{"id":"dbc0fff2-95ca-48b4-9585-7927276a8b2f","arxiv_id":"2610.26704","paper_version":1,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For d>3, the on-shell higher-spin algebra is realized explicitly on AdS-traceless polynomials, with a closed-form projector, a unique representative for each class, and an explicit differential-operator product verified on the so(d-1,2) generators.","lead":"Physicists studying theories of massless particles with every possible spin on anti-de Sitter spacetime rely on an abstract algebra of symmetries, but its multiplication was only defined implicitly. This paper writes that multiplication down as an explicit differential operator for spacetime dimensions above three and proves that the choice of representative it uses is unique.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek/deepseek-v4-flash","headline":"No significant objection identified: the quasiprojector is used only against polynomials, never squared, so the uniqueness proof stays inside the cited functional class.","rationale":"The reader's weakest assumption is the quasiprojector functional class. I agree that this is the only place where infinite series meet the star product, but I read the proof as avoiding the genuinely ill-defined product. Section 4.2 never needs Δ∗Δ; it needs Δ∗(polynomial) and associativity with one series factor, which the paper states and cites. Appendix C derives the needed identities from terminating polynomial computations. The product formula (4.30) and its associativity follow from the general representative construction once uniqueness holds; the generator computations in Section 4.5 are an independent check that the coefficients (3.33) and (4.27) are not merely formal. The d=3 and A∞ limitations are stated explicitly and do not touch Theorem 4.3's d>3 polynomial statement. I therefore see no reason to change the reader's ACCEPT; the residual uncertainty is the same moderate uncertainty about unpublished long appendices noted by the reader.","tokens_in":57613,"tokens_out":30706,"duration_ms":217075,"concrete_test":"Pick d=5 and a generic traceless monomial f̃_s (or a generic covariant A^{αβ}), and verify coefficient-wise, by computer algebra or by hand, the two identities that carry the uniqueness proof: Δ(0)∗(t^{αβ}∗A)=0 and Δ(0)∗f̃_s=Δ(t^2;s)f̃_s, expanding Δ(0) as the formal series (4.11) and using only the terminating commutation identities (C.4)-(C.7). If both hold without ever forming Δ(0)∗Δ(0), then the functional-class caveat quoted from [47] is sufficient and the proof of ker Γ∩I_sp(2)={0} stands.","verdict_should_be":"UNCHANGED","load_bearing_attack":"After tracing every use of Δ(t^2;0) in the argument for Theorem 4.3, I do not find a load-bearing gap. The uniqueness proof in Section 4.2 uses Δ only in products Δ∗(t^{αβ}∗A)=(Δ∗t^{αβ})∗A and Δ∗f̃_s=Δ(t^2;s)f̃_s; each has at most one non-polynomial factor, exactly the regime declared well defined in Section 2.4 and [47]. The divergent product Δ∗Δ never enters. The same is true of Proposition 4.1 and of the r_m extraction in Section 4.3, where Δ multiplies the polynomial (t^2)^m f̃_s. The d=3 degeneracy of Δ(s) is explicitly removed from the theorem's hypotheses. The projector coefficients solve a well-posed recurrence, and the generator checks (4.34)-(4.36) provide independent low-degree confirmation. The central isomorphism is therefore as secure as the published functional-class hypothesis it inherits; no new unproved regime is introduced by this paper.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek/deepseek-v4-flash","summary":"The paper constructs, for d > 3, an explicit realization of the polynomial on-shell higher-spin algebra hso_p(d-1,2|2) = hsgl_p(d-1,2|2)/I_sp(2) on the space of AdS-traceless sp(2)-singlet polynomials in the oscillator variables Y^α_A. The main technical steps are: a closed-form horizontal projector P onto ker Γ_{αβ} with coefficients (3.33), a proof that the splitting into a traceless representative and an element of the sp(2)-ideal is unique (Section 4.2), an explicit splitting map (4.28), and the induced product (4.30), stated as Theorem 4.3. The product is tested on the so(d-1,2) generators, reproducing the undeformed commutator (4.34), the Joseph-ideal relation (4.35), and the minimal-representation Casimir value (4.36). The paper also proves that Lorentz-traceless polynomials form another space of representatives, analyzes the degenerate d = 3 case (where uniqueness of the splitting is left open), and presents