A Proof of Selection Rules for Critical Dense Polymers

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Among the lattice loop models defined by Pearce, Rasmussen and Zuber (2006), the model corresponding to critical dense polymers (β= 0) is the only one for which an inversion relation for the transfer matrix Dₙ(u) was found by Pearce and Rasmussen (2007). From this result, they identified the set of possible eigenvalues for Dₙ(u) and gave a conjecture for the degeneracies of its relevant eigenvalues in the link representation, in the sector with d defects. In this paper, we set out to prove this conjecture, using the homomorphism of the TLₙ (β) algebra between the loop model link representation and that of the XXZ model for β= -(q+q⁻¹).