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A skew Murnaghan--Nakayama rule for Hopf dual pairs

T0 review · 2 major / 4 minor · reviewed 2026-08-02 · deepseek-v4-flash

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Pith's one-line read This paper proves that skew Murnaghan–Nakayama rules across many settings all follow from one skew Cauchy identity for Hopf dual pairs.

desk verdict The abstract skew MN package is clean and the Walker-conjecture proof is a real payoff; main fix: verify the asserted type C commutativity. read the letter →

arxiv 2606.15138 v2 pith:RKBFELPD submitted 2026-06-13 math.CO math.RT

Naihuan Jing , Ning Liu This is my paper · ORCID
classification math.COmath.RT MSC 16T0505E0505E1020C0820C2020C3017B69
keywords HopfdualpairsCauchyelementskewMurnaghan–NakayamaruleAriki–KoikealgebrasHecke–Clifford(qt)-KostkapolynomialsmodularSchurfunctionsk-cores
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

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.

What carries the argument

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.

What would settle it

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.

Watch

Extended reading notes

Core claim

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, (

Load-bearing premise

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.

Editorial extensions

If this is right

  • 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.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • 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.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

2 major / 4 minor

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.

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 (2)
  1. [§5.3 (Theorem 5.3)] 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.
  2. [§5.2–5.3] 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.
minor comments (4)
  1. [Notation / §4–§6] 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.
  2. [§7.4 / Appendix A] 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.
  3. [References] Reference [ST54] is listed as 'Shephard, J. A. Toda'; the correct name is J. A. Todd. Please correct.
  4. [Figure 6] 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.

Circularity Check

0 steps flagged · score 2.0 of 10

Abstract Hopf derivation is self-contained; only minor self-citations and an unverified commutativity assertion, neither forcing the conclusions.

full rationale

The core derivation chain (Propositions 3.8–3.10, Lemma 3.11, Theorem 3.12) is a direct consequence of the Hopf axioms, dual-basis reconstruction, and the antipode identity; no target formula is used to prove itself. The skew MN rule explicitly relies on the stated hypothesis 'We assume that H∨ is commutative' in the proof of Theorem 3.12, but this is an assumption, not a circular reuse of the conclusion. The applications to Λ, NSym/QSym, and Λ_(k) verify or state commutativity; the type C section says 'We assume H∨ is commutative (this holds in the standard type C Grassmannian setup)' (Section 5.3) without giving a proof, which is a missing verification but not a circular step. Self-citations such as [JL24, Lemma 4.1], [JL23, Corollary 3.10], and [JL24, Eq. (6.15)] supply published combinatorial evaluations used as inputs; they are not the claims being derived, and the derivation does not reduce to them. No fitted parameters are renamed as predictions; all Y-specializations are explicit and independent. Score 2 reflects the presence of several self-citations as ingredients and the load-bearing but unproved commutativity assertion, without any equation-level circularity.

Assumptions & free parameters 0 free parameters · 7 assumptions · 0 invented entities

The central claim rests on the Hopf-pairing axioms and the commutativity of the dual side; the rest are standard facts from symmetric functions, Frobenius character formulas, and the cited prior evaluations. No free parameters are fitted; the q,t,u are deformation parameters of the settings.

assumptions (7)
  • domain assumption Graded connected Hopf algebras with finite-dimensional graded components and nondegenerate graded Hopf pairing exist for each application.
    Definition 3.1/Chapter 3; needed for dual bases, completed Cauchy element, contraction maps.
  • domain assumption H^∨ is commutative in the MN-rule theorem.
    Theorem 3.12 assumption; allows commuting antipode factor with E.
  • standard math Standard symmetric-function facts: Cauchy identities, Jacobi–Trudi, plethysm, Schur/SP/Q specializations at 1, -1, q-1, roots of unity.
    Used throughout Chapters 4, 7, 8.
  • domain assumption Frobenius-type character formulas from Shoji, Wan–Wang, Dieng–Halverson–Poladian for Ariki–Koike, Hecke–Clifford, q-rook algebras.
    Chapter 6 inputs; paper assembles them into generating functions.
  • domain assumption The generalized (q,t)-binomial evaluation identity [JL24, Eq. (6.15)].
    Used in Prop 7.4 to derive skew Kostka formulas; prior published result by same authors.
  • domain assumption Known plethystic MN rule [DLT94] and the Petrie coefficient interpretation [WEK+25, CCE+23].
    Used in Lemmas 8.1 and 8.3 to identify root-of-unity specializations.
  • standard math Brauer–Nesbitt theorem and Frame–Robinson–Thrall hook-length defect-zero criterion.
    Used in Lemma 8.15 and proof of Theorem 8.11.

