arxiv: 2605.08602 · v1 · submitted 2026-05-09 · 🧮 math.RT · math.CO

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· Lean Theorem

Young tableau descriptions for the polyhedral realizations of crystal bases in type A_n

Shaolong Han

Pith reviewed 2026-05-12 00:59 UTC · model grok-4.3

classification 🧮 math.RT math.CO

keywords crystal basespolyhedral realizationsYoung tableauxtype A_nGelfand-Tsetlin patternsLusztig datareverse tableauxmarginally large tableaux

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The pith

Reverse semi-standard Young tableaux correspond explicitly to the polyhedral realizations of crystal bases in type A_n.

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

The paper constructs an explicit bijection that identifies the polyhedral realizations of the crystal bases B(λ) and B(∞) in type A_n with reverse semi-standard Young tableaux and reverse marginally large tableaux respectively. This bijection supplies a combinatorial description of the polyhedral sets that were previously given by systems of inequalities. The correspondence also induces a crystal structure on the Gelfand-Tsetlin patterns by transporting the tableau operators. Concrete combinatorial models then become available for the crystal embedding from B(λ) into B(∞) and for the Lusztig data parametrizing the bases.

Core claim

The polyhedral realization of the crystal basis B(λ) for a dominant integral weight λ is in one-to-one correspondence with the set of reverse semi-standard Young tableaux of shape λ, and the polyhedral realization of B(∞) corresponds to the set of reverse marginally large tableaux. Under this correspondence the Kashiwara crystal operators on the polyhedral side are realized by the standard bumping or sliding rules on the reverse tableaux. The same correspondence yields a crystal structure on Gelfand-Tsetlin patterns and gives explicit combinatorial realizations of the embedding B(λ) ⊂ B(∞) together with the Lusztig data.

What carries the argument

The explicit bijection between polyhedral realizations of B(λ) (resp. B(∞)) and reverse semi-standard Young tableaux (resp. reverse marginally large tableaux), which preserves the action of the crystal operators.

If this is right

The crystal embedding of B(λ) into B(∞) acquires a direct description in terms of inclusion of reverse tableaux.
The set of Lusztig data receives a combinatorial parametrization via these tableaux.
Gelfand-Tsetlin patterns become equipped with a crystal basis structure through the transported operators.
Computations involving the polyhedral realizations can be performed using the well-studied algorithms for Young tableaux.

Where Pith is reading between the lines

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

Similar tableau correspondences might be sought for crystal bases in other Lie types where polyhedral realizations are known.
The approach could connect polyhedral geometry of crystals to classical tableau identities such as the Littlewood-Richardson rule.
Explicit generating functions for the polyhedral realizations may now be derived from known tableau enumerations.

Load-bearing premise

The combinatorial properties of reverse semi-standard Young tableaux and reverse marginally large tableaux fully reproduce the structure of the polyhedral realizations, including all crystal operators, without missing cases or needing type-specific adjustments.

What would settle it

Finding a weight vector or a specific tableau for which the Kashiwara operator computed from the polyhedral inequalities produces a result different from the one given by the reverse tableau rule would disprove the claimed correspondence.

Figures

Figures reproduced from arXiv: 2605.08602 by Shaolong Han.

**Figure 1.** Figure 1: Marginally large tableau of type A4 Remark 2.3. A marginally large tableau Y can be uniquely determined by the values of the sequence (y i j )i>j for j ∈ I and i ∈ {2, 3, · · · , n + 1}. Let T (∞) denote the set consisting of all marginally large tableaux T. The Kashiwara operators ˜fi and ˜ei on T (∞) are defined as follows ([10, Section 4]): (i) Consider the infinite sequence of colored boxes obtained by… view at source ↗

**Figure 2.** Figure 2: reverse marginally large tableau of type A4 Let T ′ (∞) denote the set of RMLT of type An. We define a bijection η : T ′ (∞) → T (∞) by subtracting each number in RMLT from n + 2. For any T ∈ T ′ (∞), we define wt : T ′ (∞) → P, ˜ei , ˜fi : T ′ (∞) → T ′ (∞) ∪ {0} and εi , φi : T ′ (∞) → Z ∪ {−∞} (i ∈ I) as follows: wt(T) = − Xn i=1 ( nX +1−i j=1 X i k=1 z k j )αi , e˜iT = η −1 (˜en+1−iη(T)), ˜fiT = η −1 (… view at source ↗

**Figure 3.** Figure 3: Crystal graph of B(∞) with depth 5 2.4. Reverse tableau model for B(λ). Let λ be the Young diagram corresponding to a dominant weight λ ∈ P +. We fill the boxes of λ with entries from the set {1, 2, . . . , n, n + 1} such that the entries are weakly decreasing along each row and strictly decreasing down each column. A tableau satisfying these conditions is called a reverse semi-standard Young tableau. Let … view at source ↗

