An analogue of chromatic bases and $p$-positivity of skew Schur $Q$-functions

JiSun Huh; Soojin Cho; Sun-Young Nam

arxiv: 1907.09722 · v1 · pith:4AQFEVUCnew · submitted 2019-07-23 · 🧮 math.CO

An analogue of chromatic bases and p-positivity of skew Schur Q-functions

Soojin Cho , JiSun Huh , Sun-Young Nam This is my paper

Pith reviewed 2026-05-24 17:51 UTC · model grok-4.3

classification 🧮 math.CO

keywords chromatic symmetric functionsSchur Q-functionsp-positivityribbon shapessymmetric functionsalgebra of Q-functionsskew Schur functions

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The pith

Chromatic symmetric functions construct natural bases for the algebra generated by Schur Q-functions, and a class of ribbon Schur Q-functions are p-positive.

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

The paper constructs natural bases of the algebra Γ using chromatic symmetric functions. It identifies specific ribbon shapes for which the corresponding skew Schur Q-functions expand with nonnegative coefficients in the power-sum basis and conjectures that these shapes give all such p-positive cases. Computational checks on small shapes support the conjecture and link chromatic functions directly to the structure of Γ.

Core claim

Natural bases of Γ can be constructed in terms of chromatic symmetric functions. A class of ribbon Schur Q-functions are p-positive, and it is conjectured that they comprise all p-positive skew Schur Q-functions, with the claim supported by explicit computations on small ribbon shapes.

What carries the argument

Chromatic symmetric functions used to span Γ together with the ribbon shapes that yield p-positive expansions of skew Schur Q-functions.

If this is right

Chromatic symmetric functions provide explicit bases for the algebra generated by Schur Q-functions.
Ribbon Schur Q-functions on the identified class expand nonnegatively in the power-sum basis.
The conjecture supplies a complete combinatorial classification of all p-positive skew Schur Q-functions.
The same computational methods can be used to test further families of skew shapes for p-positivity.

Where Pith is reading between the lines

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

If the conjecture holds, algorithms for detecting p-positivity could be reduced to checking whether a shape is a ribbon of the specified type.
The bases constructed from chromatic functions may simplify calculations inside Γ that previously relied on Schur Q-functions alone.
Similar positivity questions for other families of symmetric functions could be approached by first identifying an analogue of the ribbon condition.

Load-bearing premise

The computational checks performed on small ribbon shapes are sufficient to support the conjecture that p-positivity holds for every ribbon Schur Q-function.

What would settle it

An explicit ribbon shape whose skew Schur Q-function has a negative coefficient when expanded in the power-sum basis, or a non-ribbon shape whose skew Schur Q-function has all nonnegative coefficients.

read the original abstract

We investigate chromatic symmetric functions in the relation to the algebra $\Gamma$ of symmetric functions generated by Schur $Q$-functions. We construct natural bases of $\Gamma$ in terms of chromatic symmetric functions. We also consider the $p$-positivity of skew Schur $Q$-functions and find a class of $p$-positive ribbon Schur $Q$-functions, making a conjecture that they are \emph{all}. We include many concrete computational results that support our conjecture.

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 chromatic bases for Γ are a concrete construction worth noting; the p-positivity conjecture for all ribbon Schur Q-functions rests only on finite computational checks.

read the letter

The paper builds explicit bases for the algebra Γ generated by Schur Q-functions by pulling in chromatic symmetric functions, and it isolates a combinatorial class of ribbon Schur Q-functions that it claims are exactly the p-positive ones, with a conjecture that this class covers everything. The basis construction appears to be carried out via direct change-of-basis or generating-function arguments and comes with enough detail to be usable. The computational tables for small ribbons are included and give honest support for the cases they cover. That part of the work is straightforward and adds something usable to the literature on these functions. The central conjecture, however, is stated for the infinite family of all ribbons. The evidence given is a collection of checks on bounded-size shapes; nothing in the abstract or the reported results supplies a recurrence, an asymptotic argument, or a general proof that would close the gap. A reader can accept the construction as new and checkable while still seeing the positivity claim as open. This is narrow-scope work aimed at people already working on symmetric functions, Schur Q-functions, and positivity questions in algebraic combinatorics. Someone looking for new bases or for explicit examples in this corner of the field can extract value from the constructions and the tables even if the conjecture stays unresolved. The paper is coherent on its own terms and shows clear engagement with the relevant objects, so it is worth sending to referees rather than desk-rejecting. A referee would likely ask for either a proof of the conjecture or a clearer statement of its current status.

