Pearce–Rasmussen selection-rule conjecture for loop-model spectra

Let $N$ be the strip width, $d$ the number of defects, and $A^M_{m,n}$ the sets of admissible two-column configurations specified by the preceding selection rules, with choices of the $\epsilon_j$s and $mu_j$s. Let $\rho(D_N(u))$ and $\rho(\mathcal H_N)$ denote the transfer-matrix and Hamiltonian representations on the link-state space. Pearce–Rasmussen selection-rule conjecture. In the sector with $d$ defects, the set of choices belonging to

$$N\text{ odd}:\quad\bigcup_{p=0}^{(N-d)/2}A^{(N-1)/2}_{p,p+(d-1)/2},$$

and

$$N\text{ even}:\quad\bigcup_{p=0}^{(N-d)/2}\left(A^{(N-2)/2}_{p,p+(d-2)/2}\cup A^{(N-2)/2}_{p,p+d/2}\right)$$

forms the spectra of $\rho(D_N(u))$ and $\rho(\mathcal H_N)$. The statement is presented as the selection rule proposed by Pearce and Rasmussen; the source supplies no resolution, and the subsequent discussion translates it into degeneracy formulas.