Shuffle Tableaux, Littlewood--Richardson Coefficients, and Schur Log-Concavity

Chau Nguyen; Dora Woodruff; Son Nguyen

REVIEW 5 major objections 6 minor 39 references

Shuffle Tableaux, Littlewood--Richardson Coefficients, and Schur Log-Concavity

T0 review · 5 major / 6 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read The paper establishes that Littlewood-Richardson coefficients count peelable tableaux of a given shape on a shuffle diagram, a new rule that also speeds up Temperley-Lieb immanant computations and proves a special case of the Schur…

desk verdict A new peelable-tableau rule for LR coefficients that is plausible and worth refereeing, but the unrestricted-shape claim in Corollary 2.7 and the Section 4 injection need real proofs before the results are fully established. read the letter →

arxiv 2506.00349 v1 pith:GMYQKB5O submitted 2025-05-31 math.CO

Chau Nguyen , Son Nguyen , Dora Woodruff This is my paper

classification math.CO MSC 05E0505E10

keywords Littlewood-RichardsoncoefficientsshuffletableauxpeelableTemperley-LiebimmanantsSchurlog-concavityBender-Knuthinvolutionscrystaloperatorspositivity

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 claims a new combinatorial formula for Littlewood-Richardson coefficients: given two skew shapes $\lambda/\mu$ and $\nu/\rho$, build the shuffle diagram $D$ by interlacing their squares, and let $a_i$ be the number of columns of $D$ containing squares in rows $i$ and $i+2$. Then the coefficient $c^{\kappa}_{\lambda/\mu,\nu/\rho}$ equals the number of semistandard tableaux of shape $\kappa$ that are $D$-peelable, meaning that for each $i$ there are at least $a_i$ disjoint pairs of an $i$-square and an $(i+2)$-square with the $i$-square northeast of the $(i+2)$-square. This extends the Remmel-Whitney peelable-tableau rule from products of two Schur functions to products of arbitrary skew Schur functions. It matters because the rule gives a checkable way to compute these structure constants, a faster route to the generalized Littlewood-Richardson coefficients attached to Temperley-Lieb immanants of Jacobi-Trudi matrices, and a proof of a special case of a Schur log-concavity conjecture.

What carries the argument

The carrying object is the shuffle diagram $(\lambda/\mu) \circledast (\nu/\rho)$, obtained by interlacing the squares of the two skew shapes so that rows and columns of the two factors alternate; together with its peelability condition, it packages the Littlewood-Richardson rule into a northeast-matching condition between entries differing by 2. The argument's load-bearing chain is: Yamanouchi shuffle tableaux on $D$ count $c^{\kappa}_{\lambda/\mu,\nu/\rho}$ because Temperley-Lieb crystals are type A Kashiwara crystals; the map $\varphi$ gives a bijection between Yamanouchi shuffle tableaux and $D$-compatible tableaux (standard tableaux obeying two relative-position conditions read from the diagram); and inverse standardization converts $D$-compatible tableaux into $D$-peelable tableaux, whose defining condition is precisely the standardized content condition. The Bender-Knuth involution sequence $BK_1 \circ BK_3 \circ \cdots \circ BK_{2\ell-1}$ is the symmetry mechanism, and the injection $\theta$ of Section 4 is the positivity mechanism for the Schur log-concavity special case.

What would settle it

Enumerate $D$-peelable tableaux of shape $\kappa$ for a pair of skew shapes outside the constrained family of the prior Temperley-Lieb crystal construction, for instance by brute-force enumeration over all partitions $\kappa$ of size at most 5, and compare with the classical Littlewood-Richardson rule; a single mismatch would falsify Theorem 1.2 as stated.

Watch

Extended reading notes

Core claim

The central claim is Theorem 1.2: let $D$ be the shuffle diagram of shape $(\lambda/\mu) \circledast (\nu/\rho)$, formed by placing squares of $\lambda/\mu$ at odd coordinates $(2i-1,2j-1)$ and squares of $\nu/\rho$ at even coordinates $(2i,2j)$, and let $a_i$ be the number of columns with squares on both row $i$ and row $i+2$. A semistandard Young tableau $T$ of shape $\kappa$ is $D$-peelable if for every $i$ there are at least $a_i$ disjoint pairs of an $i$-square and an $(i+2)$-square with the $i$-square northeast of the $(i+2)$-square. The theorem asserts that the Littlewood-Richardson coefficient $c^{\kappa}_{\lambda/\mu,\nu/\rho}$ counts exactly these $D$-peelable tableaux of shape $\kappa$. The proof passes through a bijection between $D$-compatible standard tableaux and Yamanouchi shuffle tableaux (shuffle tableaux on which no lowering crystal operator can be applied), then identifies $D$-peelable tableaux with inverse standardizations of $D$-compatible tableaux; because Yamanouchi shuffle tableaux on a shuffle diagram are known to count Littlewood-Richardson coefficients, the count transfers.

