1-At this point there are 3 characterizations of sort of integrable MPS. There is (2.24), there is the relation to the bYBE and there are the MPS that satisfy the reflection equation. I would like the authors to add maybe a diagram with the 3 notions and make clear what their interrelations are. It is described in the text, but a diagram would add much clarity.
2-The authors consider MPS that are invariant under some group action. Indeed, if (2.23) is satisfied, then this is the case. Can the authors comment on the other implication. Are all invariant states of the form (2.23)?
3-Below (3.3), the authors mention that they restrict to matrices of block-diagonal form. In general there can be an upper triangular part F, which does not affect the MPS. This is for example the case in MPS that appear in AdS/dCFT for higher representations , which might be cited as an example. Moreover, this implies that for a given MPS, omega is only defined up to an equivalence relation. Of course, the choice F=0 can be made and will simplify the computations. However, can the authors comment if maybe a non-trivial choice of F is needed to show that omega solves the bYBE or the square root relation?
4-Above (3.8) I would like the authors to clarify that the matrices A,B correspond to the MPS and the reflected MPS. It is at that point not really clear what corresponds to what.
5- Above Theorem 6, the authors mention that R-matrices are polynomials of the rapidity parameter, but this is only the case for rational R-matrices. Shortly after they discuss the XYZ spin chain, where the R-matrix is an infinite series in u. Can theorem 6 be extended to this case?