# On quantum $K$-groups of partial flag manifolds

## TL;DR

This paper demonstrates that the equivariant small quantum K-group of a partial flag manifold can be obtained as a quotient of that of the full flag manifold, revealing new structural insights in quantum K-theory.

## Contribution

It establishes a K-theoretic analogue of Peterson's theorem for partial flag manifolds, showing a quotient relationship that preserves Schubert classes.

## Key findings

- The quantum K-group of a partial flag manifold is a quotient of the full flag manifold's quantum K-group.
- The quotient map involves setting some Novikov variables to 1, indicating a geometric specialization.
- This behavior differs from quantum cohomology, highlighting unique features of quantum K-theory.

## Abstract

We show that the equivariant small quantum $K$-group of a partial flag manifold is a quotient of that of the full flag manifold in a way that respects the Schubert classes. This is a $K$-theoretic analogue of the parabolic version of Peterson's theorem [Lam-Shimozono, Acta Math. {\bf 204} (2010)] that exhibits a different behavior from the case of quantum cohomology. Our quotient maps send some of the Novikov variables to $1$, and the geometric meaning of this specialization is unclear in quantum $K$-theory.