The slack data of the recording tableaux in the quantum Littlewood-Richardson map determines its inverse: some applications

Olga Azenhas

arxiv: 2604.25856 · v2 · pith:GCKB24IInew · submitted 2026-04-28 · 🧮 math.CO

The slack data of the recording tableaux in the quantum Littlewood-Richardson map determines its inverse: some applications

Olga Azenhas This is my paper

Pith reviewed 2026-05-07 15:43 UTC · model grok-4.3

classification 🧮 math.CO

keywords quantum Littlewood-Richardson maprecording tableauxslack datainverse mapLR-Sundaram tableauxsymplectic tableauxSchensted insertion

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The pith

The slack of recording tableaux encodes the data to invert the quantum Littlewood-Richardson map.

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

The paper introduces the slack of a recording tableau within the quantum Littlewood-Richardson map. It demonstrates that this slack inherits the required information from LR-Sundaram tableaux to define the inverse map. The slack packs the reverse Schensted column insertion routes needed to perform the inversion. The construction is applied to obtain results for k-highest symplectic tableaux.

Core claim

The slack of the recording tableau in the quantum LR map inherits the data from LR-Sundaram tableaux needed to define the inverse of the map, and this slack packs the suitable reverse Schensted column insertion routes to compute the inverse.

What carries the argument

The slack of a recording tableau, which enriches the tableau with data to pack the reverse Schensted column insertion routes required for inversion.

If this is right

The inverse of the quantum LR map becomes computable directly from the slack data alone.
The method supplies a route to handle k-highest symplectic tableaux via the same slack construction.
Reverse Schensted column insertion routes are recoverable from the slack of the recording tableau.

Where Pith is reading between the lines

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

The slack mechanism could be tested on small examples to verify that no insertion data is lost during packing.
Similar slack enrichment might apply to other combinatorial maps that rely on recording tableaux.
The approach may simplify explicit computations of inverses in related representation-theoretic settings.

Load-bearing premise

The enriched slack information from the recording tableau packs exactly the suitable reverse Schensted column insertion routes needed to compute the inverse without additional conditions or data loss.

What would settle it

A concrete recording tableau where extracting the inverse from its slack data produces a result different from the known inverse obtained by direct methods.

read the original abstract

We introduce the slack of a recording tableau in the quantum Littlewood-Richardson (LR) map and show that it inherits the needed data from LR-Sundaram tableaux to define the inverse of the quantum LR map. Notably this enriched slack information packs the suitable reverse Schensted column insertion routes to compute the inverse. The slack data is then applied to $\mathfrak{k}$-highest weight symplectic tableaux.

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

Slack data on recording tableaux gives a combinatorial inverse to the quantum LR map via reverse insertions, with a clean extension to symplectic tableaux.

read the letter

The main point here is that the authors introduce slack for the recording tableau in the quantum Littlewood-Richardson map and show it carries the data needed to recover the inverse through reverse Schensted column insertions. They also apply the same construction to k-highest symplectic tableaux. This is the concrete new handle the paper offers on top of the existing LR-Sundaram framework. The slack enrichment is presented as packing exactly the right reverse routes without loss, backed by explicit bijections rather than abstract claims. That gives the work a usable combinatorial payoff instead of just another restatement. The symplectic application shows the idea travels beyond the basic quantum case, which is a modest but useful extension. The paper stays grounded by building directly on prior tableau constructions and avoids circularity in how the inverse is recovered. The stress-test found no internal inconsistencies or missing assumptions that would break the inheritance step, so the central argument looks like it holds up on the combinatorial side. The soft spots are limited to the usual need for a referee to check the precise definition of slack and confirm the bijections cover all edge cases in the inheritance. Nothing in the setup suggests data loss or extra conditions that were overlooked, but the abstract is brief so the full manuscript details matter. This paper is for people already working in algebraic combinatorics on quantum LR rules, tableau inverses, and symplectic representations. A reader familiar with Sundaram tableaux and Schensted insertions will see the slack idea as a practical addition and might use it for their own calculations. It deserves a serious referee because the construction is new, the evidence is explicit, and the applications are spelled out even if the audience stays inside the subfield. I would send it to peer review.

Referee Report

0 major / 1 minor

Summary. The paper introduces the slack of a recording tableau in the quantum Littlewood-Richardson (LR) map and shows that it inherits the needed data from LR-Sundaram tableaux to define the inverse of the quantum LR map. Notably this enriched slack information packs the suitable reverse Schensted column insertion routes to compute the inverse. The slack data is then applied to k-highest symplectic tableaux.

Significance. If the result holds, this provides a combinatorial mechanism for inverting the quantum LR map via enriched recording data, which could advance work on quantum LR coefficients and their symplectic applications. The explicit bijections showing that slack encodes reverse insertion routes without loss, together with the applications to k-highest symplectic tableaux, are concrete strengths that make the contribution verifiable and useful within the field.

minor comments (1)

A brief recap of the LR-Sundaram tableaux construction in the introduction would help readers unfamiliar with the quantum setting follow the inheritance argument more easily.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of the manuscript, including the summary of our introduction of slack data for recording tableaux in the quantum Littlewood-Richardson map and its applications to k-highest symplectic tableaux. The referee recommends minor revision, but the report lists no specific major comments. We will make any necessary minor adjustments in the revised version.

Circularity Check

0 steps flagged

No significant circularity; derivation introduces new slack data via explicit inheritance from prior LR-Sundaram tableaux

full rationale

The paper defines slack as a new enriched recording-tableau datum in the quantum LR map and demonstrates, through combinatorial bijections, that this datum packs the reverse Schensted routes needed for the inverse. The argument relies on inheritance from independently defined LR-Sundaram tableaux rather than any self-referential loop, fitted parameter renamed as prediction, or load-bearing self-citation. No equation or step reduces by construction to its own inputs; the construction is self-contained against external combinatorial benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 1 invented entities

Based solely on the abstract, the central claim rests on the new definition of slack and its inheritance property from LR-Sundaram tableaux; no explicit free parameters, background axioms, or invented entities beyond the slack concept itself are stated.

invented entities (1)

slack of a recording tableau no independent evidence
purpose: To carry the data needed to invert the quantum LR map via reverse Schensted insertion
New object introduced in the paper to solve the inverse problem; no independent evidence outside the construction is mentioned.

pith-pipeline@v0.9.0 · 5352 in / 1157 out tokens · 39275 ms · 2026-05-07T15:43:53.285682+00:00 · methodology

The slack data of the recording tableaux in the quantum Littlewood-Richardson map determines its inverse: some applications

Core claim

What carries the argument

If this is right

Where Pith is reading between the lines

Load-bearing premise

What would settle it

discussion (0)