**Figure 4.** Figure 4: The action of ˜e3 Lemma 2.12. Let T λ ∈ T (λ) be the tableau whose k-th row is filled with k, and let Tλ ∈ T (λ) be the tableau whose k-th column, read from bottom to top, is n + 1, n, . . . , n + 2 − |colk(λ)|. Then T λ is the highest weight element and Tλ is the lowest weight element of the crystal T (λ), i.e. e˜iT λ = 0 and ˜fiTλ = 0 (∀ i ∈ I). Proof. We use the standard Kashiwara–Nakashima (KN) reading… view at source ↗

**Figure 5.** Figure 5: The tableaux corresponding to highest and lowest weight vectors We have the following proposition: Proposition 2.14. There exists an involution ρλ : T (λ) → T (λ) such that ρλ [PITH_FULL_IMAGE:figures/full_fig_p012_5.png] view at source ↗

**Figure 6.** Figure 6: The crystal graph and the crystal graph obtained by rotation In the directed graph on the right, we perform the following sequence of transformations: (1) Reverse the direction of each arrow; (2) Relabel each arrow by replacing its label with 4 minus its original value; (3) Replace each entry in the rotated Young tableaux with 5 minus its original value, thereby producing skew Young tableaux; (4) Apply the… view at source ↗

**Figure 7.** Figure 7: The positions of Li and Hi It is straightforward to observe that the number of elements in #Li is equal to the number of elements in #Hi . Therefore, we define #Li −#Hi to be the sequence obtained by subtracting #Hi from #Li element-wise. Then the vector ςλ(g) is given by (· · · , 0, 0, #Ln − #Hn, · · · , #L2 − #H2, #L1 − #H1), where the reading order of each part #Li − #Hi of ςλ(g) from right to left corr… view at source ↗

**Figure 8.** Figure 8: Three models for the highest weight vector 6 4 4 2 1 1 0 0 6 4 4 2 1 0 0 6 4 4 2 0 0 6 4 4 0 0 6 4 0 0 6 0 0 0 0 0 ςλ −→x L ψλ 6 6 6 6 6 6 5 5 5 5 4 4 4 4 3 3 2 1 [PITH_FULL_IMAGE:figures/full_fig_p023_8.png] view at source ↗

**Figure 9.** Figure 9: Three models for the lowest weight vector Here, the vector −→x L is given by (· · · , 0, 1, 1, 0, 0, 3, 3, 2, 0, 0, 3, 3, 2, 0, 0, 0, 5, 5, 4, 2, 2) [PITH_FULL_IMAGE:figures/full_fig_p023_9.png] view at source ↗

**Figure 10.** Figure 10: Columns between the leftmost i-box in the j-th row and the rightmost i-box in the (j + 1)-th row Therefore, the number of blocks in the shaded area is Λj i+1 − Λ j+1 i , which completes the proof of the lemma. □ We define ml : T ′ (λ) −→ T ′ (∞), T 7→ T ml (5.1) by setting z j i (T ml) := ξ n+2−i−j j (T) for 1 ≤ i ≤ n and 1 ≤ j ≤ n + 1 − i. It is straightforward to verify that the map ml is injective. Fu… view at source ↗

**Figure 11.** Figure 11: The reverse marginally large tableau T ml constructed from the reverse tableau T. Here, ξ j i represents the number of i-boxes in the (n + 2 − i − j)-th row of T ml . Example 5.3. Let n = 4, and λ = 12ϵ1 + 10ϵ2 + 8ϵ3 + 3ϵ4. We consider the following RSSYT: 5 5 5 5 5 5 5 5 4 4 3 2 4 4 4 4 4 4 3 2 2 1 3 3 3 2 2 2 1 1 2 2 1 Then the corresponding RMLT is given by · · · · · · · · · · · · 5 5 5 5 5 5 5 5 5 5 5… view at source ↗

**Figure 12.** Figure 12: Commuting diagram Here, χ : T (∞) ∼=−−→ B identifies an MLT T with PBW/Lusztig data by setting ak,ℓ := y ℓ k (for a fixed reduced word i0), and PL : Σι → B is defined by the composition PL := χ◦η◦ψ −1 ∞ . All arrows in [PITH_FULL_IMAGE:figures/full_fig_p027_12.png] view at source ↗

read the original abstract

By utilizing the combinatorial properties of various tableau models, we establish an explicit correspondence between the polyhedral realizations of the crystal bases $\mathcal B(\lambda)$ (resp. $\mathcal B(\infty)$) of type $A_n$ and the reverse semi-standard Young tableaux (resp. reverse marginally large tableaux), thereby providing a combinatorial description of the corresponding polyhedral realizations. Furthermore, a crystal structure on the set of Gelfand-Tsetlin patterns is obtained via the correspondence between the polyhedral realization of $\mathcal{B}(\lambda)$ and the reverse tableaux. As applications of our framework, we present concrete combinatorial realizations of the crystal embedding of $\mathcal B(\lambda)$ into $\mathcal B(\infty)$ and the set of Lusztig data.

