Particles moving on a line and refined K-theory

The totally asymmetric simple exclusion process (TASEP) on the integers is a well-studied stochastic process, and if we start with all of the particles on negative numbers, then we can encode the states of TASEP as a (random) matrix with nonnegative integer entries. This reformulation is a last-passage percolation (LPP) model. A bijection of Yeliussizov gives a relation between the LPP model and (reverse) plane partitions, which allows us to describe the transition probabilities using skew refined dual Grothendieck polynomials introduced by Galashin, Grinberg, and Liu that arose from geometry with the study of the K-theory of the Grassmannian. In this talk, we will introduce all of these objects and the connections between them. Then, by refining the work of Johansson on LPP, we will give an algebraic proof of the Jacobi-Trudi formula for skew refined dual Grothendieck polynomials conjectured by Grinberg and an integral formula for refined dual Grothendieck polynomials. This is joint work with Kohei Motegi.