arxiv: 2603.25254 · v2 · submitted 2026-03-26 · 🧮 math.CO

Recognition: 2 theorem links

· Lean Theorem

Inverse Kazhdan--Lusztig polynomials of fan matroids

Alice L.L. Gao , Ya-Xing Li , Yun Li

Authors on Pith no claims yet

Pith reviewed 2026-05-15 00:40 UTC · model grok-4.3

classification 🧮 math.CO MSC 05B35

keywords inverse Kazhdan-Lusztig polynomialfan matroidlog-concave sequencegenerating functiongraphic matroidinverse Z-polynomialmatroid polynomial

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The pith

Fan matroids admit explicit formulas for their inverse Kazhdan-Lusztig polynomials.

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

The paper computes generating functions for the inverse Kazhdan-Lusztig polynomials of fan matroids from their recursive definition and derives closed-form expressions. Parallel results are obtained for the inverse Z-polynomials using a similar generating-function approach. As a direct consequence of the explicit formula, the coefficients of the inverse Kazhdan-Lusztig polynomial for any fan matroid form a log-concave sequence with no internal zeros. A sympathetic reader would care because these polynomials serve as combinatorial invariants whose positivity properties often signal underlying algebraic or enumerative structure.

Core claim

For the family of fan matroids, generating functions obtained from the recursive definition yield explicit formulas for both the inverse Kazhdan-Lusztig polynomials and the inverse Z-polynomials. These formulas are independently verified via the deletion formulas. The explicit expression for the inverse Kazhdan-Lusztig polynomial immediately implies that its coefficients are log-concave and have no internal zeros.

What carries the argument

The inverse Kazhdan-Lusztig polynomial, a matroid invariant defined by a recursive relation that satisfies deletion-contraction identities, which on fan matroids (graphic matroids of fan graphs) reduces to closed-form expressions via generating functions.

If this is right

The generating functions permit direct computation of the polynomials for fan matroids of arbitrary size.
The same method produces explicit expansions for the inverse Z-polynomials of fan matroids.
Deletion-contraction relations supply independent proofs of the generating functions.
Log-concavity without internal zeros follows directly from the closed-form expression.

Where Pith is reading between the lines

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

The explicit formulas may extend to other families of graphic matroids that share similar recursive deletion structures.
Log-concavity raises the possibility that these polynomials are real-rooted, a property that could be checked directly from the formula.
The results suggest testing whether analogous positivity holds for inverse polynomials of other well-structured matroid classes such as uniform or graphic matroids on trees.

Load-bearing premise

The recursive definition and deletion formulas for inverse Kazhdan-Lusztig polynomials apply without modification to fan matroids.

What would settle it

A single fan matroid whose inverse Kazhdan-Lusztig polynomial has coefficients that fail to be log-concave or that contain an internal zero.

Figures

Figures reproduced from arXiv: 2603.25254 by Alice L.L. Gao, Ya-Xing Li, Yun Li.

read the original abstract

The inverse Kazhdan--Lusztig polynomial of a matroid was introduced by Gao and Xie, and the inverse $Z$-polynomial of a matroid was introduced by Ferroni, Matherne, Stevens, and Vecchi. In this paper, we study these two polynomials for fan matroids, a family of graphic matroids associated with fan graphs. We first derive the generating functions for the inverse Kazhdan--Lusztig polynomials of fan matroids using their recursive definition, and then deduce the explicit formulas of these polynomials therefrom. For the inverse $Z$-polynomials of fan matroids, we obtain their generating functions using a parallel generating function approach, and further derive their explicit expansions based on these generating functions. Additionally, we provide alternative proofs for the above generating functions using the deletion formulas for inverse Kazhdan--Lusztig and inverse $Z$-polynomials. As an application of the explicit formula for inverse Kazhdan--Lusztig polynomials, we prove that the coefficients of the inverse Kazhdan--Lusztig polynomial of the fan matroid form a log-concave sequence with no internal zeros.