Load-bearing premise

The argument rests on an unproved assertion (Remark 2.9) that the prior construction of Yamanouchi shuffle tableaux works for all skew shapes, not only the restricted shapes where it was originally proved; if that assertion fails, the new formula for arbitrary skew shapes fails with it.

Editorial extensions

If this is right

Theorem 1.2 gives a new enumeration of $c^{\kappa}_{\lambda/\mu,\nu/\rho}$ for products of arbitrary skew Schur functions, reducing the computation to counting tableaux that satisfy a simple pairwise matching condition.
Corollary 2.23 turns the coefficient of $s_{\lambda}$ in any Temperley-Lieb immanant of a Jacobi-Trudi matrix into a count of peelable tableaux, bypassing the slower Yamanouchi-checking procedure.
Theorem 1.4 shows the symmetry $c^{\kappa}_{\lambda/\mu,\nu/\rho}=c^{\kappa}_{\nu/\rho,\lambda/\mu}$ is realized by a shape-preserving bijection given by odd-indexed Bender-Knuth involutions.
Theorem 1.7 proves Schur positivity of $s_{\nu}s_{\rho}-s_{\lambda}s_{\mu}$ for the family $\lambda=(a^k,1^{n-k-1},0)$, $\nu=\lambda-(e_m+\cdots+e_k)$, confirming a special case of the Schur log-concavity conjecture.

Reading between the lines

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

The authors do not state it, but the peelable criterion suggests a direct enumeration algorithm: count semistandard tableaux of shape $\kappa$ and check only the pairwise column-matching condition, which would avoid crystal operators entirely and could be benchmarked against the Yamanouchi count on random small skew shapes.
As an extension the paper does not pursue, the path of $\theta$ in Section 4 hints that if the path argument extends to other partitions with a block of equal rows followed by a tail, the injection would prove additional cases of the Schur log-concavity conjecture beyond the vertical-strip family.
The paper treats only the swap symmetry; because the Bender-Knuth bijection is shape-preserving and local, composing odd-indexed involutions with their inverses may yield tableau-level proofs of other symmetries of Littlewood-Richardson coefficients, such as the cyclic symmetry.
The shuffle-diagram construction naturally suggests a rule for iterated products of several skew Schur functions by interlacing more than two shapes, a generalization the paper does not pursue.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

The new thing here is a D-peelable tableau rule on shuffle diagrams, extending the Remmel–Whitney idea to products of two skew Schur functions. There is also a Bender–Knuth symmetry proof and an injection proving a special case of the Lam–Postnikov–Pylyavskyy Schur log-concavity conjecture. The paper is clearly written, with worked examples that actually help, and the bijection φ between D-compatible and Yamanouchi shuffle tableaux in Section 2 is mostly proved in detail. The BK symmetry argument is genuinely elegant, and the injection θ in Section 4 is a real construction, even if the proof is compressed.

The soft spots are in proportion to how much they carry. The biggest one is Corollary 2.7. It inherits the Yamanouchi-to-LR equality from [NP25, Theorem 5.1], and that theorem came with extra shape constraints. Remark 2.9 simply asserts those constraints are unnecessary for this section, with no argument or reference. That is load-bearing: if the crystal operators fail to be well-defined or the Temperley–Lieb crystals do not decompose as type A Kashiwara crystals for general skew shapes, the equality can fail even though the character identity holds. The proof of Theorem 1.2 never checks this independently. This is a real proof gap, not a disagreement with consensus. It may be fixable, but it has to be fixed, or the theorem must be restated under the NP25 hypotheses.

