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arxiv: 2605.17880 · v1 · pith:AEKKDSP4new · submitted 2026-05-18 · 🧮 math.CO

Thrall's problem for two rows

JiSun Huh , Woo-Seok Jung , Jang Soo Kim , Meesue Yoo This is my paper

Pith reviewed 2026-05-20 09:48 UTC · model grok-4.3

classification 🧮 math.CO
keywords Thrall's problemhigher Lie modulesSchur expansionstandard Young tableauxYamanouchi domino tableauxmajor indexspin paritytwo-row partitions
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The pith

The Schur expansion of the character of L_λ for two-row shapes λ is given by counting standard Young tableaux that satisfy major index congruence and spin-parity conditions via a bijection to Yamanouchi domino tableaux.

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

The paper solves Thrall's problem for higher Lie modules L_λ when the partition λ has exactly two rows by supplying an explicit combinatorial formula for the decomposition of its character into Schur functions. The formula counts certain standard Young tableaux of shape λ whose major indices satisfy a congruence condition and whose spin-parity is determined by mapping them bijectively to Yamanouchi domino tableaux. The same approach yields formulas for hook shapes and for partitions with distinct parts, and it extends at once to any partition in which parts larger than 2 occur at most twice.

Core claim

The Schur expansion of ch(L_λ) for two-row λ is obtained by summing, over all standard Young tableaux T of shape λ that obey a major-index congruence and a spin-parity condition coming from the Yamanouchi domino tableau bijection, the Schur function s_shape(T) with a sign given by the spin parity.

What carries the argument

The bijection between the relevant standard Young tableaux and Yamanouchi domino tableaux that preserves the major-index congruence and the spin-parity statistic.

If this is right

  • The multiplicity of any Schur function s_μ in ch(L_λ) equals the number of standard Young tableaux of shape λ that meet the major-index congruence and spin-parity conditions.
  • The same counting rule supplies the Schur expansion when λ is a hook or has distinct parts.
  • The formula continues to hold for every partition in which each part larger than 2 appears at most twice.
  • The character is therefore completely determined by a signed enumeration of ordinary standard Young tableaux under two explicit statistics.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same bijection technique may produce analogous formulas for partitions with three or more rows once a suitable higher-order domino or ribbon tableau is identified.
  • The major-index and spin-parity conditions could be reinterpreted as generating functions that relate the Lie-module character to known q-analogues of symmetric-function identities.
  • Explicit small-case computations become feasible, allowing direct verification of the decomposition for concrete two-row shapes without invoking the full representation theory.

Load-bearing premise

The bijection between standard Young tableaux and Yamanouchi domino tableaux must preserve both the major-index congruence and the spin-parity in such a way that the resulting signed count equals the multiplicity of each Schur function.

What would settle it

Compute the character of L_{(n,n)} for small n by representation-theoretic methods, extract the coefficient of a specific Schur function s_μ, and check whether it equals the number of qualifying standard Young tableaux of shape (n,n) with the stated major-index and spin-parity conditions.

Figures

Figures reproduced from arXiv: 2605.17880 by Jang Soo Kim, JiSun Huh, Meesue Yoo, Woo-Seok Jung.

Figure 1
Figure 1. Figure 1: An example of X(S, T) = (T ′ , S′ ). Cells originating from S are colored gray. For a partition λ = (λ1, λ2, . . .), we define λ □ := (2λ1, 2λ1, 2λ2, 2λ2, . . .). Carré and Leclerc showed the following combinatorial formula for the Schur expansion of h2[sλ]. Theorem 2.5. [3, Corollary 5.5] For λ ⊢ n, we have h2[sλ] = X µ⊢2n |{D ∈ YDT(λ □, µ) : spin(D) ≡ n (mod 2)}| sµ. We note that in [3, Corollary 5.5] th… view at source ↗
Figure 2
Figure 2. Figure 2: The tableaux in S µ⊢6 SYT(4,2)(µ). Above each T, the sequence (majλ,1 (T), majλ,2 (T)) is shown. We highlight in bold the entries contributing to the block-major indices, i.e., those d ∈ Des(T) such that d and d + 1 are con￾tained in the same block. By Lemma 3.3, the condition T Bλ,j ∈ Aµ (j) j is equivalent to maj(T Bλ,j ) ≡ 1 (mod λj ) and sh(T Bλ,j ) = µ (j) . Hence, summing over all partitions µ (j) ⊢ … view at source ↗
Figure 3
Figure 3. Figure 3: The tableaux in S µ⊢6 SYT(3,3)(µ) satisfying T[3] = T [4,6], together with their spin values. Lemma 4.2. There is a bijection ξ : SYT=(2n) → G λ⊢n µ⊢2n SYT(λ) × YDT(λ □, µ) such that if ξ(T) = (U, D), then U = T[n] and the weight of D is equal to sh(T). The bijection ξ in Lemma 4.2 allows us to define a spin statistic on SYT=(2n). Definition 4.3. For T ∈ SYT=(2n) with ξ(T) = (U, D), we define the spin of T… view at source ↗
Figure 4
Figure 4. Figure 4: An example of the construction of d(M) for M ∈ LR(λ ∗ λ, µ) for λ = (3, 2) and µ = (5, 3, 2). The shaded region represents the staircase partition δ5 = (4, 3, 2, 1). 4.3.2. The map Φ0. In what follows, we review the following bijection due to van Leeuwen [10, Theorem 2.2.6]: Φ0 : LR(λ ∗ λ, µ) → YDT(λ □, µ). For k ≥ 1, let δk = (k − 1, k − 2, . . . , 1) denote the staircase partition with k − 1 parts. The f… view at source ↗
Figure 5
Figure 5. Figure 5: Successive applications of the map θ to D. Solid arrows indicate open chains and dashed arrows indicate closed chains. Step 2. A chain is a maximal sequence (ζ1, . . . , ζm) of distinct dominoes such that there is an arrow from a cell in ζi+1 to a cell in ζi for all i ∈ [m − 1]. If there is an arrow from a cell in ζ1 to a cell outside α/δk, then we say that the chain is open, and otherwise the chain is clo… view at source ↗
read the original abstract

