{"id":"a2017049-b5af-4e13-ac5b-a655b587318d","arxiv_id":"2606.15138","paper_version":2,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"A completed Cauchy element in graded Hopf dual pairs yields uniform skew Murnaghan–Nakayama rules, with applications to Ariki–Koike characters, skew (q,t)-Kostka coefficients, and Walker's k-core conjecture.","lead":"This paper builds a general Hopf-algebra framework that produces skew Murnaghan–Nakayama rules for many dual pairs at once, then applies it to generate character formulas, Kostka expansions, and a proof of Walker's conjecture on modular Schur functions. A reader interested in algebraic combinatorics or Hecke algebra characters will find a unifying method plus several new formulas.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Commutativity of H∨ is the load-bearing hypothesis; type C verification is asserted, not shown","rationale":"The reader's weakest assumption is exactly the same as the only structurally substantial condition I found. I traced the proofs of Proposition 3.10 and Theorem 3.12: the skew Cauchy identity is unconditional and the algebra is coherent, and Theorem 3.12 follows cleanly from orthogonality once commutativity of H∨ is granted. The principal applications (QSym, k-Schur) do satisfy the hypothesis, though the paper could have been more explicit for Λ_(k). The type C case is asserted without demonstration, which is a gap in verification rather than an internal inconsistency. This does not undermine the central theorem, and the paper has independently useful applications and recoveries, so the reader's ACCEPT verdict should stand.","tokens_in":75105,"tokens_out":37068,"duration_ms":324767,"concrete_test":"In a small type C Grassmannian example (e.g., n=2, degrees ≤4), compute the coproduct of Γ^(n) on its Schubert generators using the known type C Schubert structure constants, and verify that the induced multiplication on the graded dual Γ_(n) is commutative (equivalently, that the coproduct on Γ^(n) is cocommutative). If cocommutativity fails, Theorem 5.3 is unsupported.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Theorem 3.12's skew MN rule (3.4.4) is proved only under the hypothesis that H∨ is commutative. The proof uses this exactly to commute 1⊗S∨(gη/µ) past E in the step after multiplying the skew Cauchy identity by 1⊗S∨(gη/µ). All advertised applications rely on this hypothesis: QSym is commutative by construction, and Λ_(k) is commutative because Λ^(k)=F[h1,...,hk] is a cocommutative Hopf subalgebra of Λ (the paper notes the antipode stability but does not explicitly spell out the commutativity argument). The type C affine Grassmannian pair, however, is treated differently: §5.3 simply states 'We assume H∨ is commutative (this holds in the standard type C Grassmannian setup)' with no verification or reference for the Hopf-dual structure constants. If Γ_(n) were not commutative, Theorem 5.3 would fail. This is the weakest point in the chain of specializations: a concrete, checkable assertion on which one of the paper's advertised new settings depends.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper develops a uniform skew Murnaghan–Nakayama theory in the setting of graded Hopf dual pairs with a nondegenerate Hopf pairing. The central object is the completed Cauchy element, and the main abstract results are a skew Cauchy identity (Prop. 3.10) and a skew Murnaghan–Nakayama rule (Thm. 3.12) valid under the hypothesis that the dual Hopf algebra H^∨ is commutative. The framework is then specialized to recover classical skew Pieri and skew MN rules for symmetric functions, and to produce formal skew MN identities for (NSym, QSym), for k-Schur/dual k-Schur functions, and for a type C affine Grassmannian pair. The later chapters apply the symmetric-function specialization to obtain generating functions for irreducible characters of Ariki–Koike, Hecke–Clifford, and q-rook algebras; ribbon/tableau formulas for skew (q,t)-Kostka polynomials and their inverses; and a proof of Walker's conjecture for odd prime k via modular Schur functions.","tokens_in":75344,"tokens_out":5552,"duration_ms":66667,"significance":"If the main theorem package is correct, it gives a genuinely common Hopf-algebraic source for a large family of skew Pieri and Murnaghan–Nakayama identities, with the abstract Cauchy element doing the work. The degreewise-finite formulation in Chapter 3 is careful, and the contraction calculation behind Prop. 3.10 is explicit and checkable. The applications are numerous and substantive: the character generating functions