The talk will focus on effective parametrisations of unitriangular Toeplitz matrices. We will first recall Thoma's parametrisation for the infinite matrices. Next, for matrices of rank k, we will see how one can recover Rietsch's parametrisation by k-tuple of nonnegative reals from the combinatorics of k-Schur functions. This approach permits to avoid delicate geometric arguments related to quantum cohomology of flag varieties and makes the parametrization completely explicit: the Toeplitz matrix corresponding to a k-tuple of nonnegative reals is given by the Perron-Frobenius vector of a natural irreducible matrix.

This is a joint work with Pierre Tarrago (Cimat Mexico).