A classical way to introduce tau functions for integrable hierarchies of
solitonic equations is by means of the Sato-Segal-Wilson infinite-dimensional
Grassmannian. Every point in the Grassmannian is naturally related to a
Riemann-Hilbert problem on the unit circle, for which Bertola proposed a tau
function that generalizes the Jimbo-Miwa-Ueno tau function for isomonodromic
deformation problems. In this paper, we prove that the Sato-Segal-Wilson tau
function and the (generalized) Jimbo-Miwa-Ueno isomonodromy tau function
coincide under a very general setting, by identifying each of them to the
large-size limit of a block Toeplitz determinant. As an application, we give a
new definition of tau function for Drinfeld-Sokolov hierarchies (and their
generalizations) by means of infinite-dimensional Grassmannians, and clarify
their relation with other tau functions given in the literature.Comment: 22 page
