Fusion in the periodic Temperley-Lieb algebra and connectivity operators of loop models

Dear Editors,

We thank the referees for their careful reading and valuable comments on our manuscript. Please find below a list of our changes and replies following their requests.

Best regards,
Yacine Ikhlef and Alexi Morin-Duchesne

Answer to the requested changes of the 1st referee:

1. The main result (Theorem 1) needs some comments: while the representation depends on the parameters x and y they disappear completely in the decomposition in right hand side. I think that authors should comment about the role of these parameters in the module decomposition.

We added a short comment below Theorem 1 and a longer paragraph in the conclusion to justify this fact. Please note that we also changed the statement of Theorem 1 to include all complex values of x and y, and not only the generic ones.

2. The part on the correlation functions (subsection 5.2) is extremely sketchy. The Gram product with defects is not really defined and there is no comments about the parameters of the Gram product pertinent for the computation of the correlation functions.

We have reworked the text below (5.15) to include a more explicit definition of the Gram product. We also added a note at the beginning of Section 5 warning the reader that this section is more concise than the rest of the paper.

3. The authors should correct the first example in (3.10c)

The rightmost diagram in the first example of (3.10c) has been fixed.

4. The authors should (probably) better explain the identification (3.11) it is not completely clear how it appears.

We expanded our explanation of the example (3.11) to make it clearer.

Answer to the requested changes of the 2nd referee:

1- Address the apparent inconsistency in Section 3.2 as per above.

We reworked the end of Section 3.2 to make it clearer why the allocation of the parameter \mu is indeed correct and yields a well-defined representation.

2- Clarify Section 3.3 as per above.

The inclusion of EPTL_{Na} \otimes EPTL_{Nb} in EPTL_{Na+Nb} is indeed not obvious. As discussed in our conclusion, this construction is given in the papers [37,38] and is however not required in our definition of fusion. We also strongly believe that keeping the notation with an x in W_{k,x}(N) is not misleading, contrary to what the referee claims. We justify this in a new paragraph at the beginning of the conclusion, that also addresses point 4 below. Moreover, we have largely reworked section 3.3 to make the characterization of the modules X in terms of the gluing clearer, stating clearly a full algebraic definition of the modules X is missing.

3- Explain the associativity of the proposed fusion (preferably at the level of Sections 2-4 rather than 5)

We added a paragraph in section 6 explaining why this question cannot be answered at this stage.

4- Theorem 1: the right-hand side is independent of x,y, which seems surprising (cf the standard modules W_{k,x} and W_{k,y} are not isomorphic). Can you comment on this?

This was also a request of the first referee, see our answer above under point 1.

Additional minor requests/comments

- Sect 1 (and perhaps 2.1): mention the related work of Al Qasimi et al, arXiv:1710.04058 and 1903.08677

We added a reference in the introduction to the first paper mentioned above.

- Sect 2.1: the definition of [k]q is somewhat unusual; although I understand the connection to the fairly common short-hand [x]=θ(x),sinh(x),sin(x),… I would be in favour of letting [k]_q=(q^k−q^{−k})/(q−q^{−1}) be the q-analogue of k∈N, which would make β=−[2]_q look simpler, whereas (2.5) and App A,B are unaffected as the values occur in ratios.

We changed the notation to the usual q-numbers, and moved this definition to Section 2.2, as these weren’t used earlier in the text.

- Jones-Wenzl: perhaps point out that these are not orthogonal

The orthogonality of the projectors appears to us to be irrelevant to the discussion and results of this paper, so we prefer not to mention this point.

- Sect 2.2: might be useful to include the relevant references (probably mentioned at the start of 2.1) for the special elements in Sect 2.2

We have added the relevant references.

- F, \bar F and Ω^N all lie in the center of EPTLN; do they generate it?

This is a question that is addressed in reference [46]. To our knowledge, this paper has not been published, so we prefer to avoid commenting on this point here.

- Sect 2.3: mention z∈C× from the start

This has been implemented.

- define 'link state' (= basis element in terms of diagrams without closed loops and up to homotopy?)

We complemented the 2nd, 3rd and 4th sentences of (2.3), which already defined link states, with an extra sentence about equivalence up to homotopy. We have added a similar sentence about homotopy of elements of EPTL_N in section 2.1.

