  A lattice model of critical dense polymers is solved exactly for finite
strips. The model is the first member of the principal series of the recently
introduced logarithmic minimal models. The key to the solution is a functional
equation in the form of an inversion identity satisfied by the commuting
double-row transfer matrices. This is established directly in the planar
Temperley-Lieb algebra and holds independently of the space of link states on
which the transfer matrices act. Different sectors are obtained by acting on
link states with s-1 defects where s=1,2,3,... is an extended Kac label. The
bulk and boundary free energies and finite-size corrections are obtained from
the Euler-Maclaurin formula. The eigenvalues of the transfer matrix are
classified by the physical combinatorics of the patterns of zeros in the
complex spectral-parameter plane. This yields a selection rule for the
physically relevant solutions to the inversion identity and explicit finitized
characters for the associated quasi-rational representations. In particular, in
the scaling limit, we confirm the central charge c=-2 and conformal weights
Delta_s=((2-s)^2-1)/8 for s=1,2,3,.... We also discuss a diagrammatic
implementation of fusion and show with examples how indecomposable
representations arise. We examine the structure of these representations and
present a conjecture for the general fusion rules within our framework.
