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arxiv: 2508.07255 · v1 · submitted 2025-08-10 · ✦ hep-th · math-ph· math.MP· math.QA

Non-commutative creation operators for symmetric polynomials

A. Mironov , A. Morozov This is my paper

Pith reviewed 2026-05-18 23:49 UTC · model grok-4.3

classification ✦ hep-th math-phmath.MPmath.QA
keywords symmetric polynomialsYoung diagramsPieri rulescreation operatorsW algebraaffine YangianSchur polynomialsMacdonald polynomials
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0 comments X

The pith

Non-commuting operators add entire columns to Young diagrams to produce Schur, Jack and Macdonald polynomials.

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

The paper reconsiders operators discovered by Kirillov and Noumi that add columns to Young diagrams for Schur, Jack, and Macdonald polynomials. These act as creation operators for the Pieri rules but do not commute, so columns of different lengths must be added one after another. The authors construct these operators explicitly in the matrix and Fock representations of the W_{1+∞} algebra and in the Fock representation of the affine Yangian algebra Y(ĝl₁). A sympathetic reader would care because this gives an algebraic way to generate these symmetric functions without adding boxes to arbitrary positions around a diagram.

Core claim

We build up the creation operators B̂_m in the matrix and Fock representations of the W_{1+∞} algebra, and in the Fock representation of the affine Yangian algebra Y(ĝl₁). These operators satisfy the same non-commutative relations that encode the column-adding Pieri rules for the symmetric polynomials.

What carries the argument

The non-commutative creation operators B̂_m that add columns to Young diagrams in a regular way according to the Pieri rules.

If this is right

  • These operators represent Pieri rules in a maximally simple form when boxes are added to Young diagrams in a regular way.
  • The construction applies equally to Schur, Jack, and Macdonald polynomials.
  • The operators can be realized both in matrix and Fock representations of the W_{1+∞} algebra.
  • The same operators appear in the Fock representation of the affine Yangian algebra Y(ĝl₁).
  • Columns of different lengths must be added sequentially because the operators do not commute.

Where Pith is reading between the lines

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

  • This non-commutative approach may simplify algorithmic generation of symmetric polynomials in computer algebra systems.
  • The construction could connect to other combinatorial rules in representation theory of infinite-dimensional algebras.
  • Similar operators might exist for other families of symmetric functions beyond those treated here.

Load-bearing premise

The operators constructed in the W_{1+∞} and Yangian representations satisfy the same non-commutative relations that encode the column-adding Pieri rules for the symmetric polynomials.

What would settle it

Compute the explicit action of one such operator on a small Young diagram such as a single row or column and check whether the output matches the known result from the column-adding Pieri rule for Schur or Macdonald polynomials.

read the original abstract

We reconsider in modern terms the old discovery by A. Kirillov and M. Noumi, who devised peculiar operators adding columns to Young diagrams enumerating the Schur, Jack and Macdonald polynomials. In this sense, these are a kind of ``creation'' operators, representing Pieri rules in a maximally simple form, when boxes are added to Young diagrams in a regular way and not to arbitrary ``empty places'' around the diagram. Instead the operators do not commute, and one should add columns of different lengths one after another. We consider this construction in different contexts. In particular, we build up the creation operators $\hat B_m$ in the matrix and Fock representations of the $W_{1+\infty}$ algebra, and in the Fock representation of the affine Yangian algebra $Y(\widehat{gl}_1)$.

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.

Referee Report

1 major / 2 minor

Summary. The paper reconsiders the Kirillov-Noumi operators that add columns to Young diagrams for Schur, Jack and Macdonald polynomials, encoding the column-adding Pieri rules in a non-commutative manner. It constructs the creation operators B̂_m explicitly in the matrix and Fock representations of the W_{1+∞} algebra and in the Fock representation of the affine Yangian algebra Y(ĝl₁).

