Mixed Products of Modified Greaves--Jing--Zhu Operators

S.-J. Lee

arxiv: 2606.28108 · v1 · pith:TD7V4RTKnew · submitted 2026-06-26 · 🧮 math.CO

Mixed Products of Modified Greaves--Jing--Zhu Operators

S.-J. Lee This is my paper

Pith reviewed 2026-06-29 03:36 UTC · model grok-4.3

classification 🧮 math.CO

keywords modified Greaves-Jing-Zhu operatorsmixed productsscalar factort-Pochhammer productsSchur Q-functionsprincipal specializationspalindromic polynomials

0 comments

The pith

The product of modified Greaves-Jing-Zhu operators with distinct parameters t and s is determined by an explicit scalar factor expressible using t-Pochhammer quotients.

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

The paper establishes that mixed products of modified Greaves-Jing-Zhu operators with two different parameters t and s produce a scalar factor with an explicit exponential expression. This factor can also be expressed as a quotient of infinite t-Pochhammer products in a completed setting. When the second parameter is a power of the first, specifically s equals t to the M, the factor simplifies to a finite quotient of Pochhammer symbols whose coefficients relate to Schur Q-functions. The resulting polynomials are palindromic and nonnegative after sign removal, and they satisfy both a Gaussian-binomial formula and a finite-order recurrence.

Core claim

The modified Greaves-Jing-Zhu operator Y(z;t) on the odd power-sum ring is obtained from the classical neutral operator by a diagonal change of variables. For parameters t and s the mixed product yields a scalar factor with an explicit exponential form that equals a quotient of infinite t-Pochhammer products. When s equals t to the power M the factor reduces to the finite quotient (u;t)_M over (-u;t)_M. The coefficients of this factor are signed principal specializations of one-row Schur Q-functions; after sign removal they become nonnegative palindromic polynomials that admit a Gaussian-binomial formula and a finite-order recurrence. Product formulas for several mixed operators and formulas

What carries the argument

The scalar factor in the mixed product of two modified Greaves-Jing-Zhu operators with parameters t and s, which admits an explicit exponential form and a representation as a quotient of t-Pochhammer products.

Load-bearing premise

The modified Greaves-Jing-Zhu operator on the odd power-sum ring arises from the classical neutral operator by a simple diagonal change of variables.

What would settle it

Direct expansion of the product of two such operators for small distinct values of t and s, followed by comparison of the resulting coefficient of the leading term against the claimed exponential or Pochhammer expression.

read the original abstract

Let $\mathcal Y(z;t)$ be the modified Greaves--Jing--Zhu operator on the odd power-sum ring. We first point out that this operator can be obtained from the classical neutral operator by a simple diagonal change of variables. We then study products in which the two deformation parameters are not necessarily the same. For two parameters $t$ and $s$, we compute the scalar factor that appears in the mixed product. This factor has an explicit exponential form and, in a completed setting, can also be written as a quotient of infinite $t$-Pochhammer products. We also give a recurrence for its coefficients, a product formula for several mixed operators, and formulas for the coefficients obtained after applying the operators to $\mathbf 1$. A particularly simple case occurs when $s=t^M$. In this case the scalar factor becomes the finite quotient $(u;t)_M/(-u;t)_M$. Its coefficients are signed principal specializations of one-row Schur $Q$-functions. As a result, after removing the signs, these coefficients are nonnegative palindromic polynomials. We also give a Gaussian-binomial formula and a finite-order recurrence.

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 computes explicit scalar factors for mixed products of modified Greaves-Jing-Zhu operators with distinct parameters t and s, plus a clean reduction when s=t^M to signed one-row Schur Q specializations.

read the letter

The main takeaway is that this paper works out the scalar factor for mixed products of the modified Greaves-Jing-Zhu operators when the two parameters differ. It gives an explicit exponential expression, rewrites it as a ratio of infinite t-Pochhammer symbols in a completed setting, supplies a recurrence for the coefficients, and extends to products of several operators. The nicest part is the case s = t^M, where the factor becomes (u;t)_M / (-u;t)_M and the coefficients turn out to be signed principal specializations of one-row Schur Q-functions. Removing the signs leaves nonnegative palindromic polynomials, and they also give a Gaussian-binomial expression plus a finite recurrence.

