Toda-Type Presentations for the Quantum K Theory of Partial Flag Varieties

Abstract

We prove a determinantal, Toda-type, presentation for the equivariant K theory of a partial flag variety Fl(𝑟₁, …, 𝑟ₖ; 𝑛). The proof relies on pushing forward the Toda presentation obtained by Maeno, Naito, and Sagaki for the complete flag variety Fl(𝑛), via Kato's KT(pt)-algebra homomorphism from the quantum K ring of Fl(𝑛) to that of Fl(𝑟₁, …, 𝑟ₖ; 𝑛). Starting instead from the Whitney presentation for Fl(𝑛), we show that the same pushforward technique gives a recursive formula for polynomial representatives of quantum K Schubert classes in any partial flag variety which do not depend on quantum parameters. In an appendix, we include another proof of the Toda presentation for the equivariant quantum K ring of Fl(𝑛), following Anderson, Chen, and Tseng, which is based on the fact that the K-theoretic J-function is an eigenfunction of the finite difference Toda Hamiltonians.