sum_{u \in W} u \left( \frac{\theta_\Pi(\cdot)}{\theta_\Pi(\hbar - \cdot)} \right).$$ This ensures that the constructed elliptic Schubert classes and their opposites form dual bases under this pairing, encoding a Frobenius algebra structure analogous to that seen in singularity theory and classical Schubert calculus.

An anti-involution $\iota$ on the twisted group algebra, satisfying $\iota(T_i) = T_i^-$, establishes an adjunction property. Explicitly,

$$Y_\Pi \bullet((T_i \bullet f) \cdot f') = Y_\Pi \bullet(f \cdot (T_i^- \bullet f')),$$

relating the geometry of Schubert calculus to representation theory at the elliptic level.

3. Parabolic Schubert Calculus and the Elliptic Poincaré Pairing

For partial flag varieties $G/P$, the theory restricts to minimal coset representatives $W^P$. Elliptic classes are then elements of a localized module,

$$\mathcal{M}^P = K_T(G/P)[H_2(G/P;\mathbb{Z}) \oplus \mathbb{Z}\hbar][[q]],$$

with localization at torus fixed points $W^P$. The parabolic elliptic Poincaré pairing generalizes the full flag variety as: $$\langle f, g \rangle_{G/P} = \sum_{u \in W^P} \frac{f(u) g(u)}{\prod_{\alpha \in \Phi^+ \setminus \Phi_P^+} \theta(-z_u)},$$ where only the non-parabolic positive roots contribute to the denominator.

Upon suitable renormalization—using products over roots mapping to negatives under $w$ and explicit theta-function factors—parabolic elliptic Schubert classes $\mathbb{E}'(X^P_w)$ are dual to the Schubert classes $E_w^P$ with respect to this pairing: $$\langle \mathbb{E}'(X^P_u), E^P_w \rangle_{G/P} = \delta_{u, w}.$$ This is proven by reduction to the full flag case and by showing compatibility with the R-matrix recursion and normalization conditions.

4. Combinatorial and Computational Models

Two fundamental combinatorial tools are established in elliptic Schubert calculus for explicit computations and positivity phenomena:

• Elliptic Billey-type localization formula: The localization of elliptic Schubert classes at $T$-fixed points is computable via explicit formulas that generalize the Billey formula from cohomology and $K$-theory. This involves summing over combinatorial data associated to the Weyl group, incorporating theta functions corresponding to the roots.
• Pipe Dream Models: In type $A$, polynomial representatives for elliptic Schubert classes admit a pipe dream interpretation, extending classical models for double Schubert and Grothendieck polynomials. The pipe dreams encode the action of Demazure operators in terms of explicit diagrams, and their weights are promoted from polynomials to rational functions of theta functions, reflecting the elliptic deformations.

5. Connections to Singularity Theory and Universal Deformations

Elliptic Schubert calculus fits into the general paradigm of the universal deformation of Schubert calculus. Starting from a universal Landau–Ginzburg potential $V$ whose Jacobi ring encodes the classical cohomology ring, one considers the versal deformation: $V_\text{def} = V + \sum_k t_k \phi_k,$ where specialization of the deformation parameters to elliptic (modular) functions yields the elliptic theory. The cohomology is realized as a Jacobi ring,

$$J(V_\text{elliptic}) = \mathbb{C}[x_1, \ldots, x_n]/(\partial_{x_1} V_\text{elliptic}, \ldots, \partial_{x_n} V_\text{elliptic}),$$

so that the structure constants and intersection numbers in elliptic Schubert calculus are accessible via residue computations involving the Hessian determinant of $V_\text{elliptic}$ (Gorbounov et al., 2010).

This singularity-theoretic perspective confirms the persistence of the Frobenius algebra structure and identifies the algebraic deformations underlying the passage from classical to elliptic cohomology.

6. Positivity, Recursion, and Degeneration to K-theory

Elliptic Schubert calculus extends positivity properties and recursive structures known from cohomology and $K$-theory:

• Positivity and Duality: The parabolic duality for elliptic Schubert calculus ensures that geometric bases and operator bases are dual under the elliptic Poincaré pairing, incorporating explicit theta-function denominators but preserving combinatorial positivity.
• Degeneration: As the elliptic parameter specializes, the formulas, polynomials, and pipe dream models degenerate to their $K$-theoretic and cohomological analogues, recovering classical results and demonstrating that the elliptic calculus provides a simultaneous refinement of previous theories (Lenart et al., 5 Oct 2025).
• Recursion and Hecke/Quantum Group Symmetry: The recursion for elliptic Schubert classes via Demazure–Lusztig or R-matrix operators reflects the underlying Hecke algebra and quantum group symmetries—these play a crucial role in categorification and in connections to integrable systems (Rimanyi et al., 2019, Kumar et al., 2019).

7. Broader Impact and Future Directions

Elliptic Schubert calculus enables explicit computation of intersection numbers, structure constants, and characteristic classes for Schubert varieties in the most general currently-developed oriented cohomology theory. It reveals new dualities (including Langlands duality lifting) and deepens connections to representation theory, combinatorics, singularity theory, and mathematical physics (Rimanyi et al., 2020). Recent algebraic frameworks for duality and explicit combinatorial models will facilitate the development of symmetries, mirror phenomena, and computational tools in both geometry and representation theory, and serve as a bridge to future work on quantum, derived, and motivic extensions of Schubert calculus.

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Key references:

• (Gorbounov et al., 2010) explains the universal deformation and Jacobi ring paradigm unifying various Schubert calculi.
• (Rimanyi et al., 2019, Kumar et al., 2019) establish the explicit recursive and geometric construction of elliptic characteristic classes using Bott–Samelson resolutions and their identification with elliptic weight functions.
• (Zhong, 19 Jun 2025) proves Poincaré duality for elliptic Schubert classes via a concrete Kostant–Kumar approach, including the construction of dual bases and explicit pairings with theta function weights.
• (Lenart et al., 5 Oct 2025) extends combinatorial models (localization formulas, pipe dreams) to elliptic Schubert calculus, describes parabolic duality, and ensures compatibility with geometric expectations.