The work is careful about giving multiple ways to express the same thing and notes the positivity. That is the kind of concrete output that people in this area can use directly.

The potential issue is the initial step. The paper starts by saying the modified operator on the odd power-sum ring comes from the classical neutral operator via a simple diagonal change of variables. All the later formulas rest on that equivalence. The abstract presents it as an observation rather than deriving it, so if the change does not preserve the exact relations needed for mixed t and s, the scalar formulas would not hold. The stress-test flags this correctly as the least secured foundation. Without seeing the full derivation in the paper, it is hard to judge how solid that base is.

This is a paper for people already working on deformed operators and symmetric function identities. Someone looking for new explicit formulas in this niche would find it useful. It is not going to change the field, but the computations look like they could be correct and worth checking.

I would send it to peer review so that experts can verify the change of variables and the coefficient identifications.

Referee Report

1 major / 1 minor

Summary. The paper studies mixed products of modified Greaves--Jing--Zhu operators on the odd power-sum ring. It claims that the modified operator is obtained from the classical neutral operator via a simple diagonal change of variables. For parameters t and s, it derives an explicit exponential form for the scalar factor in the mixed product, which can also be expressed as a quotient of infinite t-Pochhammer products in a completed setting. It provides a recurrence for the coefficients, a product formula for several operators, and formulas for coefficients after applying to 1. In the special case s = t^M, the scalar factor is (u;t)_M / (-u;t)_M, whose coefficients are signed principal specializations of one-row Schur Q-functions, leading to nonnegative palindromic polynomials after removing signs, along with a Gaussian-binomial formula and finite-order recurrence.

Significance. If the foundational equivalence holds, the paper provides valuable explicit formulas and combinatorial connections between operator products and Schur Q-functions, including nonnegativity results. This could be useful in algebraic combinatorics. The provision of recurrences and explicit forms for the scalar factors is a strength, as is the identification with principal specializations.

major comments (1)

[Introduction] The assertion that the modified Greaves--Jing--Zhu operator on the odd power-sum ring is obtained from the classical neutral operator by a simple diagonal change of variables lacks an explicit verification or derivation. As this equivalence underpins all subsequent results on mixed products, including the scalar factor formulas and the special case identifications with Schur Q-functions, a detailed check that the transformation preserves the ring and the required commutation relations is necessary to secure the claims.

minor comments (1)

Clarify the precise definition of the completed setting in which the infinite t-Pochhammer quotients are defined, and ensure all notation for the operators and the constant term 1 is consistent throughout.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and the constructive major comment. We address it directly below.

read point-by-point responses

Referee: [Introduction] The assertion that the modified Greaves--Jing--Zhu operator on the odd power-sum ring is obtained from the classical neutral operator by a simple diagonal change of variables lacks an explicit verification or derivation. As this equivalence underpins all subsequent results on mixed products, including the scalar factor formulas and the special case identifications with Schur Q-functions, a detailed check that the transformation preserves the ring and the required commutation relations is necessary to secure the claims.

Authors: We agree that the manuscript asserts the equivalence via a diagonal change of variables without supplying an explicit derivation or verification that the map preserves the odd power-sum ring and the commutation relations. In the revised version we will insert a short preliminary subsection (or an expanded paragraph in the introduction) that carries out this check in detail: we will define the diagonal change explicitly on the generators, verify that it maps the odd power-sum ring to itself, and confirm that the commutation relations with the power-sum elements are preserved. This addition will directly support the subsequent scalar-factor computations and the identification with Schur Q-functions. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation chain is self-contained.

full rationale

The paper opens by noting that the modified GJZ operator arises from the classical neutral operator via a diagonal change of variables, then proceeds to compute the mixed-product scalar factor explicitly in exponential and Pochhammer form. The special-case identification with (u;t)_M/(-u;t)_M and its link to signed principal specializations of one-row Schur Q-functions are presented as computed results, not as re-expressions of the input change of variables. No fitted parameters are renamed as predictions, no self-citation chain is load-bearing for the central formulas, and no ansatz is smuggled in. The derivation therefore remains independent of its own outputs.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claims rest on the prior definition of the Greaves-Jing-Zhu operators and the odd power-sum ring, plus the observation that the modified operator arises via diagonal change of variables. No free parameters are fitted to data; t and s are deformation parameters. No new entities are postulated.

