Whitney-Type Presentation in Quantum K-Theory

• Whitney-type presentation is a representation of the quantum K-ring of flag manifolds as a quotient of a polynomial ring with explicit generators and combinatorial relations.
• It leverages tautological bundle classes and Hirzebruch λ_y classes to establish quantum Whitney relations that generalize classical Whitney sum formulas in K-theory.
• The framework enables efficient computations in quantum K-theory, bridging geometric representation with combinatorial algorithmic methods in Schubert calculus.

A Whitney-type presentation, in the sense of geometric representation theory and (equivariant) quantum $K$-theory, is a presentation of the quantum $K$-ring of a flag manifold as a quotient of a polynomial or Laurent polynomial ring, where the generators and relations are of explicit, combinatorial nature—namely, relations satisfied by the characteristic (Hirzebruch $\lambda_{y}$) classes of tautological bundles and their associated quantum deformations. In the context of type $C$ flag manifolds, such a presentation is termed "Whitney-type" due to its direct generalization and deformation of classical Whitney sum formulae in $K$-theory. The approach has structural and computational significance, offering a concrete description of the quantum $K$-ring in terms of explicit generators and a complete, finite set of defining relations.

1. Tautological Data and $\lambda_{y}$-Class Generators

Consider the full flag manifold of type $C$, $Fl_n = Sp_{2n}(\mathbb{C})/B$, equipped with its tautological flag

$$0 = S_0 \subset S_1 \subset \cdots \subset S_n \subset \mathbb{C}^{2n}.$$

We define $L_k := S_k / S_{k-1}$ to be the $k$-th quotient line bundle, and, given a $T$-equivariant vector bundle $E$ of rank $r$, the Hirzebruch class is

$$\lambda_y(E) := 1 + y[E] + y^2[\wedge^2 E] + \cdots + y^r[\wedge^r E] \in K_T(Fl_n)[y].$$

The $\lambda_y$-classes of the $S_k$, their duals $S_k^\vee$, and the line bundles $L_k$ and $L_k^\vee$ generate the (equivariant) quantum $K$-ring algebra

$$QK_T(Fl_n) := K_T(Fl_n) \otimes_{R(T)} R(T)[Q_1, \ldots, Q_n],$$

where the $Q_k$ are Novikov variables encapsulating quantum deformations corresponding to curve classes (Kouno, 29 Dec 2025).

2. Quantum $K$-Theoretic Whitney Relations

The algebraic structure is determined by quantum deformations of the classical $K$-theory Whitney sum relations. Explicitly, for $1 \leq k \leq n$, in $QK_T(Fl_n)[y]$, the following three families of quantum Whitney relations completely determine the ideal of relations:

1. For increasing the tautological flag:

$$\lambda_y(S_{k-1}) \star \lambda_y(L_k) = \lambda_y(S_k) - y \cdot \frac{Q_{k-1}}{1-Q_{k-1}[L_k]} \star (\lambda_y(S_{k-1}) - \lambda_y(S_{k-2}))$$

2. For the dual flag:

$$\lambda_y(S_{k-1}^\vee) \star \lambda_y(L_k^\vee) = \lambda_y(S_k^\vee) - y \cdot \frac{Q_{k-1}}{1-Q_{k-1}[L_k^\vee]} \star (\lambda_y(S_{k-1}^\vee) - \lambda_y(S_{k-2}^\vee))$$

3. The “top-step” relation, particular to type $C$:

$$\lambda_y(S_n)\star \lambda_y(S_n^\vee) = \lambda_y(\mathbb{C}^{2n}) - y^2 \sum_{p=1}^n \frac{Q_p \cdots Q_n}{1-Q_{p-1}} \star (\lambda_y(S_{p-1}) - Q_{p-1} \lambda_y(S_{p-2})) \star (\lambda_y(S_{p-1}^\vee) - Q_{p-1} \lambda_y(S_{p-2}^\vee))$$

Here, $Q_0=0$ and $S_{-1}=0$ are conventionally set. At $Q_k=0$, these formulas recover the ordinary (undeformed) $K$-theoretic Whitney sum formulas (Kouno, 29 Dec 2025).

