{"id":"dd276136-a289-4029-8c1f-fd28b01f4534","arxiv_id":"2507.19896","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Markov traces on Iwahori-Hecke algebras of types B and D are represented explicitly by central symmetric polynomials in Jucys-Murphy elements.","lead":"The authors give explicit formulas for the Markov traces on Iwahori-Hecke algebras of types B and D, written as symmetric polynomials in Jucys-Murphy elements. This gives a uniform algebraic handle on link invariants associated with these Hecke algebras and on related categorified invariants.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Sign error in definition of β_n(y): coefficient should be (y−α0), not (y+α0), for Proposition 3.2.3 and the geometric specialization to work.","rationale":"The reader's conditional verdict is appropriate. I initially looked for a gap in the uniqueness or bar-involution step, but the quoted classification theorem is standard and the paper's reductions to it are plausible. The concrete algebraic inconsistency in β_n(y) is more decisive: it is a displayed formula that cannot be true with the computation printed next to it. Since the error is local and easily repaired, the right verdict is conditional, not reject; the identification with known geometric traces depends on this sign and should be verified after the correction. This is why I set agreement_with_reader to partial: the reader noted the sign and y′ confusion but did not make it the primary load-bearing assumption.","tokens_in":10161,"tokens_out":15123,"duration_ms":156081,"concrete_test":"Compute in H_{v,v0}(B_1) (or in SageMath with explicit small-rank Hecke algebras): let β_1(y)=1+c t0+a^{-1}t0^2 and T_1=t0. Use the trace pairing to evaluate ⟨β_1(y), T_1⟩; since t0^2=α0 t0+1 and t0 is orthogonal to 1 in the pairing, this equals c+α0. The identification tr^B_{1,y}(T_1)=y requires c=y−α0. The printed c=y+α0 gives y+2α0, contradicting the theorem; the corrected definition passes. For a more global check, recompute Proposition 3.2.3 for n=2 with both signs and confirm the value on T_1T_2 is y^2 only with c=y−α0.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central identification Theorem 3.2.2 is not supported by the displayed formula for β_n(y). The definition in Section 3.2 sets β_n(y)=∏(1+(y+α0)j_i^B+a^{-1}J_i^B), but the proof of Proposition 3.2.3 requires the middle coefficient to be y−α0. The paper even prints both definitions: 'y′ = y − α0, y′ = y + α0'. Tracing the computation, the first summand contributes α0, the second contributes y′·⟨β_{n−1},ι(h)⟩, and the total is written as (α0+y′)⟨β_{n−1},ι(h)⟩ = y⟨β_{n−1},ι(h)⟩; hence y′=y−α0. With the printed coefficient y+α0, the same computation gives (y+2α0) instead of y. This is not cosmetic: the value of tr_{β_n(y)} on T_1⋯T_n would be (y+2α0)^n rather than y^n, so the uniqueness theorem (Theorem 3.2.1) no longer identifies tr_{β_n(y)} with the Geck–Lambropoulou trace tr^B_{n,y}. The geometric specialization in §3.3 also uses y−α0=0 to obtain tr^B_n=tr_{ζ_n}; with the printed sign, β_n(α)=∏(1+2αj_i^B+aJ_i^B)≠ζ_n. The likely fix is to replace y+α0 by y−α0 in the definition of β_n(y); after that the arguments appear to go through. But as printed, the main formula is wrong, and the conflicting notation for y′ must be resolved.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper gives a new construction of Markov traces for Iwahori-Hecke algebras in types A, B, and D, using traces represented by central elements built from symmetric polynomials in multiplicative Jucys-Murphy elements. In type A this recovers the known construction; in types B and D the authors identify the Geck-Lambropoulou one-parameter family of Markov traces tr^X_{n,y} with traces represented by explicit central elements β_n(y) and δ_n(y) (Theorem 3.2.2 and Corollary 3.2.5). The paper then specializes the formulas to the geometric Markov traces of Gomi, Webster-Williamson, and Bezrukavnikov-Tolmachov (Section 3.3). The overall strategy is to verify the defining properties of a Markov trace and invoke the Geck-Lambropoulou uniqueness theorem, with the internal computations based on the Kazhdan-Lusztig pairing and the Serre-type property of the full-twist elements.","tokens_in":10469,"tokens_out":13932,"duration_ms":158325,"significance":"If the sign issue identified below is corrected, this is a genuinely useful contribution: it provides explicit polynomial formulas for Markov traces in types B and D, gives a uniform treatment across types A, B, and D, and connects the algebraic traces with known geometric constructions. The formulas are clean and likely to be valuable for further work, including possible categorification. The authors report having checked their formulas numerically in SageMath, which is a strength, although no code or output is included. The reliance on the Geck-Lambropoulou uniqueness classification is appropriate and is not circular.","major_comments":[{"comment":"The displayed definition of β_n(y) uses (1 + (y + α0) j_i^B + a^{-1} J_i^B), but the proof of Proposition 3.2.3 requires the coefficient y′ to be y − α0. The text even prints both options in the sentence \"y′ = y − α0, y′ = y + α0\". Tracing the computation: the first summand contributes α0⟨β_{n−1}(y), ι(h)⟩, the second contributes y′⟨β_{n−1}(y), ι(h)⟩, and the total is written as (α0 + y′)⟨β_{n−1}(y), ι(h)⟩ = y⟨β_{n−1}(y), ι(h)⟩. Hence y′ must be y − α0. With the printed coefficient y + α0, the trace on T_1⋯T_n would be (y + 2α0)^n rather than y^n, so Theorem 3.2.1 no longer identifies tr_{β_n(y)} with tr^B_{n,y}. Similarly, the geometric specialization in Section 3.3 requires y − α0 = 0; with the printed sign one would obtain β_n(α) ≠ ζ_n. The definition of β_n(y) should be changed to use y − α0, and the contradictory notation for y′ should be removed.","section":"Section 3.2, definition of β_n(y) and proof of Proposition 3.2.3"},{"comment":"The pairing ⟨ , ⟩ is A-antilinear in the first argument by Proposition 2.3.1(b). The displayed expansion in the proof of Proposition 3.2.3 pulls the scalar y′ out of the first argument without a conjugate, which implicitly requires ar{y}′ = y′. However, the extension of the bar involution to R = A(y) is not defined: the sentence \"Extend the bar involution to R, formally writing y — we assume that such an extension exists under any specialization of y we will use\" does not state what ar{y} is. If ar{y} ≠ y, the computation would acquire a conjugate on y′. Moreover, for the geometric specialization y = v − v^{-1}, the naive assignment ar{y} = y is not compatible with ar{v} = v^{-1}. The authors should specify that y is a formal variable fixed by the bar involution and explain how substitution into the resulting formulas is meant to be performed after the formal computation. This is a rigor gap in a load-bearing step, although it appears to be easily fixable.","section":"Section 3.2, base change of the Kazhdan-Lusztig pairing"}],"minor_comments":[{"comment":"In the displayed expansion, the third summand is written as a⟨β_{n−1}(y)J^B_n, ι(h)T_n⟩, but the definition of β_n(y) contains the coefficient a^{-1} for J^B_i. The term vanishes in the subsequent argument, so this does not affect the conclusion, but the coefficient should be a^{-1} for consistency.","section":"Section 3.2, proof of Proposition 3.2.3"},{"comment":"In the formula for e′_k, the symbols e_i and e_j are used without being defined. Please state that they are elementary symmetric polynomials in x_1, …, x_n, and clarify the summation convention beyond the displayed condition i + j = 2k, i, j ≥ 0.","section":"Section 3.3, Theorem 3.3.1"},{"comment":"The overline notation is used both for the Kazhdan-Lusztig involution on H(W,S) and, later, for the bar involution on the coefficient ring R. The authors should state explicitly that the latter is an extension of the former, or use a different symbol for one of the two involutions.","section":"Section 2.3"}],"recommendation":"major_revision","confidential_remarks":"The manuscript is appropriate for a representation-theory journal and the central idea is attractive. The main problem is the sign error in the definition of β_n(y), which I believe is a correctable typo rather than a fundamental obstruction. The authors should also clarify the bar-involution convention on R = A(y). If these points are addressed, I would be willing to support acceptance."