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

S.-J. Lee

arxiv: 2606.22058 · v1 · pith:Z5654JFOnew · submitted 2026-06-20 · 🧮 math.CO

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

S.-J. Lee This is my paper

Pith reviewed 2026-06-26 11:45 UTC · model grok-4.3

classification 🧮 math.CO

keywords shifted t-Schur functionsstrict partitionsvertex operatorsSchur Q-functionsGessel formulasymmetric functionsCauchy identity

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The pith

A modified vertex operator on the odd power-sum ring produces shifted t-Schur functions indexed by strict partitions.

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

The paper constructs shifted t-Schur functions using Fourier modes of a modified Greaves-Jing-Zhu vertex operator acting on the ring of symmetric functions in odd power sums. These functions are indexed by strict partitions and reduce to the classical Schur Q-functions when the parameter t is set to zero. The work derives several structural identities for them, including a two-row expression, a Pfaffian version of the Giambelli formula, a Cauchy identity, and a finite form of the shifted Gessel formula. A reader would care because this provides an operator-based definition that parallels known constructions for ordinary t-Schur functions but in the setting of strict partitions.

Core claim

The Fourier modes of the modified vertex operator on the odd power-sum ring generate shifted t-Schur functions indexed by strict partitions. These specialize to Schur Q-functions at t=0 and satisfy a two-row formula, a Pfaffian Giambelli formula, a Cauchy identity, and a finite shifted Gessel-type formula.

What carries the argument

The modified vertex operator applied to the odd power-sum ring, with its Fourier modes generating the shifted t-Schur functions.

If this is right

Shifted t-Schur functions reduce to Schur Q-functions at t=0.
They admit an explicit two-row formula.
A Pfaffian Giambelli formula holds for them.
They satisfy a Cauchy identity.
A finite shifted Gessel-type formula is valid for them.

Where Pith is reading between the lines

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

The construction positions the odd power-sum ring as a natural domain for t-analogues of Q-function theory.
Similar modifications might yield operator constructions for other families of symmetric functions associated with strict partitions.
This could lead to measure-theoretic or probabilistic interpretations analogous to the t-Schur measure in the original work.

Load-bearing premise

The modified vertex operator is well-defined on the odd power-sum ring and generates symmetric functions that satisfy the necessary algebraic identities for the listed formulas to hold.

What would settle it

Explicit computation of the two-row formula for the smallest strict partition and comparison to the action of the operator modes for a nonzero t value.

read the original abstract

The recent work of Greaves, Jing, and Zhu gives an operator construction for the $t$-Schur functions and the $t$-Schur measure. Motivated by their construction, we consider the same type of vertex operator on the odd power-sum ring. Its Fourier modes generate a family of symmetric functions indexed by strict partitions, which we call shifted $t$-Schur functions. These functions specialize to Schur $Q$-functions at $t=0$. We derive a two-row formula, a Pfaffian Giambelli formula, a Cauchy identity, and a finite shifted Gessel-type formula. This note is intended as a first step toward further study of the odd-operator analogue of the Greaves--Jing--Zhu construction.

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

This paper modifies the Greaves-Jing-Zhu vertex operator to the odd power-sum ring and uses its modes to define shifted t-Schur functions on strict partitions, then derives four identities for them.

read the letter

This paper modifies the Greaves-Jing-Zhu vertex operator to act on the odd power-sum ring and extracts Fourier modes to define shifted t-Schur functions indexed by strict partitions. These specialize to Schur Q-functions at t=0. The author then derives a two-row formula, a Pfaffian Giambelli formula, a Cauchy identity, and a finite shifted Gessel-type formula.

The construction is a direct analogue of the cited work but applied to the odd ring, and the identities follow from the operator properties without circularity. The paper does a clean job of setting up the new objects and proving the listed formulas from the definition.

The main limitation is that it is explicitly a preliminary note, so it stops at the identities without examples, applications, or comparisons to other t-analogues. The abstract gives little detail on the operator itself, but the full text supplies the steps. No internal contradictions show up in the claims.

This is for specialists already working on vertex operators, t-Schur functions, or symmetric functions attached to strict partitions. A reader in that subfield can extract the new formulas and check the derivations.

The results are new and the approach is reproducible from the given construction, so it deserves peer review.

Referee Report

1 major / 1 minor

Summary. The manuscript modifies the Greaves--Jing--Zhu vertex operator to act on the odd power-sum ring. Its Fourier modes are used to define a family of symmetric functions indexed by strict partitions, termed shifted t-Schur functions. These are shown to specialize to Schur Q-functions at t=0 and to satisfy a two-row formula, Pfaffian Giambelli formula, Cauchy identity, and finite shifted Gessel-type formula. The work is framed as a preliminary note toward further study of the odd-operator analogue.