Second, Lemma 2.19, which connects D-compatible tableaux to D-peelable tableaux via inverse standardization, is left to the reader. That is the bridge to the main theorem, and it deserves a few lines. Third, Section 4's injection θ is supported by Lemma 4.10 with a one-sentence proof, and Lemma 4.1(iii) looks garbled as printed. The construction is intricate enough that a referee will need the details. The efficiency claim for peelable versus Yamanouchi tableaux is also unbenchmarked, but that is minor.

Who is this for? People working on LR combinatorics, crystal bases, and Schur positivity. The formula, if it holds, is a useful addition to the toolkit, and the application to the LPP conjecture is a genuine special case. The gaps are substantial but look fillable, and the paper does not seem to be hiding anything—it explicitly flags the constraint issue in Remark 2.9. I would send it to a serious referee rather than desk reject. If the referee can close the Remark 2.9 gap, the paper is in decent shape; if not, the authors need to restrict the statement.

Referee Report

5 major / 6 minor

Summary. The paper proposes a new combinatorial formula for Littlewood-Richardson coefficients: for two skew shapes λ/μ and ν/ρ, the coefficient c^κ_{λ/μ,ν/ρ} is counted by D-peelable tableaux of shape κ, where D is the shuffle diagram (λ/μ) ⊛ (ν/ρ). The proof strategy is to exhibit a bijection φ between D-compatible tableaux and Yamanouchi shuffle tableaux, invoke the result of [NP25] that Yamanouchi shuffle tableaux count Littlewood-Richardson coefficients, and then construct a second bijection ψ between D-compatible and D-peelable tableaux via standardization. The paper also derives a corollary for Temperley-Lieb immanants, proves a Bender-Knuth symmetry statement, and uses the new rule to prove a special case of the Lam-Postnikov-Pylyavskyy Schur log-concavity conjecture.

Significance. If correct, Theorem 1.2 gives a new, and potentially computationally efficient, way to compute Littlewood-Richardson coefficients for products of skew Schur functions, with a direct application to Temperley-Lieb immanants in Corollary 2.23. The Bender-Knuth symmetry result is elegant, and the application to Schur log-concavity is a nontrivial step toward a known conjecture. However, the paper relies at several load-bearing points on assertions that are not fully proved, most critically the unconditional validity of the Yamanouchi criterion for arbitrary skew shapes. These gaps must be repaired before the paper can be accepted.

major comments (5)

[Section 2.1, Remark 2.9] The central equality between the number of Yamanouchi shuffle tableaux and the Littlewood-Richardson coefficient is stated in Corollary 2.7 as a consequence of [NP25, Theorem 5.1], but then Remark 2.9 asserts that the additional shape constraints imposed in [NP25] are unnecessary for this section. No proof, example, or reference is supplied for this removal. Since Theorem 1.2 and Corollary 2.23 both pass through this step for arbitrary skew shapes, this unproved assertion is load-bearing. The authors must either prove that the crystal operators remain well-defined and that the Temperley-Lieb crystals decompose as type A Kashiwara crystals without those constraints, or they must restrict the statements of Corollary 2.7, Theorem 1.2, and the subsequent applications to the shapes covered by [NP25].
[Section 2.2, Proposition 2.13] Proposition 2.13, which establishes the bijection between D-compatible tableaux and Yamanouchi shuffle tableaux, is the first pillar of the main theorem, but its proof is only a sketch. The individual lemmas (2.14-2.17) contain the main arguments, but Lemma 2.17 in particular is dense and contains unclear statements, such as 'for S to be Yamanouchi, the has to be non-(r-2,r-1)-overlapped (r-2)-squares right of h but left of i' on page 9. Given that this proposition is load-bearing for Theorem 1.2, the authors should provide a complete, self-contained proof of the bijection, with all cases in Lemma 2.17 written out clearly.
[Section 2.3, Lemma 2.19] Lemma 2.19, which asserts that inverse standardization converts D-compatible tableaux into D-peelable tableaux and standardization converts D-peelable tableaux back into D-compatible tableaux, is stated with 'the proof ... is straightforward and is left to the readers.' This lemma is the second pillar of Theorem 1.2, as it is exactly the bridge between the D-compatible/Yamanouchi bijection and the D-peelable count. Leaving this proof to the reader is not acceptable for a central step; a detailed proof should be included, or at minimum a rigorous statement of the correspondence between the NE matching condition and the inverse-standardization recipe.
[Section 4, Theorem 4.6 and Lemmas 4.10-4.14] The proof of Theorem 1.7 rests on the injection θ constructed in Definition 4.3, but the accompanying proof sketch lists five lemmas, several of which are proved only sketchily. In particular, Lemma 4.10 is proved in a single sentence: if a required square cannot be found, 'the path of θ forms a sequence of λm increasing entries, each strictly East of the previous. Hence, by the NE matching condition of peelable tableaux, µm is at least λm.' This is a nontrivial implication that is not spelled out, and it is crucial to well-definedness. Lemma 4.12 also makes several claims (e.g., the existence of tk+1, the relative position of sk+1 and tk+1, and the recursive step) that are asserted rather than fully proved. Since the injection is the core of the Schur log-concavity application, these proofs must be completed.
[Section 4, Lemma 4.1] Lemma 4.1(iii) is garbled as printed: 'µm − µℓ−1 ≥ · · · ≥µk − µℓ−1 ≤ a − 1' is not a coherent inequality, and the quantity ℓ is not defined in the lemma (it appears elsewhere as the number of parts of λ). This lemma is used repeatedly in the proof of the injection, so the statement must be corrected and proved. As written, it is not possible for a reader to verify the argument.