a toy model for lifting the factorization to the lowest A∞ higher product m_3, explicitly not covering the sp(2)-ideal.","tokens_in":57769,"tokens_out":17103,"duration_ms":123748,"significance":"If correct, the paper fills a concrete gap: it turns the conceptually defined factorization hso_p = hsgl_p/I_sp(2) into a computable differential-operator product on traceless representatives. The central derivation is self-contained and internally consistent: the projector coefficients (3.33) solve the stated recurrence, the uniqueness proof uses the quasiprojector only in products with at most one non-polynomial factor, and the independent low-degree checks (4.34)-(4.36) confirm the construction on the isometry generators. The paper is honest about its scope: the functional-class problem for non-polynomial gauge fields, the d = 3 uniqueness question, and the lift of the factorization to the full A∞ structure are all stated as open. The potential concern that the uniqueness argument squares the divergent quasiprojector does not land: every occurrence of Δ(t^2;0) in Sections 4.2-4.3 multiplies a polynomial, exactly the regime declared well defined in Section 2.4 and inherited from [47]. The main result is therefore a solid technical advance within its stated hypotheses.","major_comments":[],"minor_comments":[{"comment":"The proof of uniqueness is the only place where the quasiprojector Δ(t^2;0) is used, and the paper already states the functional-class caveat that products are defined when at most one factor is a series. It would be helpful to add one sentence making explicit that no product Δ∗Δ or any two-series product occurs in the displayed arguments, since otherwise the divergence of Δ∗Δ could distract a reader assessing the proof.","section":"Section 4.2","authors_acknowledge":null,"authors_acknowledge_p":null},{"comment":"The denominator in the prefactor of Δ(t^{μν}t_{μν};s) is typeset with an extra closing parenthesis, reading Γ(s+d/2))Γ((d-3)/2); the typographical error should be corrected.","section":"Equation (4.9)","authors_acknowledge":null,"authors_acknowledge_p":null},{"comment":"In the product formula (4.30), P is placed outside the whole sum, whereas the splitting map (4.28) has P applied inside each term. These agree because P is linear and grade-preserving and the coefficients r_m are scalar functions of the degree, but the equality is not stated. A sentence identifying the right-hand side of (4.30) with Σ_m r_m(deg) P[(ΓΓ)^m(f∗g)] would remove ambiguity about the ordering of N̂, ΓΓ, and P.","section":"Theorem 4.3 / Equation (4.30)","authors_acknowledge":null,"authors_acknowledge_p":null},{"comment":"The text says that a concrete choice of A_{αβ}(f) realizing (4.28) will be given in [57]. Since (4.28) already gives the composite map t^{αβ}∗A_{αβ}(f) explicitly, it would be clearer to state that only the two-tensor A_{αβ}(f) itself, not its star-contracted image, is deferred to future work.","section":"Conclusion / Section 4.3","authors_acknowledge":null,"authors_acknowledge_p":null},{"comment":"The proof of Theorem 5.3 says the labelled groups are 'distinguished by colour', but the arXiv text shows only labels I_k,...,VII_k and no colours. Either implement the colours or delete the remark.","section":"Appendix A","authors_acknowledge":null,"authors_acknowledge_p":null},{"comment":"The toy model is clearly labelled as not covering I_sp(2), and the introduction should perhaps state even more prominently that Section 5 is logically independent of Theorem 4.3. As written, the transition from the main theorem to the A∞ discussion may give a reader the impression that the on-shell A∞ lift has been achieved for the sp(2) ideal, which the paper explicitly says remains open.","section":"Section 5","authors_acknowledge":null,"authors_acknowledge_p":null}],"recommendation":"minor_revision","confidential_remarks":"The manuscript is within the journal's scope and I see no citation or novelty concerns beyond the acknowledged reliance on [47] for the functional class of the quasiprojector. The Section 5 material is exploratory, but the paper is explicit about its status. My recommendation of minor revision is based on local presentation issues only."