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

We prove a determinantal, Toda-type, presentation for the equivariant K theory of a partial flag variety ${\rm Fl}(r_1, \dots, r_k;n)$. The proof relies on pushing forward the Toda presentation obtained by Maeno, Naito and Sagaki for the complete flag variety ${\rm Fl}(n)$, via Kato's ${\rm K}_T({\rm pt})$-algebra homomorphism from the quantum K ring of ${\rm Fl}(n)$ to that of ${\rm Fl}(r_1, \dots, r_k;n)$. Starting instead from the Whitney presentation for ${\rm Fl}(n)$, 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 ${\rm Fl}(n)$, following Anderson, Chen, and Tseng, which is based on the fact that the ${\rm K}$-theoretic $J$-function is an eigenfunction of the finite difference Toda Hamiltonians.