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Pith. "Pith review of A skew Murnaghan--Nakayama rule for Hopf dual pairs." pith.science (2026). https://pith.science/paper/RKBFELPD

@misc{pith2026260615138,
  author       = {Pith},
  title        = {Pith review of: A skew Murnaghan--Nakayama rule for Hopf dual pairs},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/RKBFELPD}},
  note         = {Machine review of arXiv:2606.15138}
}
abstract

We develop a uniform skew Murnaghan--Nakayama theory for graded Hopf dual pairs equipped with a nondegenerate Hopf pairing. Using the completed Cauchy element, its grouplike factorization, and the resulting partial contraction operators, we establish a general skew Cauchy identity together with an abstract skew Murnaghan--Nakayama rule. Specializing this framework recovers and extends the classical skew Murnaghan--Nakayama rule for symmetric functions, and yields new skew Murnaghan--Nakayama formulas in several settings, including the dual pairs $(\mathrm{NSym}, \mathrm{QSym}) $ and $(\Lambda^{(k)}, \Lambda_{(k)}) $ arising in $k$-Schur theory, as well as the type $C$ affine Grassmannian context. As applications, we obtain generating functions for irreducible characters of Ariki--Koike algebras, including their type $A$ and type $B$ specializations, as well as Hecke--Clifford algebras and $\mathfrak q$-rook monoid algebras. We also give ribbon-tableau expansions for skew $(q,t)$-Kostka polynomials and for the entries of the inverse transition matrix, thereby answering a question of Carbonara (1998). Finally, by specializing the auxiliary alphabet $Y$ to sums of powers of primitive roots of unity, we derive a skew plethystic Murnaghan--Nakayama formula together with a Schur expansion for skew modular Schur functions; as a further consequence, we confirm Walker's conjecture (1994) by showing that if the transition from the modular Schur functions to the Schur basis is trivial in the row indexed by $\lambda$, then $\lambda$ must be a $k$-core.

Figures

Figures reproduced from arXiv: 2606.15138 by the authors.

Figure 1
Figure 1. λ = (4, 3, 1) on the left, and its conjugate partition λ t on the right. Here aλ(•) = 2 and lλ(•) = 1 [PITH_FULL_IMAGE:figures/full_fig_p016_1.png] view at source ↗
Figure 2
Figure 2. A ribbon with starting box (white dot) and ending box (black dot). Its length and height are 8 and 2, respectively. See [PITH_FULL_IMAGE:figures/full_fig_p016_2.png] view at source ↗
Figure 3
Figure 3. λ = (9, 7, 4, 3, 1, 1), µ = (6, 5, 2, 1). This generalized ribbon has three connected components with weight −q 5 (q − 1)3 . which is a ribbon. If the generalized ribbon K has m connected components (ξ1, ξ2, . . . , ξm), we define its weight by wtq(K) = (q − 1)mYm i=1 (−1)r(ξi)−1 q c(ξi)−1 . Here r(ξi) (resp. c(ξi)) denotes the number of rows (resp. columns) in the ribbon ξi . See [PITH_FULL_IMAGE:figures/full_fig_… view at source ↗
Figures from the paper (4 more)
Figure 4
Figure 4. Figure 4: λ = (15, 14, 10, 8, 7, 6, 5, 3, 1), µ = (14, 11, 8, 6, 5, 4, 3). The boxes of α(λ/µ) are marked by ∗. β(λ/µ) has three connected components. The weight of λ/µ is (−q) 4 ·2·(d2d1−d3) 2 ·(d1d3−d4) = 16q 4 (q−1)6 (q 2−3q+1)2 . Qλ/µ[q − 1] = ( wt˜ q(λ/µ), if λ/µ is a gener…
Figure 5
Figure 5. Figure 5: The left ribbon is a special ribbon, while the right one is not. Now use the standard characterization (cf. [Mac95, p. 310, (2.7)]): ⟨uλ, vµ⟩q,t = δλµ for all λ, µ ⇐⇒ X λ uλ[X] vλ[Y ] = Y n≥1 exp 1 n 1 − t n 1 − q n pn[X]pn[Y ]  , which yields (7.1.3). □ By Lemma 7.1…
Figure 6
Figure 6. Figure 6: shows a horizontal 4-ribbon of weight 5 [PITH_FULL_IMAGE:figures/full_fig_p089_6.png]
Figure 7
Figure 7. Figure 7: Let λ = (13, 12, 7, 6, 6) and µ = (9, 5, 2, 1, 1). Then λ/µ is a proper 6-ribbon with two decompositions corresponding to ν = (11, 8, 4, 2, 1) (left) and ν ′ = (13, 6, 4, 2, 1) (right), where every box of ν/µ and ν ′ /µ is marked by a cross ×. Lemma 8.3. For µ ⊂ λ, we …

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