An algebraic derivation of the eigenspaces associated with an Ising-like spectrum of the superintegrable chiral Potts model

In terms of the $\mathfrak{sl}_{2}$ loop algebra and the algebraic Bethe-ansatz method, we derive the invariant subspace associated with a given Ising-like spectrum consisting of $2^{r}$ eigenvalues of the diagonal-to-diagonal transfer matrix of the superintegrable chiral Potts (SCP) model with arbitrary inhomogeneous parameters. We show that every regular Bethe eigenstate of the $\tau_2$-model leads to an Ising-like spectrum and is an eigenvector of the SCP transfer matrix which is given by the product of two diagonal-to-diagonal transfer matrices with a constraint on the spectral parameters. We also show in a sector that the $\tau_2$-model commutes with the $\mathfrak{sl}_{2}$ loop algebra, $L(\mathfrak{sl}_{2})$, and every regular Bethe state of the $\tau_2$-model is of highest weight. Thus, from physical assumptions such as the completeness of the Bethe ansatz, it follows in the sector that every regular Bethe state of the $\tau_2$-model generates an $L(\mathfrak{sl}_{2})$-degenerate eigenspace and it gives the invariant subspace, i.e. the direct sum of the eigenspaces associated with the Ising-like spectrum.