in Chapter 6 assemble previously scattered Frobenius-formula inputs into compact product forms, the Chapter 7 ribbon/flag formulas for skew (q,t)-Kostka coefficients and their inverses go beyond the straight-shape results in the authors' prior work, and the modular Schur section provides a proof of a published conjecture of Walker under the stated parity assumption. The paper also credits prior results such as [JL24, Eq. (6.15)] and [Mac95] at the point of use, and its logical roadmap is clearly explained.","major_comments":[{"comment":"The type C affine Grassmannian specialization is advertised as a new setting, but the only support for the key hypothesis of Theorem 3.12 is the sentence 'We assume H^∨ is commutative (this holds in the standard type C Grassmannian setup)' on p. 48. This hypothesis is load-bearing: the proof of Theorem 3.12 uses commutativity of H^∨ precisely to pass 1⊗S^∨(g_{η/µ}) through E. The paper does not define the Hopf pairing ⟨·,·⟩_{Sp}, does not give the structure constants of Γ_{(n)}, and gives no reference for the claimed commutativity. Please either supply a proof or a precise reference for the commutativity of Γ_{(n)}, or state Theorem 5.3 explicitly as conditional on that property. Without this, the type C contribution is not independently verifiable from the manuscript.","section":"§5.3 (Theorem 5.3)"},{"comment":"The k-Schur and type C theorems are presented as 'formal specializations' of the abstract skew MN rule, but the paper does not provide any concrete combinatorial content or even the relevant structure constants for these bases. In particular, Theorem 5.2 and Theorem 5.3 are simply (3.4.4) rewritten after declaring dual bases; no example, no Pieri-type evaluation, and no check that the alleged dual bases have the claimed orthogonality is given. This is not an algebraic error, but it affects the advertised scope: a reader cannot test whether these formulas produce new computable identities. The authors should either add the missing verification (at least for the type C pair) or explicitly delimit Chapter 5 as a formal translation whose applications are deferred.","section":"§5.2–5.3"}],"minor_comments":[{"comment":"The notation table distinguishes the Hecke deformation parameter q from the Macdonald parameter q, but Chapter 6 repeatedly sets t = q^{-1} in Hall–Littlewood formulas while Chapter 4 uses (q,t) for Macdonald polynomials. A short paragraph at the start of Chapter 6 clarifying the local meaning of q would help avoid confusion.","section":"Notation / §4–§6"},{"comment":"In the proof of Corollary 7.19 the main text says the remaining combinatorial identity is 'postponed to Appendix A'. The appendix proves Lemma A.1, but the link could be made explicit at the point of use by saying that Lemma A.1 is exactly the claim needed.","section":"§7.4 / Appendix A"},{"comment":"Reference [ST54] is listed as 'Shephard, J. A. Toda'; the correct name is J. A. Todd. Please correct.","section":"References"},{"comment":"Figure 6 asserts that the displayed horizontal 4-ribbon has sign 1, but the intermediate ribbon decomposition τ^{(0)}⊂...⊂τ^{(5)} is not drawn. Adding the decomposition or listing the five ribbons would make the definition of sgn(ρ/λ) immediately checkable.","section":"Figure 6"}],"recommendation":"major_revision","confidential_remarks":"The central Hopf-algebraic derivation in Chapter 3 appears sound, and the symmetric-function and character-theoretic applications are substantial. The only serious obstacle is the unverified commutativity assertion for the type C dual pair in §5.3. If the authors supply that verification or explicitly make Theorem 5.3 conditional, I would be willing to accept a revised version. The other issue is that Chapter 5 is thinner than the abstract promises, but that can be addressed without changing the mathematical core."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Two things to know before you pick it up. The core package — the skew Cauchy identity and the skew MN rule for Hopf dual pairs (Prop. 3.10, Thm. 3.12) — is carefully proved, and the Walker-conjecture consequence (Thm. 8.11) is a real result settled by a clean argument: specialize Y to the root-of-unity alphabet, compare Frobenius expansions, get vanishing on k-singular classes, then defect-zero. The k=2 exception is noted honestly, and Remark 8.16 separates the full Conjecture 4.6 (partially confirmed) from the one they actually close.