  We study \tau-functions of the KP hierarchy in terms of abelian group actions
on finite dimensional Grassmannians, viewed as subquotients of the Hilbert
space Grassmannians of Sato, Segal and Wilson. A determinantal formula of
Gekhtman and Kasman involving exponentials of finite dimensional matrices is
shown to follow naturally from such reductions. All reduced flows of
exponential type generated by matrices with arbitrary nondegenerate Jordan
forms are derived, both in the Grassmannian setting and within the fermionic
operator formalism. A slightly more general determinantal formula involving
resolvents of the matrices generating the flow, valid on the big cell of the
Grassmannian, is also derived. An explicit expression is deduced for the
Pl\"ucker coordinates appearing as coefficients in the Schur function
expansion of the \tau-function.