3. Explicit Presentation as a Quotient Algebra

The full quantum $K$-ring is realized as a quotient of a Laurent polynomial ring in generators corresponding to the exterior powers and (dual) line bundles: $$R_Q^W := R(T)[Q_1, \ldots, Q_n][F_d^k, G_d^k, x_j, y_j\ |\ 1 \leq k \leq n, 1 \leq d \leq k, 1 \leq j \leq n]$$ where $F_d^k$ and $G_d^k$ correspond to $[\Lambda^d S_k]$ and $[\Lambda^d S_k^\vee]$, and $x_j, y_j$ model $[L_j]$, $[L_j^\vee]$. The relations fall into five categories:

Relation Type	Generator Pattern	Relation Formula
(a) Flag Extension ($F_d^k$)	$F_d^k$	$$F_d^k - \left(F_d^{k-1} + \frac{1}{1-Q_{k-1}}(F_{d-1}^{k-1} - Q_{k-1} F_{d-1}^{k-2}) x_k\right)$$
(b) Dual Flag ($G_d^k$)	$G_d^k$	$$G_d^k - \left(G_d^{k-1} + \frac{1}{1-Q_{k-1}}(G_{d-1}^{k-1} - Q_{k-1} G_{d-1}^{k-2}) y_k\right)$$
(c) Top-Step	$F_i^n$, $G_{d-i}^n$	$$\sum_{i=0}^d F_i^n G_{d-i}^n - \left(e_d(T_1,\ldots,T_n,T_n^{-1},\ldots,T_1^{-1}) - \sum_{p=1}^n ...\right)$$
(d) Initial Conditions	$F_1^1$, $G_1^1$, $x_1$, $y_1$	$F_1^1-x_1$, $G_1^1-y_1$
(e) Line Bundle Inverses	$x_j,y_j$	$x_j y_j - (1-Q_{j-1})(1-Q_j)$

where $e_d$ are the elementary symmetric polynomials in torus weights and the full “top-step” formula for (c) includes the previously stated explicit sum over $p$ and $r$.

This yields the presentation: $QK_T(Fl_n) \simeq R_Q^W / I_Q^W$ with the ideal $I_Q^W$ generated by the above relations. The assignment of generators to $\lambda_{y}$-classes and line bundle classes specifies the isomorphism (Kouno, 29 Dec 2025).

4. Comparison to the Borel Presentation and Classical Limit

In the purely classical regime ($Q=0$), the defining relations reduce to those in the Borel-type presentation of $K_T(Fl_n)$. The Whitney-type presentation thus generalizes the Borel presentation to the setting of quantum deformation. The completeness of the relation set is a consequence of the structure theorem asserting that these quantum $K$ Whitney relations generate the full ideal of relations, i.e., there are no additional hidden constraints in the quantum $K$-ring (Kouno, 29 Dec 2025). This aspect is a crucial distinction from the Borel presentation, as the quantum deformation is strictly governed by these Whitney-type relations.

5. Methodological Innovations and Context

The strategy for proving the Whitney-type presentation in type $C$ employs the geometry of semi-infinite flag manifolds and the Borel presentation, together with explicit manipulations of tautological bundles and their characteristic classes in quantum $K$-theory. Quantum corrections manifest through the appearance of rational expressions in $Q_k$, intertwining classical summation with quantum multiplication. The presentation is different from both the Borel and physics-inspired Coulomb branch approaches, yet diagrammatic correspondences can be established to recover the same ring structure in each case (Kouno, 29 Dec 2025).

This framework is part of a broader program of explicitly describing quantum cohomology and $K$-theory (see (Gu et al., 2023) for type $A$), where similar Whitney-type presentations appear, generalizing both the classical sum formulae and the structural algebras of tautological class-based presentations.

6. Significance and Implications

The Whitney-type presentation provides a computationally effective, geometric, and module-theoretic foundation for calculating in the quantum $K$-ring. Structural constants, multiplication rules, and ring-theoretic invariants become accessible via this algebraic model. The explicit nature of the presentation is particularly suited to algorithmic approaches and further theoretical developments in Schubert calculus, quantum $K$-theory, and connections to representation theory of semi-simple Lie groups, as well as applications in enumerative geometry.

A plausible implication is that similar presentations may be expected for additional types of (generalized) flag manifolds when quantum $K$-theoretic Whitney relations can be extracted and shown to be complete. The approach also isolates geometric features of the quantum $K$-ring in the relations among tautological data—thereby reducing complicated Gromov–Witten computations to appropriately encoded algebraic relations.

7. Non-Equivariant and Further Generalizations

Upon specializing to the non-equivariant case ($R(T)=\mathbb{Z}$), the presentation persists, with all torus weights set to 1. The general methodology potentially extends to partial flag manifolds in other types (pending identification and verification of appropriate quantum Whitney relations). Key techniques developed here—such as the use of semi-infinite manifolds and explicit models for characteristic-class relations in quantum $K$-theory—are anticipated to inform further advances across Schubert calculus, quantum $K$-theory, and related fields (Kouno, 29 Dec 2025).