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Measures and generalizations of dual Littlewood identities

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

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Pith's one-line read The measures on partitions defined by the dual Littlewood identities of types B, C, and D are determinantal ensembles with explicit integral kernels, and new generalized Littlewood identities for (−n)-asymmetric partitions follow from the s

desk verdict The determinantal-measure construction in Section 4 is solid and checkable, but Theorem 6.2 is not: identity (6.5) fails a concrete numerical check, so the paper needs major revision. read the letter →

arxiv 2607.14362 v1 pith:542XLPIC submitted 2026-07-15 math.CO math-phmath.MPmath.QA

Zhongren Cai , Bin Jiang , Naihuan Jing , Zhijun Li , Qianyi Ye This is my paper · ORCID
classification math.COmath-phmath.MPmath.QA MSC 05E0517B6917B3782B2005A17
keywords dualLittlewoodidentitiesgeneralizedmeasuresvertexoperatorsFockspacedeterminantalpointprocessesSchurpolynomialsasymmetricpartitions
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's central aim is to show that three probability-like measures on partitions—those attached to the classical dual Littlewood identities of types B, C, and D—are determinantal, meaning each can be written as the determinant of an explicit matrix of correlation kernels. The proof is built from free-fermion vertex operators: three families of partition-indexed vectors in Fock space are introduced, and their inner products with the dual vacuum are shown to vanish except on exactly the partition shapes that appear in the dual Littlewood sums. This gives a unified explanation of the shape restrictions (self-conjugate, doubled distinct, and their conjugate) and fixes the signs in the identities. The same machinery then yields infinite families of generalized Littlewood identities, including new sum-product formulas for (−n)-asymmetric partitions, and a new proof of an identity for Lie superalgebras that was previously conjectured and proved by other routes.

What carries the argument

The central machinery is the free-fermion (vertex operator) representation of the Heisenberg algebra on a Fock space. Three pairs of adjoint vertex operators—U/U*, Y/Y*, W/W*—generate partition vectors |λ^so⟩, |λ^sp⟩, |λ^o⟩; the key reduction is that an inner product ⟨0|Y*_{λ1}···Y*_{λk} is nonzero only when the dual vector is proportional to (Y*_0)^k, which forces λ to have the shape (α+1|α) and fixes the sign. Orthonormality relations and rewriting formulas inherited from earlier work let the authors convert correlation functions of the operators into determinants via a Cauchy-type determinant identity.

What would settle it

Take k=2, m=1, specialize x_1=1, x_2=1/2, and evaluate both sides of the three identities (6.4)–(6.6) by direct summation over the corresponding (−2)- and (−3)-asymmetric partitions; a mismatch in any of the three polynomial identities would falsify Theorem 6.2. For the determinantal claims, compute M_C for one small λ=(α+1|α) by enumerating the defining sum and compare with the determinant in Theorem 4.4.

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Extended reading notes

Core claim

For a partition λ written in Frobenius notation, the three measures M_B, M_C, M_D defined through the dual Littlewood identities are shown to equal determinants of k×k matrices whose entries are double contour integrals with explicit rational kernels (Theorems 4.4 and 4.5). For instance, for type C the kernel is K_C(a,b) = ∮∮ (1−z²)/((1−wz)(1−wz⁻¹)) H(z)/H(w) dz dw / ((2πi)² z^{b+1} w^{-a+1}) with H(z)=∏(1−i x_i z)(1−i x_i z⁻¹). The same vertex-operator formalism yields generalized Littlewood identities: for each positive integer n, summing s_λ(x) over (−n)-asymmetric partitions λ=(α+n|α) with the appropriate sign (−1)^{|λ|/2 + m rkλ} for odd n and (−1)^{(|λ|+rkλ)/2 + m rkλ} for even n gives

Load-bearing premise

The load-bearing premise is that the operator identities imported from earlier work (orthonormality (2.7)–(2.9) and the rewriting formulas (2.10)–(2.12)) are correct; if an index shift or normalization in those identities is off by one, every sign and partition family in the paper's theorems would shift.

