The dimer model on a strip is considered as a Yang-Baxter mbox{integrable} six vertex model at the free-fermion point with crossing parameter $lambda=tfrac{pi}{2}$ and quantum group invariant boundary conditions. A one-to-many mapping of vertex onto dimer configurations allows for the solution of the free-fermion model to be applied to the anisotropic dimer model on a square lattice where the dimers are rotated by $45degree$ compared to their usual orientation. In a suitable gauge, the dimer model is described by the Temperley-Lieb algebra with loop fugacity $beta=2coslambda=0$. It follows that the model is exactly solvable in geometries of arbitrary finite size. We establish and solve transfer matrix inversion identities on the strip with arbitrary finite width $N$. In the continuum scaling limit, in sectors with magnetization $S_z$, we obtain the conformal weights $Delta_{s}=big((2-s)^2-1big)/8$ where $s=|S_z|+1=1,2,3,ldots$. We further show that the corresponding finitized characters $chit_s^{(N)}(q)$ decompose into sums of $q$-Narayana numbers or, equivalently, skew $q$-binomials. In the particle representation, the local face tile operators give a representation of the fermion algebra and the fermion particle trajectories play the role of nonlocal degrees of freedom. We argue that, in the continuum scaling limit, there exist nontrivial Jordan blocks of rank 2 in the Virasoro dilatation operator $L_0$. This confirms that, with quantum group invariant boundary conditions, the dimer model gives rise to a {em logarithmic} conformal field theory with central charge $c=-2$, minimal conformal weight $Delta_{text{min}}=-frac{1}{8}$ and effective central charge $c_{text{eff}}=1$.Our analysis of the structure of the ensuing rank 2 modules indicates that the familiar staggered $c=-2$ modules appear as submodules.
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