},"author_rebuttal":null,"desk_editor":{"model":"deepseek/deepseek-v4-flash","letter":"This is the paper that fills a concrete gap: Vasiliev and Joung-Mkrtchyan gave the on-shell algebra as a concept and through invariants or structure constants, but nobody had written the product directly on traceless representatives. For d>3, in the polynomial sector, this paper does it. The horizontal projector (3.33) is closed-form, the trace-producing operators are classified including the linear relations (3.18)-(3.19), and uniqueness of the splitting is proved by a lowest-degree argument. The resulting product (4.30) is an explicit differential operator, and the checks on the so(d-1,2) generators land: undeformed isometry commutator, Joseph-ideal relation, and quadratic Casimir equal to the minimal-representation value. That last check is not fitted; it uses the explicit projector coefficients at low degree. The appendices are heavy, but the main line is readable and the computations look internally consistent. The scope limitations are real and openly stated: d>3 only, polynomials only, and the concrete A^{alpha beta} factorisation map deferred to [57], which is listed as in preparation. That is fine for the stated theorem, but it means the step toward actual interactions is not in this paper. Section 5 is explicitly a toy model and should be read as such; the Lorentz-traceless representatives are proved to exist, but the explicit factorisation map for them is not constructed. The place to be suspicious is the quasiprojector in the uniqueness proof. I checked every use of Delta in Sections 4.2 and 4.3: it always multiplies a polynomial and is never star-squared with itself, so the products stay inside the functional class the paper explicitly inherits from Vasiliev. The stress-test note is right about this. The paper does not re-derive that functional class, but it does not need to for Theorem 4.3. At d=3 the quasiprojector degenerates and the uniqueness proof fails; the authors say so, and the d=3 case remains open. The citation pattern looks appropriate: [47] and [51] are the right anchors, and the paper does not oversell what is new. No parameter is fitted to any target; the coefficients are solved from the defining equations of the projector and splitting. Who gets value: anyone working with Vasiliev equations in AdS_d, spin-locality, or on-shell higher-spin algebras. The paper deserves a serious referee. My recommendation: send it to peer review; the main theorem and appendix computations should be checked line by line, but I do not see a conceptual hole.","headline":"This paper actually constructs the explicit on-shell higher-spin algebra product on traceless representatives for d>3 polynomials, and the main theorem holds up under scrutiny.","tokens_in":59162,"tokens_out":962,"would_cite":true,"duration_ms":32428,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"pith_extraction":{"msc":["17B81","81R10"],"pacs":[],"model":"deepseek/deepseek-v4-flash","headline":"For $d>3$, the paper gives the on-shell higher-spin algebra of $AdS_d$ an explicit product on traceless polynomials, with a unique representative in every class.","keywords":["higher-spin gauge theory","on-shell higher-spin algebra","AdS_d space","Moyal star product","sp(2) ideal","traceless projector","unfolded formulation","minimal A-infinity algebra"],"falsifier":"For $d=4$ and each degree $2s\\le 12$, solve the linear system defining $I_{\\mathfrak{sp}(2)}\\cap\\ker\\Gamma_{\\alpha\\beta}$ using only finite polynomial arithmetic, with the ideal generated by $t_{\\alpha\\beta}\\ast A^{\\alpha\\beta}$ for polynomial $A^{\\alpha\\beta}$. The theorem predicts only the zero solution; a single non-zero traceless element of the ideal at any degree refutes it, while pushing such a finite proof to all degrees would remove the dependence on the divergent series. A weaker check is to verify the identities (4.12)\\u2013(4.13) on truncated series and confirm that the residuals at each degree in (4.15) vanish as the truncation order grows; any residual that fails to vanish would expose the functional-class gap.","tokens_in":58912,"feed_emoji":"✳","tokens_out":17942,"duration_ms":144633,"temperature":0.7,"pith_summary":"Higher-spin theories describe massless fields of every spin interacting with each other and with gravity; on an anti-de Sitter background they are formulated through an off-shell algebra whose quotient by the $\\mathfrak{sp}(2)$ ideal is the on-shell algebra that carries the actual dynamics. Until now that quotient was understood abstractly: one knew it existed, but not how to multiply two of its elements concretely. This paper fills the gap for spacetime dimension $d>3$: it proves that every class in the polynomial on-shell algebra has exactly one AdS-traceless representative, and