minor comments (6)

[Section 2.1, Example 2.8] In Example 2.8, the text says 'and κ = 3' and then 'cκ = 3', but κ should be the partition (4,3,2) as in Example 1.3, not the integer 3.
[Section 4, Definition 4.3] In Step (4) of Definition 4.3, the list of involved squares contains a typo: 'sk+2, tk+2, sk+3, tk+2, . . .' should presumably read 'sk+2, tk+2, sk+3, tk+3, . . .'.
[Section 4, Definition 4.3] There is a spelling error in Definition 4.3: 'repear' should be 'repeat'.
[Section 1.3 and Theorem 1.4] The sentence 'Assume for simplicity that ℓ(λ) = ℓ(ν) (if ℓ(λ) > ℓ(µ), we can append 0's at the end of µ to get ℓ(λ) = ℓ(ν))' is confusing, as it refers to both ℓ(λ)=ℓ(ν) and ℓ(λ)>ℓ(µ) without clear relation; the theorem statement uses ℓ = ℓ(λ). Please clarify which partitions are padded in the statement and proof of Theorem 1.4.
[Section 2.2 and 2.3, Definitions 2.12 and 2.20] The instruction 'Sort all the rows in increasing order' appears in both definitions of φ and ψ, but it is not precise about how the sorted rows are assembled into a shuffle tableau while preserving the required row and column conditions. A formal description of the sorting procedure would improve clarity.
[Section 4, Lemma 4.12] The phrase 'if there is a 2 k-square in column some c ≥ a' is ungrammatical; it should be 'in some column c ≥ a'.

Circularity Check

0 steps flagged · score 2.0 of 10

No construction-level circularity: the peelable-tableau count is a genuine bijection, but the Yamanouchi-to-Littlewood-Richardson bridge is inherited from a same-author prior paper and its unrestricted-shape extension is asserted without proof.

full rationale

Theorem 1.2 decomposes into a new bijective part and an imported part. The new part (Definitions 2.10, 2.12, 2.18, 2.20; Lemmas 2.14-2.19) establishes a shape-content-preserving bijection between D-peelable tableaux and Yamanouchi shuffle tableaux; peelability is defined independently of c^kappa, so this is not a definitional identity or a fitted-parameter count. The imported part is Corollary 2.7, which identifies Yamanouchi shuffle tableaux with highest-weight elements of a type A Kashiwara crystal and therefore with Littlewood-Richardson coefficients; the paper cites [NP25, Theorem 5.1] for this. Because one of the present authors is an author of [NP25], this is a self-citation, but it is a published, externally checkable theorem rather than an unverified premise, and no parameter in this paper is fitted to force the count. The genuine weakness is Remark 2.9: it asserts, without argument or lemma, that the additional constraints on the shapes lambda/mu and nu/rho imposed in [NP25] are unnecessary for this section. For arbitrary skew shapes, this unproved removal is load-bearing for Corollary 2.7 and hence for Theorem 1.2; that is a proof gap and a correctness risk, not circularity. The Bender-Knuth and Schur-log-concavity sections depend only on the peelable-tableau machinery and do not reintroduce c^kappa by construction. The circularity burden is therefore low.

