Bethe vectors and recurrence relations for twisted Yangian based models

The author developed the nested algebraic Bethe Ansatz for the type $Y^+(\mathfrak{gl}_{2n+1})$ twisted Yangians. This method is the generalization of [GMR19] where the algebraic Bethe Ansatz was worked out for the other twisted Yangians $Y^\pm(\mathfrak{gl}_{2n})$. The author also presented a new trace formula for Bethe vectors of $Y^\pm(\mathfrak{gl}_{N})$. Recurrence relations were also derived for the Bethe vectors.

The paper is very technical, but the derivations are clearly presented and they can be followed. I have a few suggestions to make the presentations of the results easier for the readers. Regardless of how these are implemented, I recommend the article be accepted.

I have the following questions and suggestions.

1- Why the paper focuses on the evaluation representations of $Y^\pm(\mathfrak{gl}_{N})$. It seems to me that the explicit forms of these are not used during the derivations, but there are other representations of the twisted Yangian $Y^\pm(\mathfrak{gl}_{N})$ (e.g. (5.58) in https://arxiv.org/abs/1912.09338), and the reader may find that the results do not apply to these representations.

2- The definition of the transfer matrix (51) contains twists. Usually we define double row transfer matrices which contains the solutions of the boundary Yang-Baxter equation but this matrix $M^{(2\hat{n})}$ including twists does not satisfy the boundary Yang-Baxter equation. What is the reason for these transfer matrices commute?

3- Is there any chance to diagonalize the double row transfer matrix which contains ''two boundary space''. What I mean is that the right and left M-matrices are also not one-dimensional representations of the twisted Yangian.

4- In equation (63), the subscript $j$ has two roles. It denotes the fix Bethe root $u^{(n)}_j$ but it is also a running index for the sets of Bethe roots like $\mathbf{u}^{(j)}_{III}$.

5- Perhaps it would help the reader if a simplified form of recurrence relations were included which focuses only on the numbers of Bethe roots, i.e. how many Bethe roots are contained in the Bethe vectors in the formulas, and the long and complicated coefficients are avoided. For example, the author could add extra equations like
\[
\Psi^{m_1,\dots,m_n}=\sum_{i<j}
\big[(\dots)s_{j,2n-i+1}(u^{(n)}_j)+(\dots)s_{i,2n-j+1}(u^{(n)}_j)\big]
\Psi^{m_1,\dots,m_{i-1},m_i-1,\dots,m_{j-1}-1,m_j-2,\dots,m_{n-1}-2,m_n-1}
\]
which is the compressed version of (63).

6- I also have a related question. How the recurrence relations are related to the Lie-subalgebras of the twisted Yangian? Let us consider the twisted Yangians $Y^\pm(\mathfrak{gl}_{2n})$ which have Lie-subalgebras $\mathfrak{so}(2n)$ or $\mathfrak{sp}(2n)$. The Bethe states should have well defined $\mathfrak{so}(2n)$ or $\mathfrak{sp}(2n)$ weights which depend on the numbers of Bethe roots $m_k$. The creation operators change the $\mathfrak{so}(2n)$ or $\mathfrak{sp}(2n)$ weights in a well defined way. Is it compatible with the recurrence relation?
It might be worth developing this point in the paper. I think it gives a good justification for why such a relation can exist, and what is the operator dependent part.

1- New results for open spin chains
2- Original approach

1- Very technical
2- Some inaccuracies in the definitions

The author consider the Algebraic Bethe ansatz in the case of open spin chains based on twisted Yangians $Y(gl_N)^\pm$. The case $N=2n$ was done in a previous paper by the author and collaborators, but the case $N=2n+1$ is new. A trace formula for the Bethe vectors is constructed, as well as recurrence relations.

The paper is interesting and brings new results, but it is very technical, and I think some points need to be improved before publication.

The presentation of $Y^+(gl_{2n+1})$ in term of $(2n+2)\times(2n+2)$ matrices looks new to me, and it would be nice to highlight and detail it. There are several questions on the subject (see below)

