A common algebraic language
The preprint’s central construction links stable symmetric series, shifted symmetric functions, differential operators and Jack deformations in one algebraic framework. The stable series form a commutative algebra: when products are expanded back into the same basis, their structure constants are nonnegative integers and do not depend on the ambient degree. The same algebra is identified with shifted symmetric functions through the eigenvalues attached to stable series. The work is an arXiv version 1 preprint dated 26 August 2026.
Stable is a technical way of keeping the algebra’s rules independent of a chosen ambient size. For a homogeneous symmetric function f of degree m, the associated stable series is defined using m!f divided by (1 − p1)^(m+1).
The series also have an operator form. Each homogeneous f is associated with a differential operator—an algebraic rule built from multiplication and differentiation—and that operator lies in the image of U(W1+∞). This places the construction inside a represented enveloping algebra rather than leaving it as a collection of unrelated formulas.
A Jack-polynomial extension
The Jack extension adds the parameter α to the same picture. A stable Jack lift of a partition μ of size m acts diagonally on the Jack basis Pλ′, with eigenvalue α^m P#μ(λ; α). The stable lifts are identified with shifted Jack functions. In practical terms, the operator’s action is summarized by a prescribed eigenvalue for each basis element.
Those shifted Jack functions also satisfy a Pieri relation. It combines a diagonal mP#μ term with additional terms corresponding to adding one box, giving a rule for how the functions change as the underlying partition is enlarged.
One exact case, two conjectural ones
One important boundary is between what the preprint derives exactly and what it only proposes. The degree-two deformed cut-and-join operator is exact and contains the usual join–cut terms plus a correction proportional to α − 1. That correction vanishes at α = 1, the value used for the complex specialization.
The parameter dependence is visible in one multiplication example. In the paper’s notation, pb2 multiplied by itself expands as pb22 plus 4pb3, 2αpb11 and 2(α − 1)pb2. The final contribution therefore disappears when α equals 1.
At degrees three and four, however, the status changes. The general formulas for Δ3(α) and Δ4(α) are labeled computer-assisted conjectures, and no complete term-by-term proof is supplied.
Checks in three Gaussian models
To test those proposals against concrete combinatorics, the paper uses Gaussian matrix integrals and exhaustive Wick enumeration for the real and quaternionic cases. Wick enumeration here means listing the allowed pairings of factors in the integral. For r = 2, 3 and 4, the real, complex and quaternionic models are reported to match the corresponding operators at α = 2, 1 and 1/2, respectively.
The counts behind those checks are 5!! = 15 perfect matchings in degree three and 7!! = 105 in degree four. These are finite, exhaustive enumerations of the matchings used in the two reported degrees.
Mechanical comparison found no discrepancy when it checked 49 degree-three correction-operator types and 154 degree-four types. Because the comparison is coefficientwise, it tests the individual terms in the operator expansions, not just a single final number. The reported agreement is detailed within these cases, while remaining limited to the stated specializations.
Evidence with a clear boundary
That qualification is central. Agreement at α = 2 and α = 1/2 is evidence for the real and quaternionic specializations reported, but it does not establish the formulas for an indeterminate α. The checks also cover only the reported low degrees, so the proposed degree-three and degree-four operators still require complete proofs.
The paper’s acknowledgments disclose ChatGPT assistance with manuscript organization, English editing and computer-algebra checking, while assigning responsibility for the mathematics to the author.
Paper data and sources
Original title: Stable Symmetric Series, Differential Operators, and Jack Deformations
Authors: Jean-Yves Thibon
Journal/Repository: arXiv
Status: Preprint, not yet peer-reviewed
First online: 2026-08-26
DOI: Not available
Original paper · Full text