Symplectic Grothendieck polynomials, universal characters and integrable systems

This paper is concerned with the K-theoretic analogs of symplectic Schur functions and symplectic universal character. In this paper, we first show that the linear transformations of the vertex operators presentation of symplectic Schur functions are polynomial tau-functions of symplectic KP hierarchy, and give a new symmetric function called symplectic Grothendieck polynomial, together with its vertex operators and fermions realizations, then prove that these functions are also the polynomial tau-functions of the symplectic KP hierarchy. In addition, we extend these results to universal character, and give a generalization of symplectic Grothendieck polynomial, called symplectic Grothendieck universal character.