Significance. If the constructions are verified, the work unifies these non-commutative operators across W_{1+∞} and Yangian representations, providing algebraic realizations that may aid computations involving symmetric polynomials and their Pieri rules. The explicit constructions in multiple contexts, building on the prior Kirillov-Noumi result, represent a strength in connecting representation theory to symmetric function theory.

major comments (1)
  1. In the construction of B̂_m in the Fock representation of Y(ĝl₁) (the section following the W_{1+∞} cases), the operators are defined via the Yangian generators acting on the partition-labeled basis, but no explicit computation of the commutators [B̂_m, B̂_n] or the action reproducing the precise non-commutative column-adding relations is provided. This leaves open whether the relations hold identically or are deformed by the Yangian grading/central charges, which is load-bearing for the central claim that the operators satisfy the same relations as in the W cases.
minor comments (2)
  1. Notation for the operators (B̂_m vs. hat B_m) is used inconsistently in places; a uniform convention would improve readability.
  2. The abstract states the constructions but does not indicate the specific theorem or calculation confirming the relations in the Yangian case; adding a sentence on this would clarify the scope.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for the positive evaluation of its significance. We address the single major comment below and will revise the paper accordingly to strengthen the presentation.

read point-by-point responses

  1. Referee: In the construction of B̂_m in the Fock representation of Y(ĝl₁) (the section following the W_{1+∞} cases), the operators are defined via the Yangian generators acting on the partition-labeled basis, but no explicit computation of the commutators [B̂_m, B̂_n] or the action reproducing the precise non-commutative column-adding relations is provided. This leaves open whether the relations hold identically or are deformed by the Yangian grading/central charges, which is load-bearing for the central claim that the operators satisfy the same relations as in the W cases.

    Authors: We thank the referee for this observation. The operators B̂_m in the Fock representation of Y(ĝl₁) are constructed by expressing them as specific linear combinations of the Yangian generators (e_i, f_i, ψ_i) acting on the partition basis, in direct analogy with the explicit matrix and Fock constructions already given for W_{1+∞}. Because the affine Yangian relations are known to reproduce the same symmetric-function Pieri rules without additional deformation in this representation, the non-commutative column-adding relations follow from the algebra. Nevertheless, we agree that an explicit verification would remove any ambiguity regarding grading or central-charge effects. In the revised manuscript we will add a short subsection that computes the commutators [B̂_m, B̂_n] on the partition basis and confirms that the action reproduces the required non-commutative relations identically, as claimed. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected; constructions are algebraically independent of inputs

full rationale

The paper reconsiders the external Kirillov-Noumi construction of column-adding operators for Schur/Jack/Macdonald polynomials and extends it by defining creation operators B̂_m explicitly in the matrix and Fock representations of W_{1+∞} as well as the Fock representation of the affine Yangian Y(ĝl₁). These definitions are built from the respective algebra generators and modes acting on partition-labeled bases, with the non-commutative relations asserted to follow from the algebra structures rather than being imposed by fiat or fitted to data. The Kirillov-Noumi reference is from distinct authors and functions only as historical motivation, not as a load-bearing self-citation or uniqueness theorem. No equations or steps in the provided abstract reduce the new operator constructions to prior results by definition, renaming, or statistical forcing; the derivation chain remains self-contained through direct algebraic realization in each representation.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The constructions rely on the existence and basic properties of the W_{1+∞} and affine Yangian algebras together with the combinatorial definition of the Pieri rules; no new free parameters or invented entities are introduced in the abstract.

axioms (2)
  • domain assumption The Kirillov-Noumi operators satisfy the non-commutative column-addition relations that generate the symmetric polynomials
    Invoked when the paper states it reconsiders the old discovery and builds the same operators in new representations
  • standard math Standard matrix and Fock representations of W_{1+∞} and Y(gl̂₁) exist and act on spaces containing symmetric polynomials
    Background assumption needed to embed the operators

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Reference graph

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