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arxiv: 2308.05307 · v3 · submitted 2023-08-10 · 🧮 math.AG

Seidel and Pieri products in cominuscule quantum K-theory

Anders S. Buch , Pierre-Emmanuel Chaput , Nicolas Perrin This is my paper

Pith reviewed 2026-05-24 07:37 UTC · model grok-4.3

classification 🧮 math.AG
keywords quantum K-theorySchubert classescominuscule flag varietiesSeidel representationPieri formulasGromov-Witten invariantsRichardson varieties
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The pith

In quantum K-theory of cominuscule flag varieties, Seidel class times Schubert class equals one Schubert class times a power of q.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper proves explicit product formulas for Schubert classes in the quantum K-theory ring of a cominuscule flag variety. The main result is a K-theory version of the Seidel representation, where the quantum product of any Seidel class with an arbitrary Schubert class is a single Schubert class multiplied by a power of the deformation parameter q. It also establishes new Pieri formulas for maximal orthogonal Grassmannians and Lagrangian Grassmannians, plus a new proof of the known Pieri formula for type A Grassmannians. All formulas are expressed using quantum shapes that index the basis elements q^d of the ring. Additional results include a closed formula for K-theoretic Gromov-Witten invariants of Pieri type on Lagrangian Grassmannians and a rationality statement for certain points on Richardson varieties in symplectic Grassmannians.

Core claim

In the quantum K-theory ring QK(X) of a cominuscule flag variety X, the quantum product of a Seidel class with an arbitrary Schubert class equals a single Schubert class times a power of the deformation parameter q. New Pieri formulas are proved for the quantum K-theory of maximal orthogonal Grassmannians and Lagrangian Grassmannians, and a new proof is given for the Pieri formula in the quantum K-theory of Grassmannians of type A. The formulas admit simple statements in terms of quantum shapes representing the natural basis elements q^d [O_{X^u}] of QK(X).

What carries the argument

Quantum shapes, which index the basis elements q^d [O_{X^u}] of QK(X) and make the product formulas reduce to single terms.

Load-bearing premise

The flag variety must be cominuscule, otherwise the quantum products need not reduce to single terms times a power of q.

What would settle it

Explicit computation of the quantum product between a Seidel class and a Schubert class in the quantum K-theory of the Lagrangian Grassmannian that produces either multiple nonzero terms or a coefficient that is not a pure power of q.

read the original abstract

We prove a collection of formulas for products of Schubert classes in the quantum $K$-theory ring $QK(X)$ of a cominuscule flag variety $X$. This includes a $K$-theory version of the Seidel representation, stating that the quantum product of a Seidel class with an arbitrary Schubert class is equal to a single Schubert class times a power of the deformation parameter $q$. We also prove new Pieri formulas for the quantum $K$-theory of maximal orthogonal Grassmannians and Lagrangian Grassmannians, and give a new proof of the known Pieri formula for the quantum $K$-theory of Grassmannians of type A. Our formulas have simple statements in terms of quantum shapes that represent the natural basis elements $q^d[{\mathcal O}_{X^u}]$ of $QK(X)$. Along the way we give a simple formula for $K$-theoretic Gromov-Witten invariants of Pieri type for Lagrangian Grassmannians, and prove a rationality result for the points in a Richardson variety in a symplectic Grassmannian that are perpendicular to a point in projective space.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 3 minor

Summary. The paper proves explicit product formulas for Schubert classes in the quantum K-theory ring QK(X) of cominuscule flag varieties X. This includes a K-theoretic Seidel representation asserting that the quantum product of a Seidel class with an arbitrary Schubert class equals a single Schubert class times a power of the deformation parameter q. It also establishes new Pieri formulas for the quantum K-theory of maximal orthogonal Grassmannians and Lagrangian Grassmannians, gives a new proof of the known Pieri formula in type A, and supplies a simple formula for K-theoretic Gromov-Witten invariants of Pieri type on Lagrangian Grassmannians together with a rationality result for certain points in Richardson varieties in symplectic Grassmannians. All formulas are expressed in terms of quantum shapes that form a natural basis for QK(X).

Significance. If the stated proofs hold, the results supply concrete, combinatorially simple multiplication rules in quantum K-theory precisely in the cominuscule setting where the products reduce to single terms times a power of q. This extends the Seidel representation to K-theory, furnishes new Pieri rules, and provides a direct combinatorial basis via quantum shapes, which are strengths for explicit computations and structural understanding of the ring.

minor comments (3)
  1. The abstract states that the formulas are proved, but the manuscript should include explicit section references or theorem numbers for each claimed identity (Seidel representation, each Pieri rule, and the Gromov-Witten formula) so that readers can locate the derivations without searching the full text.
  2. Notation for quantum shapes and the basis elements q^d [O_{X^u}] is introduced in the abstract; a dedicated preliminary subsection should collect all definitions of these objects and the cominuscule condition before the statements of the main theorems.
  3. The rationality result for points in Richardson varieties is mentioned only briefly in the abstract; the manuscript should clarify in which section this appears and whether it is used in the proofs of the product formulas or is an independent result.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary of the manuscript, recognition of its significance, and recommendation for minor revision. No major comments appear in the report.

Circularity Check

0 steps flagged

No significant circularity; derivations are self-contained proofs

full rationale

The paper supplies explicit mathematical proofs of product formulas (Seidel representation reducing to single Schubert class times q-power, plus Pieri formulas) that hold under the cominuscule restriction on X. Quantum shapes are introduced as the natural basis for QK(X), and the identities are established directly via algebraic geometry methods without fitted parameters, self-definitional reductions, or load-bearing self-citations that collapse the central claims to their inputs. The argument is restricted to the cominuscule setting where the formulas simplify, and the derivations remain independent of the target results.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review supplies no explicit free parameters, axioms, or invented entities; the cominuscule assumption is treated as background rather than a new postulate.

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Reference graph

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