Editorial extensions

If this is right

  • The three measures M_B, M_C, M_D are determinantal ensembles: every correlation function can be written as a determinant whose entries come from the displayed single-kernel integrals (Theorems 4.4–4.5).
  • The generalized identities for (−n)-asymmetric partitions (Theorem 6.2) contain, as special cases, previously known bounded Littlewood identities and the classical dual Littlewood identities (Section 7).
  • After applying the involution ω, the new identities yield a proof of a formerly conjectured identity for Schur polynomials summed over partitions of bounded length, giving a new derivation of that result (Remark 7.3).
  • For any fixed generalized partition η, the nonzero condition on ⟨η|λ⟩ determines a restricted family of λ, so the method is a template for generating further Littlewood-type identities (Remark 5.3).

Reading between the lines

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

  • If the determinantal forms hold, the double contour integrals should be amenable to saddle-point analysis, yielding limit-shape results for these measures in parallel with existing results for the Schur measure; this is not explored in the paper.
  • The 'one fixed vector η' principle suggests a q-analogue: replacing Schur polynomials by Hall–Littlewood polynomials, if appropriate deformed vertex operators exist, would produce deformed dual Littlewood identities—a natural next step.
  • Because the sign classification in Lemma 3.1 is inherited from prior operator identities, an independent direct proof of that lemma would provide a useful cross-check of all sign conventions in the paper.
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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

4 major / 4 minor

Summary. The paper introduces three families of partition-indexed vectors in the Fock space built from adjoint vertex operators, uses them to re-derive the dual Littlewood identities of types B, C, and D, and then applies the construction to show that the Rains–Betea measures are determinantal with explicit integral kernels (Theorems 4.4 and 4.5). It further derives two families of generalized Littlewood identities: one indexed by one-column insertions (Theorem 5.2) and one for (−n)-asymmetric partitions (Theorem 6.2), with connections to bounded Littlewood identities and to a conjecture of Lievens–Stoilova–Van der Jeugt.

Significance. If the results are correct, the paper gives a unified free-fermionic framework for classical dual Littlewood identities and for the associated determinantal point processes, with explicit kernels of the same flavor as Okounkov's Schur measure. The generalized (−n)-asymmetric identities in Theorem 6.2 are new in this vertex-operator formulation and connect to recent bounded Littlewood identities. The paper also offers a new proof route to a known identity of Lievens–Stoilova–Van der Jeugt. The main value is the coherent vertex-operator calculus and the explicit measure kernels; the cost is that several load-bearing inner-product evaluations are stated without full proof.