1- The symmetry relation eq. (10) contains a parameter $\rho$ and its role is not clear to me. First of all, I suppose it is fixed once for all, the relation is not true for all $\rho$'s? It is not clear in the text. Second, it does not appear in the usual (i.e. as $(2n+1)\times(2n+1)$ matrices) presentation of $Y^+(gl_{2n+1})$: does it means that any value of $\rho$ leads to the same algebra? That would lead to some kind of isomorphism which is not known? Finally, I don't see the use of it in the paper. If it is not used, I would strongly suggest to take $\rho=0$: it avoids to define $\tilde u$ which would be $-u$, so that the notation $\tilde R(u)$ which has nothing to do with $\tilde u$ would look less confusing. Since the paper is very very technical, and uses a lot of notation (see below) that would lighten it.
2- The $R$-matrix used for $Y^+(gl_{2n+2})$ is the same as the one of $Y^+(gl_{2n+2})$: I think it corresponds to a realization of $Y^+(gl_{2n+1})$ as a coset of $Y^+(gl_{2n+2})$ by the relations given by the doubling of row and column? I think it is new too, and it deserves details and proofs.
In particular does the symmetry relation for $Y^+(gl_{2n+2})$ implies the one for $Y^+(gl_{2n+2})$? One way to see it is to represent the coset by the relation
$S=\Pi\,S\,\Pi$ where $\Pi=1-E_{n,n+1}$.
3- Since they construct $Y^+(gl_{2n+1})$ out of $(2n+2)\times(2n+2)$ matrices, is there a way to see that $Y^-(gl_{2n+1})$ cannot be build (or is trivial in some sense)?
4- If the point above is true, and since the case $Y^+(gl_{2n})$ is known from their previous paper [GMR19], I believe that most of the properties given in the article can be deduced from the ones of $Y^+(gl_{2n+2})$. Then, I would suggest to take this point of view to get less technicalities in the proofs.
5- As I already said, the paper is very technical, introduces a lot of notation, some of them being unclear. To help the reader, there should be a section for the presentation of the results, postponing the proofs in other sections.
6- Strictly speaking, the representation used for the monodromy matrix in section 3.1 is not an evaluation representation but a tensor product of evaluation representations.
7-
$ W^{(\hat n)'}_{\dot{a}}$, $ W^{(\hat n)'}_{\ddot{a}}$, $V^{(\hat n)'}_{\dot{a}}$ and $V^{(\hat n)'}_{\ddot{a}}$ are ill-defined: you cannot have definitions that depend on the tuple $\dot{a}$ or $\ddot{a}$ you act on. You should define $\dot{W}^{(\hat n)'}_{{a}}$, $\ddot{W}^{(\hat n)'}_{{a}}$, $\dot{V}^{(\hat n)'}_{{a}}$ and
$\ddot{V}^{(\hat n)'}_{{a}}$, valid for any tuple $a$, and then consider $\dot{W}^{(\hat n)'}_{\dot{a}}$, $\ddot{W}^{(\hat n)'}_{\ddot{a}}$, $\dot{V}^{(\hat n)'}_{\dot{a}}$ and
$\ddot{V}^{(\hat n)'}_{\ddot{a}}$ in the calculation.
8- I also think that dealing with $N=2n$ and $N=2n+1$ makes things more difficult to follow. Since the even case has been already done, it would be simpler to stick to $N=2n+1$ and $Y^+(gl_{2n+1})$.
9- On line 156 there are $\hat 1$ which should be $\bar 1$.
10- It should be mentioned that partitions $\boldsymbol{u}\vdash \{\boldsymbol{u}_I,\boldsymbol{u}_{II}\}$ also includes the cases $\boldsymbol{u}_I$ or $\boldsymbol{u}_{II}$ are an emptyset (that cases are not included in the usual definition for partitions).
11- I am also confused with the sentence on line 287 stating that the union of all components in a product of functions or operators must be equal to the full set $\boldsymbol u$: this convention is not used in the papers [HL+...] and their results are used in the present paper for some of the calculations. They should show that this new convention does not affect the results they use.
12- The notation $\frac1{\boldsymbol{u}-\boldsymbol{v}}$ is not defined: one should at least say that the convention on functions also apply to $\frac1{u-v}$.

The results obtained in the paper are important for studying scalar products of Bethe vectors.

The paper does not have a concluding part containing a discussion of main results.

The paper considers a nested algebraic Bethe ansatz for one-dimensional quantum spin chains with open boundaries, whose underlying symmetry is an odd twisted Yangian.
The author constructs Bethe vectors and obtains recursion relations for them. Similar recursions for Bethe vectors with Yangian or super-Yangian symmetry played an important role in the study of scalar products. There is no doubt that the results obtained in the paper can also be used to study scalar products.

The paper is written very concisely. At the same time, all basic concepts are defined. Most of the calculations are done in sufficient detail that the reader can follow them.

I believe that the article can be published in its present form. However, I recommend that the author consider adding a short conclusion. It could briefly summarize the main results. My recommendation is optional. I leave the final decision on this issue to the author.

Optional changes are formulated in the report.

This paper is devoted to the investigation of twisted Yangian based models
in the framework of the algebraic nested Bethe ansatz.
It is a continuation of the previous work
[A. Gerrard, N. MacKay, V. Regelskis, Nested algebraic Bethe ansatz for open spin
chains with even twisted Yangian symmetry, Ann. Henri Poincare 20, 339–392 (2019)]
to the odd case when underlying Lie algebra is $\mathfrak{gl}_{2n+1}$.
Among results obtained in the present paper there are the recurrence relations for the
Bethe vectors. As it was mentioned by the author in the conclusion, these
relations provide elegant expressions when the rank is small. For the higher rank
the recurrence relations become rather complex, but nevertheless they open
a door to investigation of the norms and scalar products of the Bethe vectors
for the open spin chain models.

The paper
is rather technical and it will require a lot of work from the reader to understand
author's notations and calculations. Nevertheless it
presents new results on the twisted Yangian based
quantum integrable models and met Journal's acceptance criteria.
I would like to recommend the paper
”Bethe vectors and recurrence relations for twisted Yangian based models”
by Vidas Regelskis
for publication in SciPost Physics Journal after several corrections listed below.

{\bf Line 106}. In the definition of transposed matrix $\widehat{S}^{(2\hat{n})}$
should be $\widehat{\omega}(S^{(2\hat{n})})$, not $\widehat{\omega}(R^{(2\hat{n})})$.
{\bf Formula (20)}. What is a definition of $\sigma(i)$ and $\sigma(j)$? Below,
in the line 227, there is a definition of $\sigma_i$. Are $\sigma(i)$ and $\sigma_i$
the same? If yes, then this definition should be replaced from the page 10 to some place
earlier around equality (20). If no, then a definition of $\sigma(i)$ should be provided.
{\bf Formula (43)}. For the reader convenience, I think it is reasonable to recall
definition of $\tilde\nu$ in this formula writing, for example,
$$
p(\nu):= 1\pm\frac{1}{\nu-\tilde\nu}=1\pm\frac{1}{2\nu+\rho}\,.
$$
{\bf Line 257}. Correct typo 'repating' $\to$ 'repeating'.
{\bf Line 375}. Correct typo 'onsider' $\to$ 'consider'.