Generalized Q-functions and UC hierarchy of B-type

Abstract

We define a generalization of Schur's Q-function for an arbitrary pair of strict partitions, which is called the generalized Q-function. We prove that all the generalized Q-functions solve a series of non-linear differential equations called the UC hierarchy of B-type (BUC hierarchy). We furthermore investigate the BUC hierarchy from the viewpoint of representation theory. We consider the Fock representation of the algebra of neutral fermions and establish the boson-fermion correspondence. Using this, we discuss the relationship between the BUC hierarchy and a certain infinite dimensional Lie algebra.