- p7: it would be useful to give or summarise the rules defining the standard action diagrammatically for easy reference

We are a bit confused here as per the referee’s request, as to our sense this is precisely what the paragraph above (2.12) aims to do.

- "Loops cannot encircle the marked point in this case": perhaps clarify that the action of F, \bar F has to be expanded via (2.7)

We added a sentence below (2.8) to make this clearer.

- emphasise that z enters in separate ways depending on whether k=0 (via α=z+z^{−1}) or k>0; are these occurences completely independent?

We added a sentence above (2.11).

- (2.14) would be more clear with α rather than z+z^{−1} (easier comparison for the case of two defects later on)

We used \underbrace to include both.

- the line between (2.21) and (2.22) further uses that the standard modules are also independent under inverting q

Comment added.

- after (2.22) perhaps point out that these homomorphisms will be constructed explicitly in Sect 2.5

Note added.

- still p8: "... have identical eigenvalues..." clarify: the same eigenvalue (as opposed to the same spectrum, including multiplicities)

As is clear from (2.18), F has a unique eigenvalue on W_{k,z}, so we believe the statement is clear as is, indicating that the unique eigenvalue in each module is identical between the two modules, independently of their size.

- quotient W_{k,εq^{±l}}/... should be by the *image* of W_{l,εq^{±k}} or by the irrep I.

Change made.

- I'm confused by (2.24) and the following text: either the Is are irreducibles (quotient for the head), *or* they are summands (Grothendieck sum) that are not direct sums (as modules) so that the arrow and sentence "... can produce non-zero states" seem to make sense; please clarify

We added a short paragraph below (2.24) that clarifies what we mean by composition factors and recalls the definition of Loewy diagrams, pointing to reference [7] for a more thorough discussion.

- Sect 2.5: the discussion seems to fix k,l and vary N, while a priori it seems more natural to fix k,N and vary l; although equivalent it would help to either change this or explain the viewpoint

We added a note at the beginning of section 2.5 detailing our strategy for this section.

- p10 (twice) and in the remainder of the text (several times): the authors write that states *generate* a (sub)module, where I believe it's meant that they *span* that space -- this is potentially confusing, as there is a (different) notion of (cyclic, or finitely generated) modules that are generated by a few vectors; please correct here and elsewhere

We have corrected “generate” for “span” in many places in the paper.

- below (2.29): "the action of ... on these four states is invariant" → leaves the states invariant?

We replaced “invariant” here by “closed”.

- just below, "... vanishing result if two nodes attached to the projector ..." and "... permutes the nodes tied to the projector ...": nodes = links?

We modified the second sentence accordingly.

- preceding (3.1): what is meant precisely by 'obtained from'? Is V_k(Na)×V_l(Nb) equal to (3.1), obtained from it in some way

We changed “obtained” for “defined”.

- a little before (3.5), "... the representations depend *only* on the sum N_a+N_b": as in their dependence on N_a,N_b is on the sum only, or it does not depend on anything else (I think it also depends on k,l)

Precision added.

- just before (3.5), "the fusion ... *is read from* the decomposition": what does this mean precisely?

We reformulated this sentence.

- (3.5) or text preceding: mention that N=N_a+N_b

Mention added.

- just below (3.5): here fusion = decomposition of tensor product? (in the context of Yangians, quantum loop algebras, etc it is used for a construction involving a *projection* onto one irreducible component in such a decomposition)

We replaced the word “fusion” by “tensor product” in this sentence.

- (3.8) and preceding text: in addition to these examples it would again be useful to give or summarise the rules defining the standard action diagrammatically for easy reference

The rules defining the standard action were already given in section 2.3. No changes made here.

- cases k>0,l=0 and k=0,l>0: need to change k↔l

Here we believe the referee refers to the first sentence below (3.8b). In the entire discussion up until we discuss the case k,\ell >0, the text does not involve the parameter \ell in any way, so indeed saying k -> \ell in the first sentence is correct.

- before (3.9), "the defects do not cross * the two dashed segments* ": is meant * either* or * both* ? (likewise in the case k,l>0)

Change made.

- p14, "In this construction, the point c ...": this was already relevant in the case k=l=0

We had initially not mentioned this for the case k = \ell = 0, as the point c plays absolutely no role in determining the weights appearing in the action of the algebra. After some though, we decided to move this sentence earlier as suggested.

