Block Toeplitz determinants , constrained KP and Gelfand - Dickey hierarchies

We propose a method for computing any Gelfand-Dickey τ function living in SegalWilson Grassmannian as the asymptotics of block Toeplitz determinant associated to a certain class of symbols W(t; z). Also truncated block Toeplitz determinants associated to the same symbols are shown to be τ function for rational reductions of KP. Connection with Riemann-Hilbert problems is investigated both from the point of view of integrable systems and block Toeplitz operator theory. Examples of applications to algebro-geometric solutions are given. Introduction This paper deals with the applications of block Toeplitz determinants and their asymptotics to the study of integrable hierarchies. Asymptotics of block Toeplitz determinants and their applications to physics is a developing field of research; in recent years it has been shown how to compute some physically relevant quantities (e.g. correlation functions) studying asymptotics of some block Toeplitz determinants (see [25],[26],[27]). In particular in [25] and [26] authors, for the first time, showed effective computations for the case of block Toeplitz determinants with symbols that do not have half truncated Fourier series. This is of particular interest for us as, with our approach, we will be able to do the same for certain block Toeplitz determinants associated to algebro-geometric solutions of Gelfand Dickey hierarchies. Let us mention some theoretical results about (block) Toeplitz determinants we will use in this paper. Given a function γ(z) on the circle we denote TN (γ) the Toeplitz matrix with symbol γ given by TN (γ) := 