In this paper, we first construct an integrable system whose solutions include the orthogonal Schur functions and the symplectic Schur functions. We find that the orthogonal Schur functions and the symplectic Schur functions can be obtained by one kind of Boson-Fermion correspondence which is slightly different from the classical one. Then, we construct a universal character which satisfies the bilinear equation of a new infinite-dimensional integrable orthogonal UC hierarchy.

In this paper, we first calculate the orthogonal basis of the vector space spanned by eiejeke0|0⟩, where ej are the generators of the affine Yangian of gl(1). The elements of this orthogonal basis correspond to three dimensional bosons. Then, we calculate the Schur functions of plane partitions of 4, we find that the plane partitions become Young diagrams, and the Schur functions on plane partitions become Schur functions on Young diagrams when h1 = 1, h2 = −1, and h3 = 0.

Based on the Lax pair [Formula: see text] of the mKP hierarchy and the operator Nambu 3-bracket, we propose the generalized Lax equation with respect to the Lax triple [Formula: see text]. For different pairs [Formula: see text] in the generalized Lax equation, a generalized hierarchy including the mKP hierarchy is derived.