Difference Hierarchies for n T I" -functions

We introduce hierarchies of difference equations (referred to as nT-systems) associated to the action of a (centrally extended, completed) infinite matrix group GL∞(n) on n-component fermionic Fock space. The solutions are given by matrix elements (I"-functions) for this action. We show that the I"-functions of type nT satisfy bilinear equations of length 3, 4,...,n + 1. The 2T-system is, after a change of variables, the usual 3 term T-system of type A. Restriction from GL∞(n) to a subgroup isomorphic to the loop group LGLn, defines nQ-systems, studied earlier in [1] by the present authors for n = 2, 3.