The Orthogonal and Symplectic Schur Functions, Vertex Operators and Integrable Hierarchies

Abstract

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.

Polynomial tau-functions of the symplectic KP, orthogonal KP and BUC hierarchies

“…Generating functions and the polynomial tau-functions of the OKP hierarchy. It is known that orthogonal Schur functions are tau-functions of the OKP hierarchy [23]. Let G(u 1 , .…”

“…Under the reduction c i,l = 0, h i = −1 and l + N i = i, T ζ lead to the orthogonal Schur functions [23].…”

“…Thus the polynomial tau-functions (4.22) of the OKP hierarchy can be reduced to the solution of the OKP hierarchy in [23], which are the zero mode of an appropriate combinatorial generating functions.…”

“…Wang et.al [20] discussed the algebra of universal characters and the phase model of strongly correlated bosons. Recently, by means of the vertex operator realization of symplectic and orthogonal Schur functions [21], the authors [22,23] established generalizations of symplectic KP (SKP) and orthogonal KP (OKP) hierarchies called the symplectic and orthogonal universal character hierarchies corresponding to symplectic and orthogonal universal characters, respectively. Ogawa [24] defined a generalized Q-function expressed by the Pfaffian and constructed an integrable UC hierarchy of B-type (BUC hierarchy) characterized by the generalized Q-function.…”

Quantum fields presentation and generating functions of symplectic Schur functions and symplectic universal characters

“…Later, the generalization of symplectic Schur functions as well as the corresponding integrable systems have been well investigated. [10,11] The universal character proposed by Koike [12] is the extension of the Schur function. Tsuda [13] developed a novel integrable system called universal character (UC) hierarchy with the property of the universal character, which is a generalization of the KP hierarchy.…”