\n\nWhat is new: the completed Cauchy element with its grouplike factorization and the partial-contraction formalism. That is not in Lam–Lauve–Sottile, whose concern was skew LR structure, and it does supply the advertised family of skew Pieri/MN rules from one mechanism. The character-generating series for Ariki–Koike, Hecke–Clifford, and q-rook algebras (Thms. 6.2, 6.9, 6.12) are real, with the q↔q−1 reciprocity as a dividend; the (q,t)-Kostka ribbon/flag formulas are substantial; and the paper is explicit about what recovers [AM11], [Kon12], [JL24], [ER90] rather than claiming all of Chapter 4 as new.\n\nSoft spots, in proportion. The stress-test note is right about Theorem 3.12: the proof commutes 1⊗S∨(gη/µ) past E, which is exactly where commutativity of H∨ is used. The applications all land on the commutative side — QSym and Λ(k) are explicitly commutative, so the worry there does not really land. The exception is Section 5.3: the type C affine Grassmannian pair is introduced by 'we assume H∨ is commutative,' with no verification and no citation for the Hopf-dual structure constants. Since Theorem 5.3 is one of the advertised new settings and depends on that hypothesis, the authors need to either prove or cite it, or pull the claim back. One-section fix, not a load-bearing flaw. The other caveat: Chapter 5 is formal specialization — it shows reach but computes no structure constants and gives no combinatorial content for NSym/QSym, k-Schur, or type C. The applications in Chapters 6–8 all come from the Λ case. Also worth a referee spot-check: the inputs imported from the authors' own [JL23, JL24] (published, so not circular, but load-bearing).\n\nWho this is for: symmetric-function and Hopf-algebra people, and anyone working on Hecke-type character theory. Chapter 8 alone justifies a serious read. Send it to peer review — it deserves full referee time, and the revision is straightforward.","headline":"The abstract skew MN package is clean and the Walker-conjecture proof is a real payoff; main fix: verify the asserted type C commutativity.","tokens_in":75838,"tokens_out":7795,"would_cite":true,"duration_ms":78428,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["16T05","05E05","05E10","20C08","20C20","20C30","17B69"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper proves that skew Murnaghan–Nakayama rules across many settings all follow from one skew Cauchy identity for Hopf dual pairs.","keywords":["Hopf dual pairs","Cauchy element","skew Murnaghan–Nakayama rule","Ariki–Koike algebras","Hecke–Clifford algebras","(q,t)-Kostka polynomials","modular Schur functions","k-cores"],"falsifier":"In the self-dual case H = H∨ = Λ, take λ = (2), η = (1) and compare the coefficient of s(1)⊗s(1) on both sides of the abstract skew MN identity. A mismatch there would disprove the rule; a match, together with checks for other small shapes, would corroborate the framework.","tokens_in":74995,"feed_emoji":"🧮","tokens_out":6747,"duration_ms":69137,"temperature":0.7,"pith_summary":"The paper shows that many seemingly unrelated skew Murnaghan–Nakayama rules — the classical border-strip rule for symmetric functions, its Hecke, k-Schur, and type-C analogues, and plethystic variants for modular Schur functions — are specializations of one abstract identity. The identity lives in any graded Hopf dual pair equipped with a nondegenerate Hopf pairing: a completed Cauchy element satisfies a skew Cauchy identity, and when the dual Hopf algebra is commutative this yields a general skew Murnaghan–Nakayama rule. From this single source the authors recover known skew Pieri and skew MN formulas and derive new generating functions for irreducible characters of Ariki–Koike, Hecke–Clifford, and q-rook algebras. They also obtain ribbon-tableau expansions for skew (q,t)-Kostka polynomials and their inverses, and use a root-of-unity specialization to prove that if a modular Schur function coincides with the ordinary Schur function at odd prime k, then the indexing partition must be a k-core. The value of the framework is not any single formula but a common mechanism: Cauchy element, grouplike factorization, and partial contractions.","feed_headline":"One Hopf identity unifies skew Murnaghan–Nakayama rules","feed_subtitle":"The same completed Cauchy element drives Hecke characters, (q,t)-Kostka tables, and modular Schur functions.","key_machinery":"The completed Cauchy element E = Σλ fλ⊗gλ in the degree-completed tensor product of H and H∨. Its grouplike factorization (Δ⊗Δ∨)E = E13E14E23E24 lets products of dual bases be reorganized; partial contraction operators pick out coefficients against gμ and fλ; and the antipode orthogonality Σμ S∨(gη/μ)gμ/τ = δη,τ collapses the sums. Skew