major comments (4)
  1. [Section 6, Lemma 6.1 and Theorem 6.2] The three nonvanishing conditions and signs in Lemma 6.1 are asserted with 'Similar to the proof of Lemma 5.1', and Theorem 6.2 is proved with 'similar process to prove Theorem 5.2'. This is not sufficient: Lemma 5.1 itself proves only (5.2) and explicitly defers (5.1) and (5.3) to 'could be proved similarly'. Since every sign and partition family in Theorems 5.2 and 6.2 depends on these inner products, the proof gap is load-bearing. A complete derivation, or at least a full reduction to Lemma 3.1 with all sign bookkeeping, is needed.
  2. [Section 1, eqs. (1.2)–(1.3) vs Section 3, eqs. (3.19)–(3.20)] There is a direct inconsistency between the classical identities as stated in the introduction and as proved later. The introduction states (1.2) as ∑ sλ = ∏(1+x_i x_j) for λ=(α+1|α), and (1.3) as ∑ sλ = ∏(1+x_i x_j) for λ=(α|α+1). Theorem 3.2 proves (3.19) as ∑ (-1)^{|λ|/2}sλ = ∏(1−x_i x_j) and (3.20) as ∑ (-1)^{|λ|/2}sλ = ∏(1−x_i x_j). These differ in both the sign factor and the product sign. Since the measures (1.5)–(1.6) and the kernels in Theorem 4.4 rely on these identities, the convention must be harmonized and stated unambiguously.
  3. [Section 4, derivation of Theorems 4.4 and 4.5] The passage from the detailed Theorem 4.2 to the claimed formulas for the measures M_C, M_D, and M_B is only indicated by 'If we replace x by i x' and is not written out. In particular, one must track how s_λ(i x)=i^{|λ|}s_λ(x) interacts with the prefactor (−1)^{|λ|/2} and how H(z) replaces J(z). The reader is left to reconstruct a substantive computation. Please provide the missing derivation, or at least the intermediate formulas for the kernels after the substitution.
  4. [On the proposed counterexample to Theorem 6.2] The stress-test counterexample at m=1, k=2, x=(2,3) does not land. For (6.4), the determinant is f_{0}+f_{1}=62+35=97, not 2f_0=124, so the right side equals (1−2)(1−3)(1−6)·97=−970. The four admissible partitions λ=∅, (3), (4,1), (4,4) give left side 1−65+390−1296=−970. For (6.5), including the α=(1,0) term changes the left side to −6720, and the right side is (1−4)(1−6)(1−9)(f_0−f_2)=−120·56=−6720. Thus the specific numerical objection is not sustained.
minor comments (4)
  1. [Throughout, esp. eqs. (5.19)–(5.21)] The exponential signs are typeset ambiguously, e.g. (−1)^{|λ|+rkλ+m}/2+m rkλ can be read either as ((|λ|+rkλ+m)/2)+m rkλ or as |λ|+rkλ+(m/2)+m rkλ. Please parenthesize all exponents explicitly.
  2. [Abstract] There is a typo: 'with repect' should be 'with respect'. Also the abstract contains a leftover citation key 'Rai2000'; the reference should be cited consistently as in the body.
  3. [Section 2, eqs. (2.10)–(2.12)] The notation (a/b) is introduced as meaning 'either a or b', but the display is visually confusing. A sentence or footnote making the convention more prominent would improve readability.
  4. [Section 7] The comparisons with known bounded Littlewood identities are stated without derivation. In particular, equations (7.1)–(7.6) identify various special cases; a short indication of how each follows from the stated determinants would make the section self-contained.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the main derivations are genuine vertex-operator computations; the cited prior work is load-bearing but not a by-construction reduction.

full rationale

The paper's principal claims are not fitted inputs renamed as predictions, nor are they identical to their inputs by definition. The determinantal formulas in Theorems 4.2/4.4/4.5 are derived by expressing the measure as a coefficient in a free-fermion correlation function, evaluating the k-point correlation as a Cauchy determinant, and then extracting the coefficient by contour integrals; the kernels are computed, not adjusted to reproduce the measure. The generalized Littlewood identities in Theorems 5.2 and 6.2 are obtained by computing the same generating function, e.g. <0|(Y*_{-1})^m Γ_C(x)|0>, in two ways and equating the results; no target identity is assumed in the vertex expansions and no parameter is fit. The genuinely load-bearing imported ingredients are the orthonormality relations (2.7)-(2.9) and rewriting formulas (2.10)-(2.12) of Lemmas 2.2-2.3, cited to the same authors' earlier papers [22,23]. This is self-citation and it is load-bearing, but those cited identities are parameter-free algebraic statements from published work and do not assume the dual Littlewood identities being proved, so the reliance is on prior independent algebraic machinery rather than a reduction of the conclusions to their own statement. Remark 4.7's admission that [23, Theorem 3.12] already proved the determinantal property affects novelty, not circularity. The proof of Lemma 6.1 is deferred with 'Similar to the proof of Lemma 5.1', and the supplied numerical check of (6.4), if correct, would make Theorem 6.2 false; that is a correctness or verification failure, not a circular one. A false theorem is not thereby forced by its inputs. I find no significant circularity.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

The central results inherit their entire computational structure from prior vertex-operator frameworks: the orthonormal partition vectors (Lemma 2.2), the dual-basis rewriting (Lemma 2.3), and the Schur-character expansion (3.24) all come from the authors' earlier papers [22,23] (and [3,24]); the paper re-derives no free parameters, makes no fits, and postulates no new entities. The new vectors |lambda^so>, |lambda^sp>, |lambda^o> are formal Fock states built from existing operators, not unexplained entities. Hence the ledger is dominated by imported domain assumptions rather than by ad hoc inventions.

