Integrals over classical groups, random permutations, Toda and Toeplitz lattices

Matrix Fourier-like integrals over the classical groups O_+(n), O_-(n), Sp(n)and U(n) are connected with the distribution of the length of the longestincreasing sequence in random permutations and random involutions and thespectrum of random matrices. One of the purposes of this paper is to show thatall those integrals satisfy the Painlev\'e V equation with specific initialconditions. In this work, we present both, new results and known ones, in aunified way. Our method consists of inserting one set of time variables t=(t_1,t_2,...) inthe integrals for the real compact groups and two sets of times (t,s) for theunitary group. The point is that these new time-dependent integrals satisfyintegrable hierarchies: (i) O(n) and Sp(n) correspond to the standard Toda lattice. (ii) U(n)corresponds to the Toeplitz lattice, a very special reduction of the discretesinh-Gordon equation. Both systems, the standard Toda lattice and the Toeplitz lattice are alsoreductions of the 2-Toda lattice, thus leading to a natural vertex operator,and so, a natural Virasoro algebra, a subalgebra of which annihilates thetau-functions. Combining these equations leads to the Painlev\'e V equation forthe integrals.