- just below: "also connects the defects of b together": I believe that this should be a?

Change made.

-footnote 2: formulation unclear; "cyclic" means generated by a single vector, but here it's written that it's produced by the action on the states (plural)

We changed the formulation so that it says that the module is generated from the action of the algebra on *any* state of maximal depth.

- (3.13): clarify what "diagram : weight" means; can we replace the diagram by weight (or is it weight times the diagram with two marked points but no lines)?

We believe our new formulation around (3.15) makes this clearer.

- same equation: μ appears out of nowhere, whereas up to now the ordering of the two marked points didn't seem to matter; please clarify this before presenting those examples in (3.13)

We have reworked this subsection to make the presentation clearer.

- (3.15b), first equality: this actually seems to require something slightly stronger than the rules given so far, namely to move a defect through a dashed line (unwind it) in the case that both marked points have a defect, but without producing crossing blue lines; are we allowed to just exchange the positions of the marked points, or is this where we introduce a weight μ?

The first equality in (3.15b) uses only the rule for the winding of defects explained two paragraphs below (3.12): “A defect attached to point a crossing the segment bc is assigned a weight x^{-1}*z if b lies to its right and …”. No changes made here.

- likewise for (3.16), second equality

The rule here is likewise read from the same paragraph: “Finally, if a defect crosses more than one segment, then the resulting twist factor is the product of the corresponding weights.” Here this factor is 1/x * x/z = 1/z. To make this clearer, we have added new examples in (3.12) and referred to these examples in the third paragraph of the k,\ell>0 discussion.

- (3.16), first equality: this weight seems to differ from that in the middle diagram in the right-most column of (3.13)

Error corrected.

- Sect 3.3: mention again that N=N_a+N_b

Change made.

- (4.25) what's sign(0)?

Sentence added.

- preceding (4.32): the vector is at most unique up to nonzero scalar multiples

We changed the word “unique” for “single”.

- just above Prop 4.1, "unique state": again, it's certainly not unique without further qualification; replace e.g. by "single state"

Change made.

- right before Prop 4.1, "on the states of maximal depth": is meant any single state of maximal depth?

Yes. Correction made.

- Prop 4.1 or just below: recall that v_{k,l}(m) was defined in (4.13)

Sentence added.

- Sect 4.4, "reflect v vertically": just to clarify, is really meant that v should be reflected so that the dashed lines now meet at the top?

Yes this is correct.

- just before (4.34): is the scalar product to be extended bilinearly or sesquilinearly?

Change made. We also replaced “bilinear form” by “Gram product” in the rest of the text

- just before "Conjectural form for ...": is the radical *the* or *a* maximal submodule?

From the structure of the standard modules (described in Section 2.4), it is clear that for q generic and z non-generic, these modules have a unique non-trivial submodule, which is therefore also maximal.

- (5.2b): does this also hold for i=j+k−1 or only up to i=j+k−2?

Change made. We also corrected (5.2a).

- (5.2c): does this also hold for j=1? (seems to require some unwinding)

There was indeed a problem for j = 1. We modified this equation, with a range for the values of j appearing in the new version.

- (5.4): does this also hold when k>l, so that there will be at least on link from a to b?

We added bounds on k in (5.4).

- (5.11): there does not seem to be any need to indicate the little arcs in this context, without spectral parameter?

We removed the small arcs from the box operators.

- (A.1): for completeness consider explaining how the diagonal tile works, since the main text only uses square/rectangular tiles

We added a paragraph below A.1 describing this.

- p39, top: w_{k,z(l)} is not quite invariant (but the subspace it spans is)

We reformulated the text preceding proposition B.1.

- Prop B.1, proof: it would make more sense here to prove (B.8a) and leave (B.8b) as an exercise to be done analogously, since the proposition is actually proven from (B.8a)

In fact, both (B.8a) and (B.8b) are needed to prove the proposition. No change made here.

Finally, a few typos

- p7: "However, it will turn *out* to be more natural"

Change made.

- p8: "We say that the parameter z *of*"

Change made.

- Sect 3.1: TL_{N_a}⊗TL_{N_b} missing β

Change made.

- p13 top: no neither nor ?

Change made.

- below (3.12): "is a link state *in*"

Change made.

- p29, last line: redundant "are"

Change made.