Quantum $K$-theoretic Whitney relations for type $C$ flag manifolds

TL;DR

This paper establishes quantum $K$-theoretic Whitney relations for type $C$ flag manifolds, providing a new presentation of the quantum $K$-ring as a quotient of a polynomial ring, advancing understanding of quantum $K$-theory.

Contribution

It introduces quantum $K$-theoretic Whitney relations and offers a Whitney-type presentation of the quantum $K$-ring for type $C$ flag manifolds, different from existing Borel-type presentations.

Findings

Quantum $K$-theoretic Whitney relations are established.

The relations provide a complete set of defining relations for the quantum $K$-ring.

A new Whitney-type presentation of the quantum $K$-ring is derived.

Abstract

We study relations of $\lambda_{y}$-classes associated to tautological bundles over the flag manifold of type $C$ in the quantum $K$-ring. These relations are called the quantum $K$-theoretic Whitney relations. The strategy of the proof of the quantum $K$-theoretic Whitney relations is based on the method of semi-infinite flag manifolds and the Borel-type presentation. In addition, we observe that the quantum $K$-theoretic Whitney relations give a complete set of the defining relations of the quantum $K$-ring. This gives a presentation of the quantum $K$-ring of the flag manifold of type $C$, called the Whitney-type presentation, as a quotient of a polynomial ring, different from the Borel-type presentation.