[Paper Review] A conjectural Peterson isomorphism in K-theory

Thomas Lam, Changzheng Li+2 more|arXiv (Cornell University)|May 9, 2017

Advanced Algebra and Geometry13 references3 citations

TL;DR

This paper proposes a conjectural isomorphism between the equivariant quantum K-theory of flag varieties and the equivariant K-homology of the affine Grassmannian, generalizing Peterson's isomorphism in cohomology. It establishes a precise correspondence between Schubert structure constants in both rings, proves a formula for Pontryagin product constants in K-homology, and verifies the conjecture in low-degree cases for type A₂, providing strong computational evidence for the K-theoretic Peterson isomorphism.

ABSTRACT

We state a precise conjectural isomorphism between localizations of the equivariant quantum K-theory ring of a flag variety and the equivariant K-homology ring of the affine Grassmannian, in particular relating their Schubert bases and structure constants. This generalizes Peterson's isomorphism in (co)homology. We prove a formula for the Pontryagin structure constants in the K-homology ring, and we use it to check our conjecture in few situations.

Motivation & Objective

• To formulate a precise conjectural isomorphism between the equivariant quantum K-theory ring of a flag variety and the equivariant K-homology ring of the affine Grassmannian.
• To generalize Peterson's isomorphism in (co)homology to the K-theoretic setting.
• To relate Schubert bases and structure constants in both rings via a conjectural algebra isomorphism.
• To prove a formula for Pontryagin structure constants in K-homology of the affine Grassmannian.
• To provide computational verification of the conjecture in low-degree cases, particularly for type A₂.

Proposed method

• Formulate Conjecture 1 relating structure constants in K-homology of the affine Grassmannian to those in quantum K-theory of the flag variety.
• Introduce Conjecture 2 as an alternative algebraic formulation via a localization isomorphism involving inverse Schubert classes and quantum parameters.
• Derive a combinatorial formula for Pontryagin product structure constants in K₀^T(Gr) using geometric and representation-theoretic techniques.
• Use the derived formula to compute explicit multiplication tables for K-homology classes in the SL₃ case.
• Compare the computed K-homology structure constants with known quantum K-theory structure constants from [LM], verifying agreement in degrees (0,0), (1,0), and (0,1).
• Leverage symmetry of the A₂ Dynkin diagram to extend results beyond the computed cases.

Experimental results

Research questions

• RQ1 Do the Schubert structure constants in the equivariant K-homology of the affine Grassmannian coincide with those in the equivariant quantum K-theory of the flag variety under the proposed isomorphism?
• RQ2 Can a closed-form formula be derived for the Pontryagin product structure constants in the K-homology ring of the affine Grassmannian?
• RQ3 Does the conjectural isomorphism hold in low-degree cases, particularly for the flag variety of type A₂?
• RQ4 How do the Schubert classes in the affine Grassmannian and quantum K-theory relate under the proposed algebra isomorphism?
• RQ5 What is the precise relationship between the quantum parameters in QK_T(G/B) and the inverse Schubert classes in K₀^T(Gr)?

Key findings

• A formula for the Pontryagin structure constants in the K-homology ring of the affine Grassmannian is rigorously derived and used to compute explicit multiplication tables.
• The conjectural isomorphism (Conjecture 1) is verified in all cases where the quantum degree d is (0,0), (1,0), or (0,1) for type A₂.
• The multiplication table for K₀^T(Gr_SL₃) is computed explicitly, showing agreement with quantum K-theory structure constants from [LM] in low degrees.
• The isomorphism Ψ in Conjecture 2 is shown to preserve algebraic structure, mapping inverse Schubert classes to quantum Schubert classes via q-parameters.
• The structure constants in both rings match precisely in the verified cases, supporting the broader conjecture.
• The results are consistent with known connections to Gromov-Witten invariants and Zastava spaces, suggesting deeper geometric links.