writes the induced product as an explicit differential operator combining the Moyal star product with a closed-form horizontal projector. The product is tested on the $\\mathfrak{so}(d-1,2)$ isometry generators, where it leaves the commutators undeformed and produces the quadratic Casimir of the minimal representation, and a toy model indicates how the same factorisation could be lifted from the binary product to the first higher product of an $A_\\infty$-algebra. The construction is proved for $d>3$ and for polynomials only; the $d=3$ case degenerates and is left open, as is the functional class of series needed for genuine interactions.","feed_headline":"On-shell higher-spin algebra in AdS_d gets an explicit product","feed_subtitle":"Every class has one traceless polynomial representative; the induced product is a concrete differential operator.","key_machinery":"The load-bearing objects are two operators built from the generating variables $Y^{\\alpha A}$, an $\\mathfrak{sp}(2)$ doublet of $AdS_d$ vectors. The horizontal projector $P$ maps any $\\mathfrak{sp}(2)$-singlet polynomial to its AdS-traceless part while preserving degree; it is written as a sum over trace-producing operators $T^{(k,m)}=(t^{\\mu\\nu})^k(t^{\\sigma\\lambda}t_{\\sigma\\lambda})^m(\\Gamma_{\\mu\\nu})^k(\\Gamma_{\\sigma\\lambda}\\Gamma^{\\sigma\\lambda})^m$, with gamma-function coefficients (3.33) fixed by requiring $P$ to be the unique $\\mathfrak{so}(d-1,2)$-covariant, grade-preserving projection onto $\\ker\\Gamma_{\\alpha\\beta}$. The quasiprojector $\\Delta(t^{\\mu\\nu}t_{\\mu\\nu};0)$ is an infinite series in the central invariant $t^{\\mu\\nu}t_{\\mu\\nu}$ that kills the ideal generators through $t_{\\alpha\\beta}\\ast\\Delta=0$; its decisive property is that on traceless polynomials the star product collapses to multiplication by the known series $\\Delta(t^{\\mu\\nu}t_{\\mu\\nu};s)$, which turns the uniqueness question into a degree-by-degree comparison. The splitting map (4.28) combines these two operators, and the induced product (4.30) composes the Moyal product with the projector, the gamma ratios and the trace operator $(\\Gamma_{\\alpha\\beta}\\Gamma^{\\alpha\\beta})^m$ supplying precisely the corrections that remove the ideal.","core_discovery":"The central claim is Theorem 4.3: for $d>3$, the polynomial on-shell higher-spin algebra $\\mathrm{hso}_p(d-1,2|2)$, defined as the quotient $\\mathrm{hsgl}_p(d-1,2|2)/I_{\\mathfrak{sp}(2)}$ of the off-shell algebra by the two-sided $\\mathfrak{sp}(2)$ ideal, is isomorphic to the algebra of $\\mathfrak{sp}(2)$-singlet, AdS-traceless polynomials with the product $$f\\circ g = P\\left[\\sum_{m=0}^\\infty \\frac{\\Gamma((d+\\hat N)/2)}{\\Gamma((d+\\hat N)/2+m)}\\,\\frac{(\\Gamma_{\\$\\alpha$\\$\\beta$}\\$Gamma^{{\\alpha\\beta}}$/8)^m}{m!}\\,(f\\ast g)\\right],$$ where $\\ast$ is the Moyal star product, $\\hat N$ is the degree operator, and $P$ is the horizontal projector (3.33) that sends each polynomial to its traceless part. The proof rests on three pieces: a closed-form projector derived from the multiplicity-free branching of two-row $\\mathfrak{gl}(d+1)$ diagrams under $\\mathfrak{so}(d+1)$; a uniqueness theorem, $\\ker \\Gamma_{\\alpha\\beta}\\cap I_{\\mathfrak{sp}(2)}=\\{0\\}$, proved with the quasiprojector $\\Delta(t^{\\mu\\nu}t_{\\mu\\nu};0)$, an infinite series whose star products with traceless polynomials reduce to multiplication by known series; and an explicit splitting map (4.28), so every class has exactly one traceless representative. As a consistency test the product is evaluated on the generators $M_{AB}$: the commutator is undeformed (4.34), one Joseph-ideal relation is recovered, and the quadratic Casimir comes out as $(d+1)(d-3)/2$, the value of the minimal representation. The authors leave two boundaries explicit: at $d=3$ the quasiprojector family degenerates and uniqueness of the splitting remains open, and the whole construction lives on polynomials, whereas genuine interactions require a functional class closed under the induced product.","pith_inferences":["The uniqueness argument is the part most exposed to the divergent-series caveat: the identities $t_{\\alpha\\beta}\\ast\\Delta=0$ and $\\Delta\\ast\\tilde f_s=\\Delta(t^2;s)\\tilde f_s$ are invoked for series whose self-products are ill defined, with validity quoted from [47]. A finite, truncated test of the degree-by-degree cancellation (4.15)\\u2013(4.16) for the first several degrees would show whether t","With the product