First, we consider the group integrals where integrands are the monomials of matrix elements of irreducible representations of classical groups. These group integrals are invariants under the group action. Based on analysis on Young tableaux, we investigate some related duality theorems and compute the asymptotics of the group integrals for fixed signatures, as the rank of the classical groups go to infinity. We also obtain the Viraosoro constraints for some partition functions, which are power series of the group integrals. Second, we show that the proof of Witten's conjecture can be simplified by using the fermion-boson correspondence, i.e., the KdV hierarchy and Virasoro constraints of the partition function in Witten's conjecture can be achieved naturally. Third, we consider the partition function involving the invariants that are intersection numbers of the moduli spaces of holomorphic maps in nonlinear sigma model. We compute the commutator of the representation of Virasoro algebra and give a fat graph(ribbon graph) interpretation for each term in the diferential operators.