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

Explicit formulas for inverse Kazhdan-Lusztig polynomials on fan matroids plus a direct log-concavity proof.

read the letter

The main thing to know is that this paper derives closed-form expressions for the inverse Kazhdan-Lusztig polynomials of fan matroids and shows that the coefficient sequence is log-concave with no internal zeros. It does the same for the inverse Z-polynomials in parallel. They start from the recursive definition to produce generating functions, pull out the explicit polynomials, and then give a second derivation of the generating functions via deletion formulas. The log-concavity statement follows immediately once the formula is written down. All of this is new for the fan family. The work is methodical and stays within the tools supplied by the earlier papers that defined these polynomials. Fan matroids are graphic, so the general recursion and deletion rules apply without modification or extra conditions. The alternative deletion proof is a useful consistency check rather than a deep new idea. No load-bearing gaps appear in the argument from what is described. The only limitation worth noting is that the paper is narrowly focused on one concrete family, so it supplies useful examples and verifies a property rather than reshaping the broader theory. Within algebraic combinatorics this is still worthwhile because explicit formulas make it easier to test further conjectures or look for patterns across matroid classes. A reader working on matroid polynomials, positivity questions, or log-concavity would find the results directly usable. I would bring the paper to a reading group on combinatorial invariants. It deserves peer review because the calculations are concrete, the methods are standard but applied cleanly, and the log-concavity application is a clear, verifiable payoff for this family.

Referee Report

0 major / 3 minor

Summary. The paper studies inverse Kazhdan-Lusztig polynomials and inverse Z-polynomials for fan matroids. It derives generating functions for the inverse Kazhdan-Lusztig polynomials from their recursive definition, deduces explicit formulas, obtains parallel generating functions and explicit expansions for the inverse Z-polynomials, supplies alternative derivations via deletion formulas, and applies the explicit inverse Kazhdan-Lusztig formula to prove that the coefficient sequence is log-concave with no internal zeros.

Significance. The explicit formulas and the log-concavity result supply concrete, verifiable data for a natural infinite family of graphic matroids. The dual use of recursion and deletion formulas illustrates standard combinatorial techniques and may serve as a benchmark for broader conjectures on these polynomials.

minor comments (3)

The abstract and introduction should explicitly state the indexing convention for fan matroids (e.g., number of blades) to avoid ambiguity when comparing formulas across sections.
In the proof of log-concavity (application of the explicit formula), the verification that there are no internal zeros should include a short inductive or direct argument rather than being left implicit from the closed form.
The generating-function derivations would benefit from a displayed base-case computation for the smallest fan matroid (e.g., F_2 or F_3) to make the recursion step fully transparent.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary of our work on inverse Kazhdan--Lusztig and inverse Z-polynomials for fan matroids, including the explicit formulas, generating functions, and the log-concavity result. The recommendation of minor revision is noted; however, no specific major comments were raised in the report.

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The paper takes the recursive definition and deletion formulas for inverse Kazhdan-Lusztig and inverse Z-polynomials from prior literature (Gao-Xie and Ferroni et al.) and applies them to the new family of fan matroids. It computes generating functions, extracts explicit formulas, supplies an alternative deletion-based derivation, and then uses the explicit formula to prove log-concavity with no internal zeros. These steps consist of direct algebraic computation on a concrete matroid class; none reduce the claimed results to the inputs by definition or by a self-citation chain that itself lacks independent grounding. The self-citation of the original definition is ordinary and does not render the fan-matroid calculations circular.

Axiom & Free-Parameter Ledger

0 free parameters · 3 axioms · 0 invented entities

No new free parameters or invented entities are introduced. The work relies on the recursive definition and deletion formulas from the cited papers that introduced the polynomials.

axioms (3)

domain assumption The inverse Kazhdan-Lusztig polynomial of a matroid satisfies the recursive definition given by Gao and Xie.
Invoked to derive generating functions for fan matroids.
domain assumption The inverse Z-polynomial satisfies the parallel recursive relations introduced by Ferroni et al.
Used for the generating-function approach.
domain assumption Deletion formulas hold for both families of polynomials.
Employed for alternative proofs.

pith-pipeline@v0.9.0 · 5506 in / 1425 out tokens · 34739 ms · 2026-05-15T00:40:03.111815+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

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unclear
Relation between the paper passage and the cited Recognition theorem.

We first derive the generating functions for the inverse Kazhdan–Lusztig polynomials of fan matroids using their recursive definition... explicit formula (3)... coefficients form a log-concave sequence with no internal zeros.
IndisputableMonolith/Foundation/AbsoluteFloorClosure.lean absolute_floor_iff_bare_distinguishability unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

alternative proofs... using the deletion formulas for inverse Kazhdan–Lusztig and inverse Z-polynomials

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

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