Toda-Darboux maps and vertex operators

The process of Borel decomposing a matrix as a product of lower- and upper-triangular matrices and multiplying them in opposite order (or conversely) is called the Backlund-Darboux map. The matrices considered here are bi-infinite tridiagonal matrices or, more generally, 2p + 1-band matrices L - lambda I, where lambda is a free parameter; their Borel decomposition and thus also the corresponding Darboux map will depend on 2p - 1 parameters, in addition to the spectral parameter lambda. Letting these matrices evolve according to the standard (commuting) Toda vector fields introduces a dependency on a time-parameter t is an element of C-infinity. Then we show that, upon adjusting appropriately the free parameters, the Darboux transformed matrix evolves according to the Toda lattice, whereas, in the tridiagonal case, each of the factors evolves according to the KM lattice. As is well known, the entries and the eigenvectors of the t-dependent matrix can entirely be expressed in terms of a single vector of tau-functions (...,tau(-1)(t), tau(0)(t), tau(1)(t),...). Given such a Darboux map, how are the new tau-functions and eigenvectors expressed in terms of the old ones? The formulae so obtained involve certain vertex operators, which depend on the spectral parameter lambda and which turn out to be very useful even after setting t = 0; indeed, the tau-functions are often well-known quantities like matrix integrals, determinants of moments, Fredholm determinants, etc.