Recursion relations for unitary integrals, combinatorics and the Toeplitz lattice

In a discussion in spring 2001, Alexei Borodin showed us recursion relations for the Toeplitz determinants going with the symbols e(t(z + z-1)) and (1 - xiz)(alpha) (1 - xiz(-1))(beta). Borodin obtained these relations using Riemann-Hilbert methods; see the recent work of Borodin B and Baik Baik. The nature of Borodin's recursion relations pointed towards the Toeplitz lattice and its Virasoro algebra, introduced by us in [3]. In this paper, we take the Toeplitz lattice and Virasoro algebra approach for a fairly large class of symbols, leading to a systematic way of generating recursion relations. The latter are very naturally expressed in terms of the L-matrices appearing in the Toeplitz lattice equations. As a surprise, we find, compared to Borodin's, a different set of relations, except for the 3-step relations associated with the symbol e(t(z + z-1)). The Painleve analysis of the Toeplitz lattice enables us to show the "singularity confinement'' for these recursion relations.