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague,\n\nThe short version: the paper has a good idea and the main theorem is probably true, but the central formula as printed has a sign error. The stress-test is correct: β_n(y) is defined with (y+α0), while the proof of Proposition 3.2.3 needs (y−α0). The text even prints both “y′ = y − α0” and “y′ = y + α0” in the same paragraph. With the displayed coefficient, the computation gives (y+2α0)⟨β_{n−1}, ι(h)⟩, not y⟨⋯⟩, so the identification with the Geck–Lambropoulou trace fails at the uniqueness step. That is load-bearing: Theorem 3.2.2 and Corollary 3.2.5 rest on it. The likely fix is a one-character change, but as submitted the formula is wrong.\n\nWhat is genuinely new and good: the uniform ζ_n Markov trace in all classical types (Theorem 3.1.4) is a clean construction, and the proof of the Markov properties via the Kazhdan–Lusztig pairing and the Serre property is elegant. The idea of representing the Geck–Lambropoulou traces by symmetric polynomials in multiplicative Jucys–Murphy elements is natural and, as far as I know, not in the literature. The type D reduction by even/odd degree parts is also neat. The connection to the geometric traces of Gomi, Webster–Williamson, and Bezrukavnikov–Tolmachov at the specializations y=α (type B) and y^2=−aα^2 (type D) is the right kind of payoff. The citation pattern is fine; self-citations are motivational, not load-bearing.\n\nOther soft spots are minor by comparison. Proposition 2.4.2’s displayed proof is too terse and has notation slips, but the statement is standard. The paper assumes without discussion that the bar involution extends to R=A(y) at the needed specializations; for the classification theorem that deserves a sentence or a citation. Small-n embeddings are excluded from the definition of Markov trace and handled ad hoc in Section 3.3; acceptable but should be more explicit.\n\nWho is this for: people working on Iwahori–Hecke algebras, Markov traces, and link invariants in solid tori, and people in the Soergel-bimodule/Khovanov–Rozansky world who care about categorifying these traces. They should read it after the sign fix. I would send it to a serious referee: the construction is explicit, checkable, and worth having in the literature, and the flaw looks repairable. Just make sure the referee knows to verify the sign in β_n(y) and to ask for a cleaned-up proof of the Serre property.","headline":"Good idea, likely fixable, but the central formula for β_n(y) has a sign error as printed; the Geck–Lambropoulou identification does not go through until it is corrected.","tokens_in":11064,"tokens_out":5476,"would_cite":true,"duration_ms":57214,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["20C08","20F55","57K10"],"pacs":[],"model":"deepseek-v4-flash","headline":"Jucys-Murphy polynomials give all Markov traces in types B and D","keywords":["Markov traces","Iwahori-Hecke algebras","Jucys-Murphy elements","types B and D","link invariants","central elements","Khovanov-Rozansky homology","HOMFLY-PT polynomial"],"falsifier":"Compute, in a computer algebra system, the trace $\\operatorname{tr}^B_{3,y}(T_1T_2T_3)$ using the explicit pairing $\\langle\\beta_3(y),T_1T_2T_3\\rangle$ and compare it with $y^3$ for generic $y,v,a$ over $\\mathbb{Q}(v,a)$; any deviation would falsify Theorem 3.2.2, and a similar check of $\\operatorname{tr}^D_{4,y}(U_1U_2U_3U_4)=y^4$ against $\\delta_4(y)$ would falsify Corollary 