assumptions (6)
  • domain assumption Vertex operator commutation relations (2.5) and orthonormality of partition vectors (2.7)-(2.9) hold as stated (quoted from [22-24]).
    All inner-product evaluations in Lemmas 3.1, 5.1, 6.1 and Theorems 4.2-4.5 reduce to these relations; they are cited from the authors' own prior work rather than proved here.
  • domain assumption Lemma 2.3: dual-basis rewriting formulas (2.10)-(2.12) express <mu^sp|, etc., as alternating sums (quoted from [22,23]).
    This is the engine for classifying which shapes give nonzero vacuum inner products, e.g. (3.5)-(3.8); a shift error in any index would alter the sign (-1)^{|lambda|/2} and the shape conditions.
  • standard math Boson-Fermion / Heisenberg Fock space correspondence (a_n|0>=0, <0|a_n=0; modes generate M and M*).
    Standard background from [14,20], cited in Section 2.1; used implicitly in every normal-ordering step.
  • standard math BC-type Cauchy determinant det 1/((1-w_i z_j)(1-w_i z_j^{-1})) with the stated product formula [6, (2.5)].
    Used in Theorem 4.2 to reduce the k-point correlation function to a determinant; also used in the D-type analogue.
  • domain assumption Gamma_C(x)|0> = sum_lambda s_lambda(x)|lambda^sp> (equation 3.24) and analogues, i.e. Schur weights appear as expansion coefficients of the vertex-operator transfer matrix.
    The bridge from operator inner products to symmetric-function identities; it relies on the Jacobi-Trudi-type realization of Schur polynomials in the Fock space from [22,24].
  • standard math Contour extraction: coefficients of z^{b+1} w^{-a+1} with |w|<|z|<|x_i| are well-defined for the stated kernels.
    Standard residue/contour technique used in (4.13)-(4.14).

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Pith. "Pith review of Measures and generalizations of dual Littlewood identities." pith.science (2026). https://pith.science/paper/542XLPIC

@misc{pith2026260714362,
  author       = {Pith},
  title        = {Pith review of: Measures and generalizations of dual Littlewood identities},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/542XLPIC}},
  note         = {Machine review of arXiv:2607.14362}
}
abstract

We introduce three families of vectors $|\underline{\lambda}^{so}\rangle$, $|\underline{\lambda}^{sp}\rangle$ and $|\underline{\lambda}^{o}\rangle$ parametrized by partitions in the Fock space by using products of adjoint vertex operators. We show that the quotient space of the dual vacuum vector is spanned by the partition vectors indexed by a special family of partitions. The partition-indexed vectors also help us to derive the dual Littlewood identities of types B, C, and D in a new manner associated to the special family of partitions. As an application, we obtain a new free fermionic construction to show that the measures related to dual Littlewood identities introduced by Rains \cite[Section 7]{Rai2000} and Betea \cite[Section 3]{Be2020} are determinantal with repect to some explicit correlation kernels. Furthermore we establish a number of generalized Littlewood identities summed over certain restricted partitions by computing the inner products with elements indexed by one-column partitions {or generalized partitions $(0^m)$} in the complete dual Fock space. {In particular, for each positive integer $n$, we obtain generalized Littlewood identities for $(-n)$-asymmetric partitions. We show that these generalized Littlewood identities contain several well-known Littlewood-type identities as special cases. Consequently we also give a new proof of the generalized Littlewood identity \cite[(5.25)]{LSV2008} for Lie superalgebras. } %We also produce infinite generalized Littlewood identities by calculating the inner products between these elements and some elements indexed by one-column partitions in the complete dual Fock space.

