Generalized Q-Functions and UC Hierarchy of B-Type

Abstract

A generalization of the Schur Q-function is introduced. This generalization, called generalized Q-function, is indexed by any pair of strict partitions, and can be expressed by the Pfaffian. A connection to the theory of integrable systems is clarified. Firstly, the bilinear identities satisfied by the generalized Q-functions are given and proved to be equivalent to a system of partial differential equations of infinite order. This system is called the UC hierarchy of B-type (BUC hierarchy). Secondly, the algebraic structure of the BUC hierarchy is investigated from the representation theoretic viewpoint. Some new kind of the boson-fermion correspondence is established, and a representation of an infinite dimensional Lie algebra, denoted by $\mathfrak{go}_{2\infty}$, is obtained. The bilinear identities are translated to the language of neutral fermions, which turn out to characterize a $\boldsymbol{G}$-orbit of the vacuum vector, where $\boldsymbol{G}$ is the group corresponding to $\mathfrak{go}_{2\infty}$.