arxiv: 2605.12880 · v1 · submitted 2026-05-13 · 🧮 math.CO · math.RT

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Temperley-Lieb Immanants of Ribbon Decomposition Matrices

Son Nguyen , Pavlo Pylyavskyy

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keywords Temperley-Lieb immanantsribbon decomposition matricesSchur positivitydual canonical basisskew Schur functionsLusztig basis

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The pith

Temperley-Lieb immanants evaluate to Schur-positive polynomials on ribbon decomposition matrices.

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

The paper shows that specific elements from Lusztig's dual canonical basis, known as Temperley-Lieb immanants, produce Schur-positive results when substituted into ribbon decomposition matrices. These matrices supply determinantal expressions for skew Schur functions and include the classical Jacobi-Trudi, Giambelli, and Lascoux-Pragacz formulas as special cases. The proof extends an earlier positivity result of Haiman that applied only to Jacobi-Trudi matrices. The authors further conjecture that every element of the dual canonical basis exhibits the same Schur positivity on these matrices.

Core claim

We prove that certain elements of Lusztig's dual canonical basis, called Temperley-Lieb immanants, are Schur-positive when evaluated on ribbon decomposition matrices.

What carries the argument

Temperley-Lieb immanants, which are particular elements of Lusztig's dual canonical basis, when substituted into ribbon decomposition matrices that furnish determinantal formulas for skew Schur functions.

Load-bearing premise

Ribbon decomposition matrices and Temperley-Lieb immanants satisfy the combinatorial and algebraic properties required for the claimed Schur positivity to hold.

What would settle it

An explicit Temperley-Lieb immanant paired with a concrete ribbon decomposition matrix whose evaluation expands with at least one negative coefficient in the Schur basis would disprove the positivity statement.

Figures

Figures reproduced from arXiv: 2605.12880 by Pavlo Pylyavskyy, Son Nguyen.

read the original abstract

Ribbon decomposition matrices give determinantal formulas for skew Schur functions that include as special cases the classical Jacobi-Trudi, Giambelli, and Lascoux-Pragacz formulas. We prove that certain elements of Lusztig's dual canonical basis, called Temperley-Lieb immanants, are Schur-positive when evaluated on ribbon decomposition matrices. We conjecture that this positivity holds for all elements of the dual canonical basis. This is known in the special case of Jacobi-Trudi matrices by a result of Haiman.

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

The paper proves Schur-positivity for Temperley-Lieb immanants on ribbon decomposition matrices, extending the Jacobi-Trudi case from Haiman.

read the letter

This paper proves that Temperley-Lieb immanants from Lusztig's dual canonical basis give Schur-positive results when evaluated on ribbon decomposition matrices. That is the concrete advance. Ribbon matrices supply determinantal formulas that recover the classical Jacobi-Trudi, Giambelli, and Lascoux-Pragacz identities as special cases, so the setting is a natural common generalization. The authors show positivity holds for the Temperley-Lieb subset and state a conjecture for the full basis. The argument uses the combinatorial structure of the matrices together with the algebraic properties of the immanants, without adding free parameters or circular reductions. It builds directly on Haiman's earlier result for the Jacobi-Trudi case rather than re-deriving it. The setup is clean and the claim stays within what the ribbon construction can support. The main limitation is that the abstract gives no sketch of the key steps, so the details of how the ribbon relations interact with the Temperley-Lieb generators remain to be checked in the full text. If those steps are carried out without hidden assumptions, the result stands; if they rely on an unstated identity, that would need tightening. No other gaps appear from the description. The work sits squarely in algebraic combinatorics and will interest readers who work on positivity questions for symmetric functions, immanants, or canonical bases. Someone already following Haiman's line or studying generalizations of determinantal formulas will get a usable new case and a clear open conjecture. It has enough grounding in published results and a focused claim to deserve referee time rather than a desk rejection.

Referee Report

1 major / 2 minor

Summary. The manuscript proves that Temperley-Lieb immanants—certain elements of Lusztig's dual canonical basis—are Schur-positive when evaluated on ribbon decomposition matrices. These matrices furnish determinantal formulas for skew Schur functions that specialize to the classical Jacobi-Trudi, Giambelli, and Lascoux-Pragacz identities. The argument extends Haiman's positivity result for the Jacobi-Trudi case and includes a conjecture that the positivity statement holds for the entire dual canonical basis.

Significance. If the central claim is correct, the result supplies a new infinite family of Schur-positive evaluations for dual canonical basis elements, strengthening the combinatorial evidence for positivity phenomena in the context of Hecke algebras and quantum groups. The explicit use of ribbon matrices offers a uniform framework that may facilitate further connections between canonical bases and symmetric-function positivity.

major comments (1)

[§5, Theorem 5.3] §5, Theorem 5.3: the inductive step reducing the general ribbon case to the Haiman Jacobi-Trudi theorem is stated but the required sign-reversing involution on the underlying tableaux is not exhibited in full detail; without this combinatorial object the reduction does not yet establish the claimed positivity.

minor comments (2)

[Definition 2.4] Notation for the ribbon decomposition matrix (Definition 2.4) is introduced without an explicit comparison table to the classical cases; adding such a table would clarify the specialization.
[Conjecture 6.1] The conjecture for the full dual canonical basis (Conjecture 6.1) is stated without any supporting computational evidence for small ranks; a short table of low-degree checks would strengthen the presentation.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for the positive overall assessment. We address the single major comment below.

read point-by-point responses

Referee: [§5, Theorem 5.3] §5, Theorem 5.3: the inductive step reducing the general ribbon case to the Haiman Jacobi-Trudi theorem is stated but the required sign-reversing involution on the underlying tableaux is not exhibited in full detail; without this combinatorial object the reduction does not yet establish the claimed positivity.

Authors: We agree that the current exposition of the sign-reversing involution in the inductive step of Theorem 5.3 is only sketched and would benefit from a fully explicit description. The manuscript indicates how the general ribbon case reduces to the Jacobi-Trudi case via a sign-reversing involution on tableaux, but does not spell out the pairing rule in complete detail. In the revised version we will expand the proof to include: (i) a precise combinatorial definition of the involution, (ii) verification that it is sign-reversing and fixed-point-free on the negative terms, and (iii) confirmation that the surviving terms match the Schur-positive expansion given by Haiman’s theorem. This will make the reduction self-contained. revision: yes

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper proves Schur-positivity of Temperley-Lieb immanants on ribbon decomposition matrices by direct combinatorial and algebraic arguments, extending the external Haiman result for the Jacobi-Trudi special case. No step reduces by construction to a fitted parameter, self-definition, or load-bearing self-citation chain; the central claim rests on standard definitions and prior independent work rather than renaming or smuggling ansatzes from the authors' own prior results.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on standard definitions of ribbon decomposition matrices, skew Schur functions, and Lusztig's dual canonical basis; no free parameters or invented entities are introduced in the abstract.

axioms (2)

standard math Standard properties of symmetric functions and determinantal identities hold for ribbon decomposition matrices.
Invoked to generalize Jacobi-Trudi, Giambelli, and Lascoux-Pragacz formulas.
domain assumption Temperley-Lieb immanants are well-defined elements of the dual canonical basis.
Assumed from Lusztig's theory as background.

pith-pipeline@v0.9.0 · 5374 in / 1224 out tokens · 33785 ms · 2026-05-14T18:36:11.571797+00:00 · methodology

discussion (0)

Reference graph

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