Assumptions & free parameters 0 free parameters · 3 assumptions · 0 invented entities

No independent physical or mathematical entities are postulated; shuffle diagrams, D-compatible tableaux, and D-peelable tableaux are definitions within the paper, not externally evidenced entities. No free parameters are fitted to data.

assumptions (3)

standard math Standard facts about Schur functions, skew Schur functions, standardization of semistandard tableaux, and crystal operators.
Used throughout, for example in Section 2.1 where s_{lambda/mu} s_{nu/rho} = sum_kappa c^kappa s_kappa and in the definition of Yamanouchi tableaux.
domain assumption The theorem of Nguyen-Pylyavskyy [NP25, Theorem 5.1] that Temperley-Lieb crystals are type A Kashiwara crystals, and Corollary 2.7 identifying Yamanouchi shuffle tableaux with Littlewood-Richardson coefficients.
This is the bridge connecting the paper's new peelable-tableaux count to actual Littlewood-Richardson coefficients. It is a separate published result by one co-author, not derived in this paper.
standard math Bender-Knuth involutions preserve semistandardness and swap contents of i and i+1.
Used in Theorem 1.4 without proof; this is a standard property of Bender-Knuth involutions.

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Cite this review

Pith. "Pith review of Shuffle Tableaux, Littlewood--Richardson Coefficients, and Schur Log-Concavity." pith.science (2026). https://pith.science/paper/GMYQKB5O

@misc{pith2026250600349,
  author       = {Pith},
  title        = {Pith review of: Shuffle Tableaux, Littlewood--Richardson Coefficients, and Schur Log-Concavity},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/GMYQKB5O}},
  note         = {Machine review of arXiv:2506.00349}
}

read the original abstract

We give a new formula for the Littlewood--Richardson coefficients in terms of peelable tableaux compatible with shuffle tableaux, in the same fashion as Remmel--Whitney rule. This gives an efficient way to compute generalized Littlewood--Richardson coefficients for Temperley--Lieb immanants of Jacobi--Trudi matrices. We will also show that our rule behaves well with Bender--Knuth involutions, recovering the symmetry of Littlewood--Richardson coefficients. As an application, we use our rule to prove a special case of a Schur log-concavity conjecture by Lam--Postnikov--Pylyavskyy.

Figures

Figures reproduced from arXiv: 2506.00349 by the authors.

**Figure 1.** The Young diagram of (3, 2)/(1, 0) (left) and (3, 3)/(1, 0) (center) and their shuffle diagram (right) Thus, for a SSYT T to be D-peelable, we need one compatible pair of 1 and 3, and two compatible pairs of 2 and 4. Let κ = (4, 3, 2), [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗

**Figure 2.** D-peelable tableaux [PITH_FULL_IMAGE:figures/full_fig_p002_2.png] view at source ↗

**Figure 3.** Image under BK1 ◦BK3 of the peelable tableaux in [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗

Figures from the paper (4 more)

**Figure 4.** Figure 4: λ (left) and ν (right) Acknowledgement We thank Alex Postnikov and Pavlo Pylyavskyy for telling us about their Schur log-concavity conjecture. We thank Vic Reiner for telling us about peelable tableaux. We thank Daniel Soskin for helpful conversations. 2. Littlewood–Ri…

**Figure 5.** Figure 5: SSYTs of shape λ/µ (left) and ν/ρ (center) and their corresponding shuffle tableau (right) Definition 2.2. Given a shuffle tableau T, an (i, i + 1)-overlap is a pair of squares (s, t) on the same column such that s contains an i and t contains an i + 1. We define the …

**Figure 6.** Figure 6: Yamanouchi shuffle tableaux Remark 2.9. Shuffle tableaux were originally introduced to study Temperley–Lieb immanants, hence the name Temperley–Lieb crystal. In [NP25], there are additional constraints on the shapes λ/µ and ν/ρ so that the crystal operators behave ni…

**Figure 7.** Figure 7: Labeling of D 1 1 12 2 2 3 4 5 6 7 8 9 3 4 5 6 7 8 9 3 4 5 6 7 8 9 [PITH_FULL_IMAGE:figures/full_fig_p007_7.png]

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A generalization of the L ittlewood- R ichardson rule and the R obinson- S chensted- K nuth correspondence

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Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.

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Parametrizations of canonical bases and totally positive matrices

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