axioms (1)

domain assumption The modified Greaves--Jing--Zhu operator can be obtained from the classical neutral operator by a simple diagonal change of variables.
Stated explicitly in the abstract as the initial observation before studying mixed products.

pith-pipeline@v0.9.1-grok · 5728 in / 1640 out tokens · 54616 ms · 2026-06-29T03:36:22.785753+00:00 · methodology

discussion (0)

Forward citations

Cited by 2 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

A Two-Color Lift of the Shifted $t$-Schur Measure
math.PR 2026-07 unverdicted novelty 6.0

Introduces a two-color lift of the shifted Schur measure on pairs of partitions and derives its normalization, marginals, transition kernel, and independence of color volumes.
A Shifted $t$-Schur Weight from the Modified Odd Operator
math.CO 2026-07 unverdicted novelty 5.0

Defines shifted t-Schur weight via modified odd operator on strict partitions, derives normalization, Pfaffian correlation kernel, Fredholm Pfaffian for largest part, and size cumulants, with positive measure for t eq...

Reference graph

Works this paper leans on

7 extracted references · 3 canonical work pages · cited by 2 Pith papers · 1 internal anchor

[1]

Greaves, N

G. Greaves, N. Jing, and H. Zhu, Vertex operators, infinite wedge representations, and correlation functions of the𝑡-Schur measure, arXiv:2602.14190, 2026

work page arXiv 2026
[2]

V. N. Ivanov, Interpolation analogues of Schur𝑄-functions, J. Math. Sci. (N.Y.)131(2005), no. 3, 5495–5507

2005
[3]

2, 340–398

N.Jing, Vertexoperators,symmetricfunctions,andthespingroup Γ𝑛, J.Algebra138(1991), no. 2, 340–398

1991
[4]

A Modified Greaves--Jing--Zhu Operator and a Shifted $t$-Gessel Formula

S.-J. Lee, A modified Greaves–Jing–Zhu operator and a shifted 𝑡-Gessel formula, arXiv:2606.22058, 2026

work page internal anchor Pith review Pith/arXiv arXiv 2026
[5]

I. G. Macdonald,Symmetric Functions and Hall Polynomials, second ed., Oxford University Press, Oxford, 1995

1995
[6]

Rozhkovskaya, Linear transformations of vertex operator presentations of Hall– Littlewood polynomials, arXiv:2202.01544, 2022

N. Rozhkovskaya, Linear transformations of vertex operator presentations of Hall– Littlewood polynomials, arXiv:2202.01544, 2022

work page arXiv 2022
[7]

J. R. Stembridge, Shifted tableaux and the projective representations of symmetric groups, Adv. Math.74(1989), no. 1, 87–134. 11

1989

[1] [1]

Greaves, N

G. Greaves, N. Jing, and H. Zhu, Vertex operators, infinite wedge representations, and correlation functions of the𝑡-Schur measure, arXiv:2602.14190, 2026

work page arXiv 2026

[2] [2]

V. N. Ivanov, Interpolation analogues of Schur𝑄-functions, J. Math. Sci. (N.Y.)131(2005), no. 3, 5495–5507

2005

[3] [3]

2, 340–398

N.Jing, Vertexoperators,symmetricfunctions,andthespingroup Γ𝑛, J.Algebra138(1991), no. 2, 340–398

1991

[4] [4]

A Modified Greaves--Jing--Zhu Operator and a Shifted $t$-Gessel Formula

S.-J. Lee, A modified Greaves–Jing–Zhu operator and a shifted 𝑡-Gessel formula, arXiv:2606.22058, 2026

work page internal anchor Pith review Pith/arXiv arXiv 2026

[5] [5]

I. G. Macdonald,Symmetric Functions and Hall Polynomials, second ed., Oxford University Press, Oxford, 1995

1995

[6] [6]

Rozhkovskaya, Linear transformations of vertex operator presentations of Hall– Littlewood polynomials, arXiv:2202.01544, 2022

N. Rozhkovskaya, Linear transformations of vertex operator presentations of Hall– Littlewood polynomials, arXiv:2202.01544, 2022

work page arXiv 2022

[7] [7]

J. R. Stembridge, Shifted tableaux and the projective representations of symmetric groups, Adv. Math.74(1989), no. 1, 87–134. 11

1989