In this paper, we study Thrall's problem for the higher Lie modules $L_\lambda$. Our main result provides a tableau-theoretic description of the Schur expansion of the character of $L_\lambda$ when $\lambda$ has two rows, thereby solving Thrall's problem in this case. This formula is expressed in terms of standard Young tableaux with major index congruence conditions and a spin-parity condition defined through bijections with Yamanouchi domino tableaux. We also obtain tableau formulas for hook shapes and partitions with distinct parts, and these results extend to all partitions in which each part greater than $2$ occurs at most twice.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper studies Thrall's problem for the higher Lie modules L_λ. Its main result gives a tableau-theoretic description of the Schur expansion of ch(L_λ) when λ has two rows, expressed via standard Young tableaux of shape λ satisfying a major-index congruence condition together with a spin-parity condition obtained by transporting the sum through an explicit bijection to Yamanouchi domino tableaux. The paper also supplies formulas for hook shapes and for partitions with distinct parts, and states that the results extend to all partitions in which each part greater than 2 occurs at most twice.

Significance. If the central bijection is shown to preserve the required major-index congruence class and spin-parity statistic, the result would solve Thrall's problem for the two-row case and supply an explicit combinatorial formula for the Schur multiplicities. This is a concrete advance in combinatorial representation theory; the explicit map and the extension to hooks and nearly-distinct parts are positive features that could support further computations or generalizations.

major comments (2)
  1. [§3] §3 (construction of the SYT-to-Yamanouchi-domino bijection): the claim that the map simultaneously preserves the congruence class of maj and the parity of the spin statistic is load-bearing for the equality between the signed sum and the Schur coefficients. The verification must be made fully explicit—either by direct computation of how the statistics transform under the map or by a complete inductive argument with all base cases checked—rather than left as a case-by-case or outline argument.
  2. [Proof of main theorem] Proof of the main theorem: the collected coefficients after transport must be shown to equal the multiplicity of each Schur function in ch(L_λ). A small explicit example (e.g., λ = (4,2) or λ = (3,3)) computing both sides independently would confirm that the preserved statistics produce the correct multiplicities; without such a check the central equality remains unverified.
minor comments (2)
  1. [Abstract] The abstract refers to the spin-parity condition being 'defined through bijections' but does not give even a one-sentence description; a brief parenthetical definition or forward reference to the precise definition in the text would improve readability.
  2. [§2] Notation for the spin statistic and the precise meaning of 'major-index congruence' should be introduced once in a dedicated preliminary subsection rather than only inside the proof.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive suggestions. We address the two major comments below and have revised the manuscript accordingly to make the arguments fully explicit and to include a concrete verification example.

read point-by-point responses

  1. Referee: [§3] §3 (construction of the SYT-to-Yamanouchi-domino bijection): the claim that the map simultaneously preserves the congruence class of maj and the parity of the spin statistic is load-bearing for the equality between the signed sum and the Schur coefficients. The verification must be made fully explicit—either by direct computation of how the statistics transform under the map or by a complete inductive argument with all base cases checked—rather than left as a case-by-case or outline argument.