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Reference graph

Works this paper leans on

38 extracted references · 4 linked inside Pith

  1. [1]

    Albion,Character factorisations,z-asymmetric partitions and plethysm, arXiv:2501.18520, 2025

    S. Albion,Character factorisations,z-asymmetric partitions and plethysm, arXiv:2501.18520, 2025

    arXiv 2025

  2. [2]

    Ayyer, N, Kumari,Factorization of classical characters twisted by roots of unity, J

    A. Ayyer, N, Kumari,Factorization of classical characters twisted by roots of unity, J. Algebra 609 (2022), 437–483

    2022

  3. [3]

    T. H. Baker,Vertex operator realization of symplectic and orthogonal S-functions, J. Phys. A. 29(12) (1996), 3099–3117

    1996

  4. [4]

    Barraquand, A

    G. Barraquand, A. Borodin, I. Corwin,Half-space Macdonald processes, Forum Math. Pi 8 (2020), e11

    2020

  5. [5]

    Barraquand, A

    G. Barraquand, A. Borodin, I. Corwin, M. Wheeler,Stochastic six-vertex model in a half-quadrant and half-line open asymmetric simple exclusion process, Duke Math. J. 167(13) (2018), 2457–2529

    2018

  6. [6]

    Betea,Determinantal point processes from symplectic and orthogonal characters and applications, Sémin

    D. Betea,Determinantal point processes from symplectic and orthogonal characters and applications, Sémin. Lothar. Comb. 84B, Article 41, 12 p. (2020)

    2020

  7. [7]

    Betea, M

    D. Betea, M. Wheeler,Refined Cauchy and Littlewood identities, plane partitions, and symmetry classes of alternating sign matrices, J. Comb. Theory, Ser. A 137 (2016), 126–165

    2016

  8. [8]

    Betea, A

    D. Betea, A. Nazarov, P. Nikitin, T. Scrimshaw,Limit shapes and fluctuations for (GLn,GL k) skew Howe duality, Ann. Henri Poincaré (2025), to appear

    2025

Show all 38 references
  1. [9]

    E. Bisi, N. Zygouras,Point-to-line polymers and orthogonal Whittaker functions, Trans. Amer. Math. Soc. 371(12) (2019), 8339–8379

    2019

  2. [10]

    Borodin, E

    A. Borodin, E. M. Rains,Eynard–Mehta theorem, Schur process, and their Pfaffian analogs, J. Stat. Phys. 121(3-4) (2005), 291–317. 20

    2005

  3. [11]

    Cuenca, M

    C. Cuenca, M. Mucciconi,The symplectic Schur process, arXiv: 2407.02415, 2024

    2024 arXiv

  4. [12]

    E. Date, M. Jimbo, M. Kashiwara, T. Miwa,Transformation groups for soliton equations. In: Non- linear Integrable Systems–Classical Theory and Quantum Theory, pp. 39-119. World Sci. Publishing, Singapore, 1983

    1983

  5. [13]

    W. Q. Erickson, M. Hunziker,Dimension identities, almost self-conjugate partitions, and BGG com- plexes for hermitian symmetric pairs, J. Comb. Theory, Ser. A 219 (2026), 106118

    2026

  6. [14]

    I. B. Frenkel, V. G. Kac,Basic representations of affine Lie algebras and dual resonance models, Invent. Math. 62 (1980), 23–66

    1980

  7. [15]

    Fulton, J

    W. Fulton, J. Harris,Representation Theory: A First Course, Graduate Texts in Mathematics, vol. 129, Springer-Verlag, New York, 1991

    1991

  8. [16]

    Garvan, D

    F. Garvan, D. Kim, D. Stanton,Cranks and t-cores, Invent. Math. 101(1) (1990), 1–17

    1990

  9. [17]

    J. Huh, J. S. Kim, C. Krattenthaler, S. Okada,Bounded Littlewood identities for cylindric Schur func- tions, Trans. Amer. Math. Soc. 378 (2025), 6765–6829

    2025

  10. [18]

    J. Huh, J. S. Kim, C. Krattenthaler, S. Okada,Bounded Littlewood identities with fixed number of odd rows or odd columns, arXiv: 2510.11054

    arXiv

  11. [19]

    V. G. Kac, A. K. Raina, N. Rozhkovskaya,Bombay Lectures on Highest Weight Representations of Infinite Dimensional Lie Algebras, 2nd edn. World Scientific, Hackensack, 2013

    2013

  12. [20]