elements fλ/μ are defined by Δ(fλ) = Σμ fλ/μ⊗fμ. These three ingredients convert the Hopf-pairing axioms into the skew Cauchy identity and then into the skew Murnaghan–Nakayama rule.","core_discovery":"The central claim is the abstract skew Murnaghan–Nakayama rule: in a graded Hopf dual pair (H,H∨) with nondegenerate Hopf pairing and dual homogeneous bases fλ, gλ, the completed Cauchy element E = Σλ fλ⊗gλ satisfies E(fλ/η⊗1) = Σρ,μ fρ/μ ⊗ S∨(gη/μ)gρ/λ, provided H∨ is commutative. Together with the skew Cauchy identity E Στ fλ/τ⊗gμ/τ = Σρ fρ/μ⊗gρ/λ, this is the formal engine of the paper. The authors prove these identities from three properties of E: its grouplike factorization (Δ⊗Δ∨)E = E13E14E23E24, partial contraction operators induced by the pairing, and an orthogonality relation for the antipode. Every subsequent result — classical skew Pieri/MN rules, character generating functions, (","pith_inferences":["The framework suggests that any graded Hopf dual pair with a nondegenerate pairing and commutative dual should admit a skew MN rule of the same shape; testing pairs beyond the paper, such as Heisenberg or quantum-group duals, could reveal further identities.","The character generating functions make reciprocity laws visible as plethystic transformations; analogous one-kernel reciprocity may hold for other Hecke-type algebras using the same argument.","Specializing the auxiliary alphabet to other virtual alphabets, beyond root-of-unity sums, may yield new plethystic skew rules for yet other families of symmetric functions, and the k-core criterion might extend to composite k with a modified defect argument.","The ribbon-tableau formulas for inverse (q,t)-Kostka coefficients could be used computationally to determine truncated tensor-decomposition matrices in modular representation theory without computing plethysms directly."],"forward_implications":["Classical skew Pieri and skew Murnaghan–Nakayama rules for symmetric functions, including their Schur-P/Q analogues, are recovered as the Λ = Λ∨ specialization with the auxiliary alphabet chosen appropriately.","New skew MN formulas hold in the (NSym, QSym), k-Schur, and type-C affine-Grassmannian Hopf dual pairs, settings where no such uniform rule previously existed.","Frobenius-type character data for Ariki–Koike, Hecke–Clifford, and q-rook algebras assemble into closed symmetric-function generating functions; the Ariki–Koike series specializes to type A and type B Hecke algebras.","Skew (q,t)-Kostka polynomials and their inverses admit special-ribbon-tableau and flag expansions; the q=0 case gives skew Kostka–Foulkes formulas and at t=1 recovers the inverse Kostka formula, resolving a 1998 open question.","A root-of-unity specialization yields a skew plethystic MN rule and a Schur expansion for skew modular Schur functions, from which the paper proves that a trivial modular-to-Schur transition at odd prime k forces the partition to be a k-core."],"fun_headline_variants":["One skew MN rule for all Hopf dual pairs","Hopf dual pairs unify skew Murnaghan–Nakayama","Skew MN identity settles Walker's 1994 conjecture","Same Cauchy element drives Hecke and modular Schur","Abstract skew MN rule yields new character formulas"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The abstract skew Murnaghan–Nakayama rule is proved only for dual Hopf algebras that are commutative; the step that moves the antipode factor past the Cauchy element uses this commutativity, and the rule is not established without it.","fun_headline_variants_meta":{"raw":{"variants":["One skew MN rule for all Hopf dual pairs","Hopf dual pairs unify skew Murnaghan–Nakayama","Skew MN identity settles Walker's 1994 conjecture","Same Cauchy element drives Hecke and modular Schur","Abstract skew MN rule yields new character formulas"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.00027,"raw_usage":{"total_tokens":1556,"prompt_tokens":932,"completion_tokens":624,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":676,"completion_tokens_details":{"reasoning_tokens":545}},"tokens_in":676,"tokens_out":624,"duration_ms":6448,"temperature":1.0,"reasoning_tokens":545,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-02T11:22:57.594911+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"In the self-dual case H = H∨ = Λ, take λ = (2), η = (1) and compare the coefficient of s(1)⊗s(1) on both sides of the abstract skew MN identity. A mismatch there would disprove the rule; a match, together with checks for other small shapes, would corroborate the framework.","supporting_citations":[],"review_version":1}