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