Difference Hierarchies for nT τ-functions

Abstract

We introduce hierarchies of difference equations (referred to as [Formula: see text]-systems) associated to the action of a (centrally extended, completed) infinite matrix group [Formula: see text] on [Formula: see text]-component fermionic Fock space. The solutions are given by matrix elements ([Formula: see text]-functions) for this action. We show that the [Formula: see text]-functions of type [Formula: see text] satisfy bilinear equations of length [Formula: see text]. The [Formula: see text]-system is, after a change of variables, the usual [Formula: see text] term [Formula: see text]-system of type [Formula: see text]. Restriction from [Formula: see text] to a subgroup isomorphic to the loop group [Formula: see text], defines [Formula: see text]-systems, studied earlier in [1] by the present authors for [Formula: see text].