3.2.5.","tokens_in":9912,"feed_emoji":"🔗","tokens_out":11007,"duration_ms":102643,"temperature":0.7,"pith_summary":"This paper proves that the general Markov traces on Iwahori-Hecke algebras of types B and D, which had previously been classified only by existence and uniqueness, are represented by explicit central elements built from commuting Jucys-Murphy elements. In type B the representing element is $\\beta_n(y)=\\prod_{i=1}^n(1+(y+\\alpha_0)j_i^B+a^{-1}J_i^B)$; in type D it is the even-degree part of $\\prod_i(1+yx_i+a^{-1}x_i^2)$ evaluated at the elements $j_i^D$. The same mechanism yields a uniform Markov trace in types A, B and D with constants $\\rho=1+a$ and $\\mu=v-v^{-1}$. This matters because Markov traces are what turn Hecke algebras into link invariants, and the geometric Markov traces studied via categorification arise as specializations of these formulas.","feed_headline":"Jucys-Murphy polynomials give all Markov traces in types B and D","feed_subtitle":"Markov traces in types B and D are now explicit Jucys-Murphy polynomials; geometric traces are special cases.","key_machinery":"The engine is the family of multiplicative Jucys-Murphy elements $J^X_n=S(X_n)^{-1}S(X_{n-1})$ in $H(X_n)$, where $S(X_n)=t_{w_0}^{-2}$ is the inverse square of the longest-element basis element. These elements commute, and any symmetric polynomial in them is central. A functional is represented by a central element $z$ when $\\varphi(h)=\\langle z,h\\rangle$ for the natural pairing $\\langle h,h'\\rangle=\\tau(i(h)h')$. The proof of the Markov-move properties rests on the Serre property $\\langle h_1,Sh_2\\rangle=\\langle h_2,h_1\\rangle$, which lets a factor $J^X_n=S(X_n)^{-1}S(X_{n-1})$ move across the pairing, and on the Geck-Lambropoulou uniqueness theorem, which reduces verification to normalization and to the values on $T_1\\cdots T_n$ or $U_1\\cdots U_{2n}$.","core_discovery":"The paper's central claim is that the Geck-Lambropoulou Markov traces in types B and D, uniquely determined by normalization $(1+a)^n$ and by prescribed values on $T_1\\cdots T_n$ (type B) or $U_1\\cdots U_{2n}$ (type D), coincide with traces written as pairings with explicit central elements. In type B the representing element is $\\beta_n(y)=\\prod_{i=1}^n(1+(y+\\alpha_0)j_i^B+a^{-1}J_i^B)$ in $H(B_n)\\otimes A(y)$, and the equality is proved by checking the defining properties of the classified trace. In type D the representing element is $\\delta_n(y)$, the even-degree part of $\\prod_{i=1}^n(1+yx_i+a^{-1}x_i^2)$ evaluated at the commuting elements $j_i^D$. At the specializations $y=v-v^{-1}$ (type B) and $y^2=-(v-v^{-1})^2a$ (type D), these formulas reproduce the geometric Markov traces studied in [Gom06], [WW11] and [BT22]; coefficient-wise, type B's $k$th piece is represented by $e_k(J_1^B,\\ldots,J_n^B)$, and type D's by $(-1)^k e'_k(j_1^D,\\ldots,j_n^D;\\alpha)$, where $e'_k$ is the degree-$2k$ part of $\\prod_i(1+\\alpha x_i-x_i^2)$.","pith_inferences":["The same proof pattern may extend to other finite Coxeter types that carry Jucys-Murphy elements and a uniqueness theorem for Markov traces; the paper does not assert this.","The type D formula's even-degree projection suggests that odd symmetric polynomials in $j_i^B$ are invisible to traces on $H(D_n)$, which could explain why the geometric D-type trace differs from the uniform $\\zeta$-trace.","If a categorification in types B and D is constructed, these explicit representatives predict filtrations of Khovanov-Rozansky homology indexed by products of Jucys-Murphy elements, as the paper's closing remark hopes for."],"forward_implications":["In type B, every Markov trace with constants $\\rho=1+a$, $\\mu=v-v^{-1}$ and normalization $(1+a)^n$ is represented by $\\beta_n(y)$, giving a closed Jucys-Murphy formula for the whole one-parameter family.","In type D, the same conclusion holds with $\\delta_n(y)$, the even-degree part of $\\prod_i(1+yx_i+a^{-1}x_i^2)$ evaluated