explicit, the natural next step is to compute on-shell cubic vertices in generic $d$ by applying (4.30) to the bilinear and trilinear forms of [51]; agreement with known results in $d=4$ and $d=5$ would be an independent check of the whole construction.","The $d=3$ failure is structural\\u2014the traceless module splits into self-dual halves invisible to the ambient oscillators\\u2014so recovering the one-parameter family of three-dimensional higher-spin algebras likely requires an enlarged oscillator set; if it is reproduced, the parameter value at the undeformed point would locate the ambient construction within that family.","The syzygy (2.32)\\u2013(2.33) among the ideal generators means the splitting map is fixed only up to a non-trivial freedom; the $A_\\infty$ analysis of Section 5 turns this freedom into concrete consistency requirements on the first interaction product, giving a checklist that any future construction of $m_3$ for $I_{\\mathfrak{sp}(2)}$ must pass."],"forward_implications":["Every element of the polynomial off-shell algebra splits uniquely as a traceless representative plus an element of the $\\mathfrak{sp}(2)$ ideal, so the on-shell algebra is a computable space of traceless polynomials rather than an abstract quotient.","Because the product is an explicit differential operator, structure constants and ideal relations of $\\mathrm{hso}_p(d-1,2|2)$ can be evaluated by direct differentiation, as demonstrated on the $\\mathfrak{so}(d-1,2)$ generators with the undeformed commutator (4.34), the Joseph-ideal relation (4.35), and the minimal-representation Casimir (4.36).","Lorentz-traceless polynomials form a second space of representatives for the same ideal, giving an alternative model of the algebra that is better adapted to the Weyl sector of the theory.","In the $A_\\infty$ framework, factorisation of an associative algebra by an ideal lifts from the binary product to the first higher product $m_3$ under the explicit conditions (5.17)\\u2013(5.18), indicating a concrete route from off-shell interaction vertices to on-shell ones.","Because $P$ is an explicit differential operator in the generating variables, the factor-algebra can be handled directly in terms of representatives, which the authors note may be convenient within the sources formalism of the spin-locality analysis."],"supporting_citations":[{"why":"Establishes the off-shell algebra, the $\\mathfrak{sp}(2)$ ideal, the quasiprojector with $t_{\\alpha\\beta}\\ast\\Delta=0$, the existence of the splitting, and the functional class in which quasiprojector star-products are defined; the paper's uniqueness and product construction build directly on it.","marker":"[47]"},{"why":"Introduces the nonlinear generating equations for interacting higher-spin fields, whose gauge-sector algebra is the target object of the factorisation.","marker":"[2]"},{"why":"Defines the on-shell algebra as the quotient by the $\\mathfrak{sp}(2)$ ideal in any dimension, the object the paper realises explicitly.","marker":"[5]"},{"why":"Identifies the on-shell algebra with higher symmetries of the Laplacian, the interpretation the explicit traceless realisation makes concrete.","marker":"[49]"},{"why":"Supplies the earlier computation of bilinear and trilinear invariant forms via the quasiprojector, the baseline the explicit product extends, and the comparison case for the $d=3$ degeneration.","marker":"[51]"},{"why":"Provides the $\\mathfrak{gl}(d+1)$-to-$\\mathfrak{so}(d+1)$ branching rules for two-row diagrams that fix the form and coefficients of the horizontal projector.","marker":"[60]"},{"why":"Gives the AdS$_3$ deformed-oscillator realisation whose one-parameter family the ambient construction at $d=3$ does not reproduce.","marker":"[48]"},{"why":"Constructs all-order off-shell vertices whose factorisation motivates the explicit on-shell product and the sources-formalism application.","marker":"[12]"}],"fun_headline_variants":["Explicit product for AdS higher-spin algebra, traceless reps for d>3","Traceless projector yields explicit on-shell higher-spin product in AdS_d","AdS_d higher-spin algebra product made explicit on traceless reps","One traceless representative per class, explicit product in AdS_d"],"cache_read_input_tokens":0,"weakest_assumption_plain":"The proof that no traceless polynomial lies in the $\\mathfrak{sp}(2)$ ideal multiplies polynomials by the quasiprojector $\\Delta(t^{\\mu\\nu}t_{\\mu\\nu};0)$, an infinite series whose star-square is ill-defined, and relies on the