Bethe vectors and recurrence relations for twisted Yangian based models

I'd like to thank the referees for their valuable comments and suggestions. Responses to the queries raised by the referees are given below.

Referee 1: Thank you for spotting the misprints and notation issues. These have been corrected. The notation $\tilde{v} = -v-\rho$ has been recalled before formula (3.35) (previously (43)).

Referee 2: Both Introduction and Conclusions sections were extended and a summary of the main results is now included.

Referee 3:

1. I agree with the referee that the parameter $\rho$ does not play a significant role in the paper, but it is important to applications of the results of this paper. This is now explained at the beginning of Section 2.3. Please see lines 140-145.

2. I have simplified Sections 2.3 and 2.4. I have removed the confusing part about the presentation of $Y^+(\mathfrak{gl}_{2n+1})$ in terms of the $(2n+2)\times (2n+2)$-dimensional generating matrix. Instead, I have defined "overlapping" matrix operators A, B, C and D in (2.15). Their crucial property is that in the $N=2n+1$ case they satisfy exchange relations of the same form as their non-overlapping counterparts in the $N=2n+2$ case. Taking matrix coefficients of (2.16)–(2.20) one obtains relations among generating series that coincide with those given by the defining relations (2.10) and (2.11). Please see lines 161-162.

3. It is not possible to construct $Y^-(\mathfrak{gl}_{2n+1})$ because it is not possible to anti-symmetrise the middle index $i=n+1$ giving $\theta_{n+1}=0$ and thus trivialising the series $s_{ij}(u)$ with $i=n+1$ or $j=n+1$.

4. The new Remark 2.1 explains the issues emerging in the $N=2n+1$ case that can not be deduced from the results of [GMR19].

5. The Introduction has now a summary of the main results of the paper.

6. I have clarified this point in Section 3.1. Please see lines 183-184.

7. I have rewritten the second half of Section 3.2 that introduces the nested quantum spaces. I added an explanation of the tuple labelling. Please see lines 207-210.

8. I agree with the referee that some statements in the paper could be omitted by referring to [GMR19] instead. However, the current paper uses a slightly different notation and a row-nesting instead of a column-nesting after the first (top) level of nesting. The two approaches give the same result, but the row-nesting is more natural for the purposes of the present paper. This is why many things are recalled (and reformulated to fit the row-nesting approach) from [GMR19].

9. The misprint has been corrected.

10. I have updated Section 4.1 discussing partitions of tuples. Please see lines 399-403.

11. This has been clarified now. The new Appendix A.3 recalls the relevant recurrence relation from [HL+17b] in terms of the notation used in this paper. Please see Proposition A.2 and Example A.3.

12. This notation is now included in (3.9).

Referee 4:

1. I agree with the referee that it is possible to choose a different full quantum space than the one defined in (3.1). For the nesting to work it is crucial that the individual quantum spaces can be restricted to irreducible representations of the reduced algebra. A tensor product of evaluation modules does have the wanted property (see lines 189-196). The paper is already technical, thus we do not consider more general constructions of the full quantum space.

2. This has been clarified. See lines 330-333 and Appendix A.2.

3. It is possible to consider situations when both boundary spaces are not one-dimensional. In such cases one would have to find a way to diagonalised one of the boundary spaces (see Remark 4.1 in [GMR19]) for the (usual) nested Bethe ansatz to work, or to employ off-diagonal Bethe ansatz techniques (see the book "Off-diagonal Bethe ansatz for exactly solvable models" by Y. Wang, W.-L. Yang, J. Cao and K. Shi). Alternatively, one could try employing separation of variables techniques (see new references [GLMS17] and [RV21]).

4. This is correct. The index $j$ does appear in both places. I have now included a derivation of formula (4.3) (previously (61)) where this property originates due to a repeated application of formula (4.2). Please see Proof of Corollary 4.2.

5. I thank the referee for this suggestion. I have updated the Introduction accordingly. Please see lines 85-99.

6. I thank the referee for this suggestion. The new Appendix A.1 explains this property.

- I have extended Introduction and Conclusions sections with additional references, a summary of the main results, and some open problems.
- I have simplified Section 2.3 and updated Section 2.4. A new Remark 2.1 was added.
- I have updated the discussion of quantum spaces in Section 3.1 and clarified Sections 3.2 and 3.3 by emphasizing the specificities of the N=2n+1 case.
- I have updated the explanation of partitions in Section 4.1.
- The previous expansion formula (61) is now presented in the new Corollary 4.2. A proof of this formula and an example of such an expansion when N=3 was added.
- An appendix was added. Appendix A.1 shows compatibility of recurrence relations with the weight grading of twisted Yangians. Appendix A.2 shows commutativity of transfer matrices. Appendix A.3 recalls a Y(n)-type recurrence relation used in this paper.