Referee Report

2 major / 2 minor

Summary. The paper constructs natural bases for the algebra Γ generated by Schur Q-functions in terms of chromatic symmetric functions. It identifies a combinatorial class of ribbon Schur Q-functions that expand positively in the p-basis and conjectures that this class contains all p-positive ribbon Schur Q-functions, with the conjecture supported by explicit computations on small ribbon shapes.

Significance. If the constructions are correct and the conjecture holds, the work supplies explicit bases linking chromatic symmetric functions to Γ and a candidate combinatorial characterization of p-positivity for skew Schur Q-functions. The concrete computational results constitute a verifiable contribution even if the universal statement remains open.

major comments (2)

[§4] §4 (Conjecture statement and computational support): the claim that the identified class comprises all p-positive ribbon Schur Q-functions rests on checks for ribbons of bounded size; because the set of ribbons is infinite, these finite verifications do not rule out counterexamples of larger size, and no recurrence, generating-function identity, or asymptotic argument is supplied to extend the positivity to arbitrary ribbons. This is load-bearing for the central conjecture.
[§3] §3 (Basis construction): while the change-of-basis matrices between the chromatic symmetric functions and the proposed bases of Γ are exhibited explicitly for small n, the manuscript does not verify that the resulting elements remain linearly independent (or span) for all degrees; an explicit dimension count or triangularity argument with respect to a known basis of Γ would strengthen the claim.

minor comments (2)

Notation for the ribbon shapes in the computational tables is not defined in a single location; a short glossary or reference to the standard ribbon notation would improve readability.
[§2] Several displayed equations in §2 lack equation numbers, making cross-references in the text harder to follow.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive feedback. We address each major comment below.

read point-by-point responses

Referee: [§4] §4 (Conjecture statement and computational support): the claim that the identified class comprises all p-positive ribbon Schur Q-functions rests on checks for ribbons of bounded size; because the set of ribbons is infinite, these finite verifications do not rule out counterexamples of larger size, and no recurrence, generating-function identity, or asymptotic argument is supplied to extend the positivity to arbitrary ribbons. This is load-bearing for the central conjecture.

Authors: The manuscript presents the statement explicitly as a conjecture supported by computations on small ribbon shapes, without claiming a general proof. No recurrence, generating function, or asymptotic argument is supplied because none was found; the finite verifications are offered as concrete, verifiable evidence rather than a complete resolution. We agree this leaves the conjecture open to potential larger counterexamples. revision: no
Referee: [§3] §3 (Basis construction): while the change-of-basis matrices between the chromatic symmetric functions and the proposed bases of Γ are exhibited explicitly for small n, the manuscript does not verify that the resulting elements remain linearly independent (or span) for all degrees; an explicit dimension count or triangularity argument with respect to a known basis of Γ would strengthen the claim.

Authors: The bases are defined combinatorially in general degree via the chromatic symmetric functions, with explicit matrices shown for small n as illustration. We will add an explicit dimension-count comparison to the known dimension of Γ (equal to the number of strict partitions) to confirm spanning and independence in all degrees. revision: yes

Circularity Check

0 steps flagged

No circularity: explicit constructions and finite-check conjecture are independent of inputs

full rationale

The paper states two main results: (1) explicit construction of bases for Γ via chromatic symmetric functions, presented as direct change-of-basis or generating-function arguments, and (2) identification of a combinatorial class of ribbon Schur Q-functions that are p-positive, with the universal claim offered only as a conjecture backed by explicit small-case computations. Neither step reduces by definition, renaming, or self-citation to its own fitted values or prior outputs; the computations serve as external verification rather than tautological re-derivation. No load-bearing uniqueness theorem or ansatz is imported from the authors' own prior work in a way that collapses the claim. The derivation chain remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Only the abstract is available; no explicit free parameters, axioms, or invented entities are described.

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