Finite-dimensional tau functions of the universal character hierarchy

Using the so-called Schubert decomposition, we present a finite-dimensional twisted description of the tau functions of the universal character (UC) hierarchy using Grassmannians. Moreover, from the standpoint of the relation between the UC and Kadomtsev–Petviashvili hierarchies, we study the expansion of this tau function in terms of the action of Abelian groups on finite-dimensional Grassmannians. We use the Gekhtman–Kasman determinant formula involving exponentials of finite-dimensional matrices, which naturally leads to the structure of two finite Grassmannians. Using the Gekhtman–Kasman-type formula, we consider some concrete examples: rational, soliton, and mixed solutions.