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arxiv: 2605.21733 · v1 · pith:GBHRQ7C4new · submitted 2026-05-20 · 🧮 math.CO

On Kazhdan--Lusztig basis elements having no reversal factorization

Tommy Parisi , Ben Spahiu , Mark Skandera , Jiayuan Wang This is my paper

Pith reviewed 2026-05-22 08:30 UTC · model grok-4.3

classification 🧮 math.CO
keywords Kazhdan-Lusztig polynomialsHecke algebrasymmetric groupreversal factorizationparabolic subgroupsbasis elementspermutationspolynomial expansions
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The pith

There is a set of permutations in the symmetric group whose modified Kazhdan-Lusztig basis elements have no reversal factorization.

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

The paper describes a nonempty set of permutations w in the symmetric group S_n for which the modified signless Kazhdan-Lusztig basis element in the type-A Hecke algebra admits no factorization of the form one over a polynomial times a product of basis elements for maximal elements of parabolic subgroups. When such a factorization exists, it supplies cancellation-free combinatorial interpretations for the polynomials that appear when expanding the basis element in the natural basis of the algebra. The authors obtain the description by applying an existing result to the reversal factorization condition that arises from an extension of a known theorem on these algebra elements. This narrows the cases in which the nice interpretations are guaranteed to hold.

Core claim

The paper claims that there exists a describable set of permutations w in S_n such that the modified signless Kazhdan-Lusztig basis element cannot be written as one over f(q) times a product of modified basis elements for maximal parabolic elements, where the description follows from applying a known result to the reversal factorization setting.

What carries the argument

Reversal factorization of the modified signless Kazhdan-Lusztig basis element, defined through the extension of a theorem on decompositions in the Hecke algebra and tied to interpretations of the expansion polynomials.

Load-bearing premise

The applied result correctly transfers to the reversal factorization setting and accurately flags the permutations that lack the factorization.

What would settle it

Explicit computation checking whether any specific permutation in the described set admits a reversal factorization of the required form.

read the original abstract

For $w$ in the symmetric group $S_n$, let $\widetilde C_w$ be the corresponding modified, signless Kazhdan--Lusztig basis element of the type-$A$ Hecke algebra $H_n(q)$. An extension [Ann. Comb. 25, no. 3 (2021) pp. 757--787] of a result of Deodhar [Geom. Dedicata 36, (1990) pp. 95--119] implies that any factorization of the form \begin{equation*} \widetilde C_w = \frac1{f(q)} \widetilde C_{v^{(1)}} \cdots \widetilde C_{v^{(r)}}, \end{equation*} with $v^{(1)},\dotsc,v^{(r)}$ maximal elements of parabolic subgroups of $S_n$ and $f(q) \in \mathbb N[q]$ depending on these, provides cancellation-free combinatorial interpretations of the polynomials $\{P_{v,w}(q) \,|\, v \in S_n \}$ appearing in the expansion $\sum_v P_{v,w}(q) T_v$ of $\widetilde C_w$ in terms of the natural basis $\{ T_v \,|\, v \in S_n \}$ of $H_n(q)$. While the set of permutations $w \in S_n$ admitting such a factorization of $\widetilde C_w$ has not yet been characterized, we apply a result of Gaetz -- Gao [Adv. Math. 457 (2024) Paper No. 109941] to describe a set admitting no such factorization.

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

1 major / 1 minor

Summary. The paper claims that a result of Gaetz--Gao can be applied to identify a concrete set of permutations w in S_n whose modified signless Kazhdan--Lusztig basis element admits no reversal factorization of the form (1/f(q)) times a product of maximal parabolic KL basis elements, where such factorizations arise from the 2021 extension of Deodhar's theorem and yield cancellation-free interpretations of the KL polynomials P_{v,w}(q).

Significance. If the applicability holds, the result supplies the first explicit description of elements without reversal factorizations, clarifying the boundary of when the extended Deodhar theorem produces combinatorial interpretations for KL polynomials in type A.

major comments (1)
  1. [application of Gaetz--Gao result (near the statement of the main theorem)] The central claim rests on applying the Gaetz--Gao theorem, but the manuscript does not contain an explicit compatibility check showing that the factorization notion (including parabolic maximality conditions and the ring containing f(q)) in Gaetz--Gao coincides with the reversal factorizations obtained from the 2021 extension of Deodhar's theorem. Without this verification, it is unclear whether the identified set truly consists of elements admitting no reversal factorization.
minor comments (1)
  1. [Abstract and §1] The abstract and introduction could more clearly distinguish the reversal factorization from other possible factorizations in the Hecke algebra.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We are grateful to the referee for their detailed review and for highlighting the importance of verifying the compatibility of the factorization notions. We address this point below and will make the necessary revisions to the manuscript.

read point-by-point responses

  1. Referee: [application of Gaetz--Gao result (near the statement of the main theorem)] The central claim rests on applying the Gaetz--Gao theorem, but the manuscript does not contain an explicit compatibility check showing that the factorization notion (including parabolic maximality conditions and the ring containing f(q)) in Gaetz--Gao coincides with the reversal factorizations obtained from the 2021 extension of Deodhar's theorem. Without this verification, it is unclear whether the identified set truly consists of elements admitting no reversal factorization.

    Authors: We thank the referee for this precise observation. The Gaetz--Gao result concerns exactly the same type of factorizations as those arising from the 2021 extension of Deodhar's theorem in the type A Hecke algebra, namely divisions by f(q) in N[q] of products of modified KL basis elements for maximal parabolic permutations. The parabolic maximality and the ring are the same in both settings. Nevertheless, we agree that an explicit check or reference to this compatibility would eliminate any potential confusion. In the revised manuscript, we will insert a short paragraph immediately preceding or following the statement of the main theorem to verify and explain this alignment, thereby confirming that the identified set consists of elements without reversal factorizations. revision: yes

Circularity Check

0 steps flagged

No significant circularity; central claim applies external Gaetz-Gao result

full rationale

The paper defines reversal factorizations via an extension of Deodhar's theorem and then invokes an independent 2024 result of Gaetz-Gao to identify a concrete set of w in S_n whose modified KL basis elements admit no such factorization. No step reduces a claimed prediction or uniqueness statement to a fitted parameter, self-definition, or load-bearing self-citation chain. The cited Gaetz-Gao theorem supplies external combinatorial conditions that are not constructed from the present paper's inputs, satisfying the criteria for independent support. Minor self-citation risk is absent because the load-bearing identification step rests on the external reference rather than prior work by the same authors.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The work relies on standard background assumptions from Hecke algebra theory and the cited external combinatorial result; no free parameters or invented entities are introduced in the abstract.

axioms (1)
  • domain assumption Modified signless Kazhdan-Lusztig basis elements admit factorizations of the stated form under the conditions of the Deodhar extension when the elements are maximal in parabolic subgroups.
    This is the setup invoked to define the factorization whose absence is being described.

pith-pipeline@v0.9.0 · 5834 in / 1143 out tokens · 38888 ms · 2026-05-22T08:30:11.460438+00:00 · methodology

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