    Authors: We agree that the preservation of the major-index congruence class and spin parity under the bijection requires a fully explicit argument. In the revised manuscript we replace the outline with a complete induction on the number of dominoes. The inductive step tracks how each local replacement in the bijection affects the major index (modulo the relevant congruence) and the spin parity; all base cases for shapes with at most two dominoes are enumerated directly. We also include a short direct computation of the statistic transformation for the generating step of the map. revision: yes

  2. Referee: [Proof of main theorem] Proof of the main theorem: the collected coefficients after transport must be shown to equal the multiplicity of each Schur function in ch(L_λ). A small explicit example (e.g., λ = (4,2) or λ = (3,3)) computing both sides independently would confirm that the preserved statistics produce the correct multiplicities; without such a check the central equality remains unverified.

    Authors: We accept the suggestion and have added an explicit verification subsection for λ = (4,2). Using the known decomposition of the higher Lie module into irreducibles (via the Lie representation theory of the symmetric group), we compute the Schur coefficients of ch(L_{(4,2)}) independently. We then enumerate the qualifying SYT of shape (4,2) satisfying the major-index congruence and spin-parity condition obtained from the Yamanouchi-domino bijection, and confirm that the resulting multiplicities match exactly. This check is now included in the revised proof of the main theorem. revision: yes

Circularity Check

0 steps flagged

No circularity: direct combinatorial bijection for two-row Thrall problem

full rationale

The paper constructs an explicit bijection between standard Young tableaux of two-row shape and Yamanouchi domino tableaux, then verifies that this map preserves the major-index congruence class and spin-parity statistic. The resulting signed sum is shown to equal the Schur coefficients in ch(L_λ) by direct transport of the statistics. No parameter is fitted to a subset of the target multiplicities and then re-used as a prediction; no self-citation supplies a uniqueness theorem that forces the form of the answer; the derivation therefore remains independent of its own output.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The work rests on standard facts about Young tableaux, major index, and known bijections to domino tableaux; no new free parameters or invented entities are introduced.

axioms (1)
  • standard math Standard combinatorial properties of standard Young tableaux, major index statistic, and Yamanouchi domino tableaux hold as previously established in the literature.
    Invoked to define the congruence and parity conditions in the main formula.

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Reference graph

Works this paper leans on

11 extracted references · 11 canonical work pages

  1. [1]

    Ahlbach and J

    C. Ahlbach and J. P. Swanson. Cyclic sieving, necklaces, and branching rules related to Thrall’s problem. Electron. J. Combin., 25(4):Paper No. 4.42, 38, 2018

    work page 2018

  2. [2]

    Benkart, F

    G. Benkart, F. Sottile, and J. Stroomer. Tableau switching: algorithms and applications.J. Combin. Theory Ser. A, 76(1):11–43, 1996

    work page 1996

  3. [3]

    Carré and B

    C. Carré and B. Leclerc. Splitting the square of a Schur function into its symmetric and antisymmetric parts. J. Algebraic Combin., 4(3):201–231, 1995

    work page 1995

  4. [4]

    A. A. Klyachko. Lie elements in a tensor algebra.Sibirsk. Mat. Ž., 15:1296–1304, 1430, 1974

    work page 1974

  5. [5]

    Kraśkiewicz and J

    W. Kraśkiewicz and J. Weyman. Algebra of coinvariants and the action of a Coxeter element.Bayreuth. Math. Schr., (63):265–284, 2001

    work page 2001

  6. [6]

    I. G. Macdonald.Symmetric functions and Hall polynomials. Oxford Mathematical Monographs. The Claren- don Press Oxford University Press, New York, second edition, 1995

    work page 1995

  7. [7]

    R. P. Stanley.Enumerative combinatorics. Vol. 2, volume 208 ofCambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge, second edition, 2024. With an appendix by Sergey Fomin

    work page 2024

  8. [8]

    R. M. Thrall. On symmetrized Kronecker powers and the structure of the free Lie ring.Amer. J. Math., 64:371–388, 1942

    work page 1942

  9. [9]

    M. A. A. van Leeuwen. Edge sequences, ribbon tableaux, and an action of affine permutations.European J. Combin., 20(2):179–195, 1999

    work page 1999

  10. [10]

    M. A. A. van Leeuwen. Some bijective correspondences involving domino tableaux.Electron. J. Combin., 7:Research Paper 35, 25, 2000

    work page 2000

  11. [11]

    M. A. A. van Leeuwen. The Littlewood-Richardson rule, and related combinatorics. InInteraction of combi- natorics and representation theory, volume 11 ofMSJ Mem., pages 95–145. Math. Soc. Japan, Tokyo, 2001. Department of Mathematics, University of Seoul, Seoul 02504, South Korea Email address:hyunyjia@yonsei.ac.kr Department of Mathematics, University of...

    work page 2001