claim\\u2014quoted from [47], not re-derived here\\u2014that star products are well behaved whenever at most one factor is such a series.","fun_headline_variants_meta":{"raw":{"variants":["Explicit product for AdS higher-spin algebra, traceless reps for d>3","Traceless projector yields explicit on-shell higher-spin product in AdS_d","AdS_d higher-spin algebra product made explicit on traceless reps","One traceless representative per class, explicit product in AdS_d"]},"model":"deepseek/deepseek-v4-flash","effort":"low","cost_usd":9.8e-05,"raw_usage":{"cost":9.7825e-05,"is_byok":false,"cost_details":{"upstream_inference_cost":9.7825e-05,"upstream_inference_prompt_cost":3.2851e-05,"upstream_inference_completions_cost":6.4974e-05},"total_tokens":1542,"prompt_tokens":1185,"completion_tokens":357,"prompt_tokens_details":{"audio_tokens":0,"video_tokens":0,"cached_tokens":1030,"cache_write_tokens":0},"completion_tokens_details":{"audio_tokens":0,"image_tokens":0,"reasoning_tokens":281}},"tokens_in":1185,"tokens_out":357,"cost_source":"openrouter_usage","duration_ms":7296,"image_pages":[],"temperature":1.0,"billed_attempts":1,"reasoning_tokens":281,"transport_provider":"openrouter","cache_read_input_tokens":0,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-10-20T06:55:36.181330+00:00","model_set":{"reader":"deepseek/deepseek-v4-flash"},"falsifier":"For $d=4$ and each degree $2s\\le 12$, solve the linear system defining $I_{\\mathfrak{sp}(2)}\\cap\\ker\\Gamma_{\\alpha\\beta}$ using only finite polynomial arithmetic, with the ideal generated by $t_{\\alpha\\beta}\\ast A^{\\alpha\\beta}$ for polynomial $A^{\\alpha\\beta}$. The theorem predicts only the zero solution; a single non-zero traceless element of the ideal at any degree refutes it, while pushing such a finite proof to all degrees would remove the dependence on the divergent series. A weaker check is to verify the identities (4.12)\\u2013(4.13) on truncated series and confirm that the residuals at each degree in (4.15) vanish as the truncation order grows; any residual that fails to vanish would expose the functional-class gap.","supporting_citations":[{"cited_title":"Higher spin superalgebras in any dimension and their representations,","cited_arxiv_id":null,"evidence_quote":"Establishes the off-shell algebra, the $\\mathfrak{sp}(2)$ ideal, the quasiprojector with $t_{\\alpha\\beta}\\ast\\Delta=0$, the existence of the splitting, and the functional class in which quasiprojector star-products are defined; the paper's uniqueness and product construction build directly on it."},{"cited_title":"Consistent equation for interacting gauge fields of all spins in (3+1)- dimensions,","cited_arxiv_id":null,"evidence_quote":"Introduces the nonlinear generating equations for interacting higher-spin fields, whose gauge-sector algebra is the target object of the factorisation."},{"cited_title":"Nonlinear equations for symmetric massless higher spin fields in (A)dS(d),","cited_arxiv_id":null,"evidence_quote":"Defines the on-shell algebra as the quotient by the $\\mathfrak{sp}(2)$ ideal in any dimension, the object the paper realises explicitly."},{"cited_title":"Higher symmetries of the Laplacian,","cited_arxiv_id":null,"evidence_quote":"Identifies the on-shell algebra with higher symmetries of the Laplacian, the interpretation the explicit traceless realisation makes concrete."},{"cited_title":"Notes on higher-spin algebras: minimal representations and structure constants,","cited_arxiv_id":null,"evidence_quote":"Supplies the earlier computation of bilinear and trilinear invariant forms via the quasiprojector, the baseline the explicit product extends, and the comparison case for the $d=3$ degeneration."},{"cited_title":"The unitary representations of the Poincar´ e group in any spacetime dimension,","cited_arxiv_id":null,"evidence_quote":"Provides the $\\mathfrak{gl}(d+1)$-to-$\\mathfrak{so}(d+1)$ branching rules for two-row diagrams that fix the form and coefficients of the horizontal projector."},{"cited_title":"Quantization on sphere and high spin superalgebras,","cited_arxiv_id":null,"evidence_quote":"Gives the AdS$_3$ deformed-oscillator realisation whose one-parameter family the ambient construction at $d=3$ does not reproduce."},{"cited_title":"All vertices for unconstrained symmetric gauge fields,","cited_arxiv_id":null,"evidence_quote":"Constructs all-order off-shell vertices whose factorisation motivates the explicit on-shell product and the sources-formalism application."}],"review_version":1}