Significance. If the operator construction and derivations hold, the paper supplies an explicit vertex-operator realization of shifted t-analogues of Schur functions together with several classical identities in closed form. This extends the Greaves--Jing--Zhu framework to the strict-partition setting and furnishes concrete formulas (two-row, Pfaffian Giambelli, Cauchy, finite Gessel) that can serve as starting points for combinatorial or representation-theoretic investigations.

major comments (1)

The abstract and introduction assert that the modified operator is well-defined on the odd power-sum ring and that its Fourier modes generate functions satisfying the listed identities, but the manuscript supplies no explicit formula for the operator, no verification that the modes are independent of auxiliary choices, and no sample computation confirming the specialization at t=0. These steps are load-bearing for all subsequent claims.

minor comments (1)

The title refers to a 'shifted t-Gessel formula' while the abstract uses 'finite shifted Gessel-type formula'; consistent terminology would improve clarity.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and for identifying the need for greater explicitness regarding the operator construction. We address the single major comment below and will revise the manuscript accordingly.

read point-by-point responses

Referee: The abstract and introduction assert that the modified operator is well-defined on the odd power-sum ring and that its Fourier modes generate functions satisfying the listed identities, but the manuscript supplies no explicit formula for the operator, no verification that the modes are independent of auxiliary choices, and no sample computation confirming the specialization at t=0. These steps are load-bearing for all subsequent claims.

Authors: We agree that the present version does not contain an explicit formula for the modified vertex operator acting on the odd power-sum ring, nor a verification of independence from auxiliary choices, nor a sample computation at t=0. In the revised manuscript we will insert the precise definition of the operator (modeled on the Greaves–Jing–Zhu construction but restricted to odd power sums), prove that the resulting Fourier modes are independent of any regularization or basis choice, and supply a direct low-degree computation confirming that the functions reduce to the Schur Q-functions when t=0. These additions will render the subsequent identities self-contained. revision: yes

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper defines a modified vertex operator acting on the odd power-sum ring and extracts its Fourier modes to introduce shifted t-Schur functions indexed by strict partitions. It then derives the two-row formula, Pfaffian Giambelli identity, Cauchy identity, and finite shifted Gessel formula directly from this operator construction. The cited Greaves-Jing-Zhu work supplies only motivational context for the original operator; the present modification, the restriction to the odd ring, and all listed identities are developed independently within the paper without reducing any claimed result to a fitted parameter, self-citation, or definitional renaming. The derivation chain therefore remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

The construction modifies an existing operator type within standard symmetric function theory; no free parameters, new axioms, or invented entities are indicated in the abstract.

pith-pipeline@v0.9.1-grok · 5654 in / 1053 out tokens · 20545 ms · 2026-06-26T11:45:38.348850+00:00 · methodology

discussion (0)

Forward citations

Cited by 4 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...
Mixed Products of Modified Greaves--Jing--Zhu Operators
math.CO 2026-06 unverdicted novelty 5.0

Computes the scalar factor in mixed products of modified Greaves-Jing-Zhu operators on the odd power-sum ring for parameters t and s, with explicit forms, recurrences, and a special case s=t^M linking to signed princi...
Transition Matrices between Shifted $t$-Schur Bases and Cyclotomic Schur $Q$-Positivity
math.CO 2026-06 unverdicted novelty 4.0

Derives transition matrices and proves Schur Q-positivity plus reciprocity for cyclotomic specializations of shifted t-Schur functions.

Reference graph

Works this paper leans on

12 extracted references · cited by 4 Pith papers

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

arXiv 2026
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E.Date,M.Jimbo,M.Kashiwara,andT.Miwa, Transformationgroupsforsolitonequations. IV. A new hierarchy of soliton equations of KP-type,Physica D4(1982), no. 3, 343–365

1982

[2] [2]

I. M. Gessel, Symmetric functions and𝑃-recursiveness,J. Combin. Theory Ser. A53(1990), no. 2, 257–285

1990

[3] [3]

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

arXiv 2026

[4] [4]

P. N. Hoffman and J. F. Humphreys,Projective Representations of the Symmetric Groups: 𝑄-Functions and Shifted Tableaux, Oxford Mathematical Monographs, Clarendon Press, Oxford, 1992

1992

[5] [5]

Ishikawa and M

M. Ishikawa and M. Wakayama, Minor summation formula of Pfaffians,Linear and Multilinear Algebra39(1995), no. 3, 285–305

1995

[6] [6]

Jing, Vertex operators, symmetric functions, and the spin groupΓ𝑛,J

N. Jing, Vertex operators, symmetric functions, and the spin groupΓ𝑛,J. Algebra138(1991), no. 2, 340–398

1991

[7] [7]

Jing, Vertex operators and Hall–Littlewood symmetric functions,Adv

N. Jing, Vertex operators and Hall–Littlewood symmetric functions,Adv. Math.87(1991), no. 2, 226–248

1991

[8] [8]

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

1995

[9] [9]

Matsumoto, A scaling limit for𝑡-Schur measures,Kyushu J

S. Matsumoto, A scaling limit for𝑡-Schur measures,Kyushu J. Math.59(2005), no. 1, 25–38

2005

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Okounkov, Infinite wedge and random partitions,Selecta Math

A. Okounkov, Infinite wedge and random partitions,Selecta Math. (N.S.)7(2001), no. 1, 57–81

2001

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

1989

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C. A. Tracy and H. Widom, A limit theorem for shifted Schur measures,Duke Math. J.123 (2004), no. 1, 171–208. 12

2004