Magic in the spectra of the XXZ quantum chain with boundaries at Δ=0 and Δ=−1/2

We show that from the spectra of the Uq(sl(2)) symmetric XXZ spin-1/2 finite quantum chain at Δ=−1/2 (q=eπi/3) one can obtain the spectra of certain XXZ quantum chains with diagonal and non-diagonal boundary conditions. Similar observations are made for Δ=0 (q=eπi/2). In the finite-size scaling limit the relations among the various spectra are the result of identities satisfied by known character functions. For the finite chains the origin of the remarkable spectral identities can be found in the representation theory of one and two boundaries Temperley-Lieb algebras at exceptional points. Inspired by these observations we have discovered other spectral identities between chains with different boundary conditions.