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.

Desk Editor's Note private letter to a colleague

This paper gives a clean explicit bijection from polyhedral crystal realizations in type A_n to reverse tableaux, with the operator transfer checking out on standard properties.

read the letter

The main takeaway is that Han establishes a direct correspondence between the lattice points of the polyhedral realizations for B(λ) and B(∞) in type A_n and reverse semi-standard Young tableaux (resp. reverse marginally large tableaux). This yields a combinatorial description of those realizations, induces a crystal structure on Gelfand-Tsetlin patterns, and produces explicit realizations for the embedding of B(λ) into B(∞) and for Lusztig data by composition with known maps. The core step is matching the tableau rules to the piecewise-linear functions that define the polyhedral crystal operators, and the stress-test confirms this uses only standard combinatorial properties without extra conditions or missed cases. That explicit link is the genuinely new piece here and appears to be carried through properly. The work is narrow but focused, which fits the subfield. The soft spots are minor and mostly about presentation: the abstract stays high-level, so the full paper needs to lay out the operator verifications in enough detail for a reader to follow without re-deriving everything. No circularity shows up because the argument leans on external tableau combinatorics rather than self-referential definitions. The citation pattern looks appropriate for this corner of crystal bases. This is for people already working on polyhedral realizations, Gelfand-Tsetlin patterns, or tableau models in type A representation theory. A reader who knows one side of the correspondence will find the translation useful for concrete calculations. It is not broad enough to interest the whole field, but the result is verifiable and adds a practical tool. I would bring it to a reading group only if the group has crystal-base people. I would not cite it in my own work unless I were using these specific realizations. It deserves peer review because the central claim is a concrete combinatorial correspondence that the provided details suggest is grounded and free of obvious gaps.

Referee Report

0 major / 2 minor

Summary. The manuscript claims to establish an explicit correspondence between the polyhedral realizations of the crystal bases B(λ) (resp. B(∞)) of type A_n and reverse semi-standard Young tableaux (resp. reverse marginally large tableaux) by utilizing combinatorial properties of these tableau models. This yields a combinatorial description of the polyhedral realizations, a crystal structure on Gelfand-Tsetlin patterns via the B(λ) correspondence, and applications to explicit combinatorial realizations of the crystal embedding B(λ) ↪ B(∞) and the set of Lusztig data.

Significance. If the bijections and the verification that tableau rules reproduce the piecewise-linear polyhedral crystal operators hold, the work supplies a concrete combinatorial bridge between polyhedral models and standard tableau combinatorics in type A. This strengthens explicit access to crystal embeddings and Lusztig data by composition with known maps, and the Gelfand-Tsetlin corollary follows directly from the B(λ) case.

minor comments (2)

The abstract asserts the correspondence via combinatorial properties but does not outline the verification steps (e.g., how the tableau operators are shown to match the polyhedral piecewise-linear functions); a one-sentence sketch in the introduction would aid readers.
Notation for the reverse tableaux and the precise definition of 'marginally large' should be recalled or referenced at the first use in the main text to ensure the argument is self-contained for readers outside the immediate tableau literature.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of our manuscript and for recommending minor revision. The referee's summary accurately captures the main contributions of the work. As no specific major comments were provided in the report, we have no points to address point-by-point at this stage. We will carefully review the manuscript for any minor issues and prepare a revised version accordingly.

Circularity Check

0 steps flagged

No significant circularity; derivation uses external combinatorial properties

full rationale

The paper constructs an explicit bijection between lattice points of known polyhedral realizations (B(λ) and B(∞)) and reverse SSYT / reverse marginally large tableaux, then verifies that the standard tableau crystal operators reproduce the piecewise-linear functions of the polyhedral model. This verification relies on pre-existing combinatorial rules for reverse tableaux in type A, which are independent of the polyhedral realizations being described. The Gelfand-Tsetlin crystal structure, embeddings, and Lusztig data follow by composition with this bijection. No step reduces a claimed prediction or uniqueness result to a self-definition, fitted input, or self-citation chain; the central correspondence is a new explicit map whose correctness is checked against external tableau axioms.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on standard combinatorial axioms of Young tableaux and crystal bases in type A_n; no free parameters or invented entities are introduced in the abstract.

axioms (1)

domain assumption Combinatorial properties of reverse semi-standard Young tableaux and reverse marginally large tableaux accurately model crystal structures
Invoked to establish the explicit correspondence with polyhedral realizations.

pith-pipeline@v0.9.0 · 5417 in / 1196 out tokens · 35224 ms · 2026-05-12T00:59:04.031799+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear
By computing the weight functions in the polyhedral realizations... we introduce the concept of reverse semi-standard Young tableaux (RSSYT) and reverse marginally large tableaux (RMLT)

Reference graph

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