Yang-Baxter algebra, higher rank partition functions and $K$-theoretic Gysin map for partial flag bundles

We investigate the $K$-theoretic Gysin map for type $A$ partial flag bundles from the viewpoint of integrability. We introduce several types of partition functions for one version of $q=0$ degeneration of $U_q(\widehat{sl_n})$ vertex models on rectangular grids which differ by boundary conditions and sizes, and can be viewed as Grothendieck classes of the Grothendieck group of a nonsingular variety and partial flag bundles. By deriving multiple commutation relations for the $q=0$ $U_q(\widehat{sl_n})$ Yang-Baxter algebra and combining with the description of the $K$-theoretic Gysin map for partial flag bundles using symmetrizing operators, we show that the $K$-theoretic Gysin map of the first type of partition functions on a rectangular grid is given by the second type whose boundary conditions on one side are reversed from the first type. This generalizes the author's previous result from Grassmann bundles to partial flag bundles. We also discuss the inhomogenous version of the partition functions and applications to the $K$-theoretic Gysin map.