Rigorous quantum K-theory relations for general partial flag varieties

Establish rigorous ring relations for the quantum K-theory rings of general partial flag varieties.

Background

The paper derives Bethe ansatz equations for a GL(n) asymmetric five-vertex integrable model and identifies their specializations with quantum cohomology and quantum K-theory relations for flag varieties. For quantum K-theory, the proposed relations are supported by results from gauged linear sigma models and are known in certain special cases, including incidence varieties and full flag varieties. However, the paper notes that a rigorous general presentation of the quantum K-theory ring relations for arbitrary partial flag varieties is still unavailable. Establishing such relations would provide a mathematical foundation for the proposed Bethe-ansatz and gauge-theoretic descriptions in the general partial-flag setting.

References

Though ring relations of quantum K-theory ring of general partial flag varieties have not been rigorously established, it was checked in that the ring relations obtained from the GLSMs match those of the special cases, such as incidence varieties and full flag varieties, in which exact results have been proved.

— Quantum integrable model for the quantum cohomology/K-theory of flag varieties and the double $β$-Grothendieck polynomials  (2502.03768 - Guo, 6 Feb 2025) in Remark following Section 2, Section 2 (Bethe ansatz equations and the equivariant quantum cohomology/K-theory ring of flag varieties)