Quantum K Whitney relations for partial flag varieties

TL;DR

This paper mathematically analyzes conjectural presentations of the equivariant quantum K ring of partial flag varieties, proving key cases and conditions under which the conjectures hold, thus advancing understanding of quantum K theory.

Contribution

It proves the conjectured presentation for incidence varieties and establishes conditions for the full flag variety, linking physics-inspired conjectures with rigorous mathematical results.

Findings

Proved the conjectured presentation for incidence varieties.

Established that a quantum K divisor axiom implies the conjecture for complete flag varieties.

Revisited the change of variables connecting mathematical and physical formulations.

Abstract

In a recent paper, we stated conjectural presentations for the equivariant quantum K ring of partial flag varieties, motivated by physics considerations. In this companion paper, we analyze these presentations mathematically. We start by proving a Nakayama type result for quantum K theory: if the conjectured set of relations deforms a complete set of relations of the classical K theory ring, then they must form a complete set of relations for the quantum K ring. We prove the conjectured presentation in the case of the incidence varieties, and we show that if a quantum K divisor axiom holds (as conjectured by Buch and Mihalcea), then the conjectured presentation also holds for the complete flag variety. Finally, we briefly revisit the change of variables relating the mathematics and physics presentations.