  Solvable critical dense polymers is a Yang-Baxter integrable model of
polymers on the square lattice. It is the first member LM(1,2) of the family of
logarithmic minimal models LM(p,p'). The associated logarithmic conformal field
theory admits an infinite family of Kac representations labelled by the Kac
labels r,s=1,2,.... In this paper, we explicitly construct the conjugate
boundary conditions on the strip. The boundary operators are labelled by the
Kac fusion labels (r,s)=(r,1) x (1,s) and involve a boundary field xi. Tuning
the field xi appropriately, we solve exactly for the transfer matrix
eigenvalues on arbitrary finite-width strips and obtain the conformal spectra
using the Euler-Maclaurin formula. The key to the solution is an inversion
identity satisfied by the commuting double-row transfer matrices. The transfer
matrix eigenvalues are classified by the physical combinatorics of the patterns
of zeros in the complex spectral-parameter plane. This yields selection rules
for the physically relevant solutions to the inversion identity which takes the
form of a decomposition into irreducible blocks corresponding combinatorially
to finitized characters given by generalized q-Catalan polynomials. This
decomposition is in accord with the decomposition of the Kac characters into
irreducible characters. In the scaling limit, we confirm the central charge
c=-2 and the Kac formula for the conformal weights Delta_{r,s}=((2r-s)^2-1)/8
for r,s=1,2,3,... in the infinitely extended Kac table.