at $j_i^D$.","The geometric Markov traces of [Gom06], [WW11] and [BT22] are the specializations $y=v-v^{-1}$ in type B and $y^2=-(v-v^{-1})^2a$ in type D; their coefficient-at-$a^k$ pieces are represented by $e_k(J^B)$ and $(-1)^k e'_k(j^D;\\alpha)$.","The uniform trace $\\operatorname{tr}_{\\zeta_n}$ with $\\zeta_n=\\prod_i(1+a^{-1}J_i^X)$ is Markov in all classical types A, B and D, so the classical type A Markov trace of [Jon87] is a special case of the same formula.","The top coefficient of each trace is represented by the full twist $S^{-1}$, reproducing the HOMFLY-PT symmetry observed in [Kál09]."],"supporting_citations":[{"why":"Supplies the type B classification and uniqueness result that Theorem 3.2.2 matches.","marker":"[GL97]"},{"why":"Extends the uniqueness and trace classification to type D, used in Corollary 3.2.5.","marker":"[Gec98]"},{"why":"Provides the type A Markov trace construction and uniqueness that this paper generalizes.","marker":"[Jon87]"},{"why":"Establishes that the center of the affine Hecke algebra consists of symmetric Laurent polynomials, underpinning centrality of the JM-polynomial representatives.","marker":"[Lus83]"},{"why":"Supplies the treatment of affine Hecke algebras and multiplicative Jucys-Murphy elements used in Section 2.5.","marker":"[RR03]"},{"why":"Gives the standard trace and pairing properties (Proposition 2.3.1) that all computations of $\\langle z,h\\rangle$ rely on.","marker":"[GP00]"},{"why":"Defines the geometric Markov traces in types B and D that the paper recovers as specializations.","marker":"[Gom06]"},{"why":"Studies the same geometric traces via the equivariant Hecke category, one of the specializations matched in Section 3.3.","marker":"[WW11]"},{"why":"Categorifies Markov traces and motivates the JM-polynomial representatives; its geometric traces are the $y$-specializations identified here.","marker":"[BT22]"}],"fun_headline_variants":["Explicit Jucys-Murphy formulas for type B and D Markov traces","Markov traces in B and D are Jucys-Murphy polynomials","New proof: all Markov traces in B and D via Jucys-Murphy elements","Jucys-Murphy elements describe all Markov traces in types B and D","Type B and D Markov traces realized by Jucys-Murphy polynomials"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The argument leans on the Geck-Lambropoulou classification, which asserts that a Markov trace in type B or D with fixed constants, normalization, and prescribed values on $T_1\\cdots T_n$ (or $U_1\\cdots U_{2n}$) is unique; if that theorem does not apply at the chosen specializations, the equalities identifying the explicit central elements with the known traces would not follow.","fun_headline_variants_meta":{"raw":{"variants":["Explicit Jucys-Murphy formulas for type B and D Markov traces","Markov traces in B and D are Jucys-Murphy polynomials","New proof: all Markov traces in B and D via Jucys-Murphy elements","Jucys-Murphy elements describe all Markov traces in types B and D","Type B and D Markov traces realized by Jucys-Murphy polynomials"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.00048,"raw_usage":{"total_tokens":2358,"prompt_tokens":912,"completion_tokens":1446,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":528,"completion_tokens_details":{"reasoning_tokens":1340}},"tokens_in":528,"tokens_out":1446,"duration_ms":11404,"temperature":1.0,"reasoning_tokens":1340,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T13:52:48.412348+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute, in a computer algebra system, the trace $\\operatorname{tr}^B_{3,y}(T_1T_2T_3)$ using the explicit pairing $\\langle\\beta_3(y),T_1T_2T_3\\rangle$ and compare it with $y^3$ for generic $y,v,a$ over $\\mathbb{Q}(v,a)$; any deviation would falsify Theorem 3.2.2, and a similar check of $\\operatorname{tr}^D_{4,y}(U_1U_2U_3U_4)=y^4$ against $\\delta_4(y)$ would falsify Corollary 3.2.5.","supporting_citations":[],"review_version":1}