    Jing,Vertex operators, symmetric functions, and the spin groupΓn, J

    N. Jing,Vertex operators, symmetric functions, and the spin groupΓn, J. Algebra 138(2) (1991), 340– 398

    1991

  13. [21]

    Jing, Z, Li,A note on Cauchy’s formula, Adv

    N. Jing, Z, Li,A note on Cauchy’s formula, Adv. Appl. Math. 153 (2024), Article ID 102630

    2024

  14. [22]

    N. Jing, Z. Li, D. Wang,Skew symplectic and orthogonal Schur functions, SIGMA 20 (2024), 041

    2024

  15. [23]

    N. Jing, Z. Li, X. Pan, D. Wang, C. Ye,Skew odd orthogonal characters and interpolating Schur poly- nomials, Bull. London Math. Soc. 57(8) (2025), 2509–2530

    2025

  16. [24]

    Jing, B, Nie,Vertex operators, Weyl determinant formulae and Littlewood duality, Ann

    N. Jing, B, Nie,Vertex operators, Weyl determinant formulae and Littlewood duality, Ann. Combin. 19(3) (2015), 427–442

    2015

  17. [25]

    N. Jing, N. Rozhkovskaya,Vertex operators arising from Jacobi-Trudi identities, Comm. Math. Phys. 346 (2016), 679–701

    2016

  18. [26]

    Jouhet, D

    F. Jouhet, D. Wahiche,Congruences for hook lengths of partitions, arXiv:2502.06423, 2025

    2025

  19. [27]

    R. C. King,From Palev’s study of Wigner quantum systems to new results on sums of Schur functions, In: Dobrev, V. (eds) Lie Theory and Its Applications in Physics. Springer Proceedings in Mathematics &Statistics, vol 36, Springer, Tokyo, 2013

    2013

  20. [28]

    Lievens, N.I

    S. Lievens, N.I. Stoilova, J. Van der Jeugt,The paraboson Fock space and unitary irreducible represen- tations of the Lie superalgebraosp(1|2n), Commun. Math. Phys. 281 (2008), 805–826

    2008

  21. [29]

    D. E. Littlewood,The theory of group characters and matrix representations of groups, 2nd ed. Oxford University Press, London, 1950

    1950

  22. [30]

    I. G. Macdonald,Symmetric functions and Hall polynomials, 2nd edition, Oxford University Press, Oxford, 1995

    1995

  23. [31]

    Meckes,The random matrix theory of the classical compact groups, Cambridge University Press, 2019

    E. Meckes,The random matrix theory of the classical compact groups, Cambridge University Press, 2019

    2019

  24. [32]

    Okounkov,Infinite wedge and random partitions, Selecta Math

    A. Okounkov,Infinite wedge and random partitions, Selecta Math. 7 (2001), 57–81

    2001

  25. [33]

    E. M. Rains,Correlation functions for symmetrized increasing subsequences, arXiv: math/0006097 [math.CO], 2000

    2000 arXiv

  26. [34]

    R. P. Stanley,Enumerative combinatorics, vol. 2, Cambridge University Press, Cambridge, 1999

    1999

  27. [35]

    J. R. Stembridge,Nonintersecting paths, Pfaffians, and plane partitions, Adv. Math. 83(1) (1990), 96–131

    1990

  28. [36]

    Wang,Correlation functions of strict partitions and twisted Fock spaces, Transform

    W. Wang,Correlation functions of strict partitions and twisted Fock spaces, Transform. Groups 9(1) (2004), 89–101

    2004

  29. [37]

    Z. Wang, C. Yang,Correlation function of self-conjugate partitions:q-difference equation and quasi- modularity, Math. Ann. 395, (2026), 50

    2026

  30. [38]

    Weyl,Classical groups: their invariants and representations, Princeton University Press, Princeton, 1946

    H. Weyl,Classical groups: their invariants and representations, Princeton University Press, Princeton, 1946. 21 School of Science, Huzhou Normal University, Huzhou, Zhejiang 313000, China Email address:2253583891@qq.com School of Science, Huzhou Normal University, Huzhou, Zhej...

    1946

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