Two types of universal characters and the integrable hierarchies

In this paper, we propose two types of universal characters corresponding to partition shapes π = (3) and π = (2, 1) and construct their vertex operators realizations. It is proved that (3)-type and (2, 1)-type universal characters can be derived by the products of vertex operators acting on the identity. Furthermore, we investigate (3)-type and (2, 1)-type universal characters by means of Hamiltonian and fermions expectation values. In addition, based upon bilinear equations, we present the (3)-type and (2, 1)-type universal characters hierarchies whose τ functions can be derived from (3)-type and (2, 1)-type universal characters.