Virasoro action on Schur function expansions, skew Young tableaux and random walks

It is known that some matrix integrals over U(n) satisfy an sl(2,R)-algebra of Virasoro constraints. Acting with these Virasoro generators on 2-dimensional Schur function expansions leads to difference relations on the coefficients of this expansions. These difference relations, set equal to zero, are precisely the backward and forward equations for non-intersecting random walks. The transition probabilities for these random walks appear as the coefficients of an expansion of U(n)-matrix integrals (of the type above), by inserting in the integral the product of two Schur polynomials associated with two partitions; the latter are specified by the initial and final positions of the non-intersecting random walk. An essential ingredient in this work is the generalization of the Murnaghan-Nakayama rule to the action of Virasoro on Schur polynomials.