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arxiv: 2606.22585 · v1 · pith:K3LMQIVR · submitted 2026-06-21 · math.CO

Normal ordering in the (p,q)-deformed generalized Weyl algebra. I: Algebraic Framework and Combinatorial Identities

Toufik Mansour , Lahcen Oussi , Matthias Schork This is my paper

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keywords normal ordering(p,q)-deformed Weyl algebraYoung diagramscombinatorial identitiesgeneralized Stirling numberscommutation relationsWeyl algebra
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The pith

The (p,q)-deformed generalized Weyl algebra admits normal ordering of arbitrary words encoded by Young diagrams, yielding combinatorial identities that specialize to generalized Stirling numbers.

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

The paper establishes a procedure to reorder arbitrary products of the generators X, Y and Z_p satisfying the given commutation relations into normal form. Young diagrams track each commutation step arising from XY minus q YX equals h Y to the s times Z_p together with the p-scaling rules for Z_p. The coefficients that result after reordering satisfy explicit combinatorial identities. These identities reduce to known relations for generalized Stirling numbers under parameter specialization. A sympathetic reader would care because the method supplies a systematic, visual bookkeeping device for calculations that otherwise grow combinatorially complex.

Core claim

The (p,q)-deformed generalized Weyl algebra generated by X, Y, Z_p with relations XY - q YX = h Y^s Z_p, X Z_p = p Z_p X and Z_p Y = p Y Z_p allows any word to be brought to normal order by a procedure whose steps are encoded by Young diagrams; the resulting coefficients obey combinatorial identities that specialize to generalized Stirling numbers.

What carries the argument

Young diagrams that serve as a bookkeeping device to encode the reordering steps arising from the commutation relations.

If this is right

  • Explicit normal-ordered expressions exist for arbitrary words via summation over the corresponding Young diagrams.
  • Special cases of the parameters recover the defining relations of generalized Stirling numbers.
  • The method applies directly to several families of special words treated in the paper.
  • The algebraic relations translate into a family of combinatorial summation identities.

Where Pith is reading between the lines

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

  • The diagram method could be used to evaluate expectation values of operator products in physical models whose commutation rules match the given deformations.
  • Similar visual encodings might apply to normal-ordering problems in other multiparameter algebras.
  • Direct enumeration of small words offers an immediate numerical check of the derived identities.

Load-bearing premise

Young diagrams provide a complete bookkeeping device that captures every reordering sequence generated by the commutation relations without missing or double-counting terms.

What would settle it

A concrete word in X, Y, Z_p whose expansion after normal ordering cannot be reproduced by any sum over the Young diagrams given in the paper, or a numerical mismatch with a known generalized Stirling number identity after parameter specialization.

Figures

Figures reproduced from arXiv: 2606.22585 by Lahcen Oussi, Matthias Schork, Toufik Mansour.

Figure 1
Figure 1. Figure 1: The Young diagrams in I4. Definition 2.4. Let fpyq “ řs k“0 αky k be a polynomial. For a partition λ with d ě 1 parts, we define gdpλ; Y q ” gdpλ1, λ2, . . . , λd; Y q :“ ÿs i1,i2,...,id“0 ´ q řd j“1 ijλj ź d j“1 αij d ź´1 j“1 ri1 ` ¨ ¨ ¨ ` ij ` 1 ´ jsp,q¯ Y i1`i2`¨¨¨`id`1´d . Moreover, for d “ 0, we set g0pH; Y q :“ Y. Note that gdpλ; Y q depends on the polynomial f, but to simplify the notation, we do no… view at source ↗
read the original abstract

The $(p,q)$-deformed generalized Weyl algebra is generated by variables $X, Y$ and $Z_p$ which satisfy the commutation relations $XY-qYX=h Y^sZ_{p}, XZ_p=pZ_pX$, and $Z_pY=pYZ_p$, with $s\in \mathbb{N}_0$. We investigate the problem of normal ordering arbitrary words in these letters with the help of Young diagrams, and we treat certain special cases explicitly. In particular, the connection to generalized Stirling numbers is considered in detail.

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

2 major / 1 minor

Summary. The paper defines the (p,q)-deformed generalized Weyl algebra generated by X, Y, Z_p subject to the relations XY - q YX = h Y^s Z_p, X Z_p = p Z_p X and Z_p Y = p Y Z_p (s a non-negative integer). It claims that normal ordering of arbitrary words can be encoded via Young diagrams, yielding explicit combinatorial identities for the coefficients; special cases are treated in detail and shown to recover generalized Stirling numbers.

Significance. If the Young-diagram encoding is rigorously shown to be exhaustive for arbitrary monomials, the work supplies a combinatorial calculus for normal ordering in an inhomogeneous deformed algebra. This would give parameter-dependent generalizations of Stirling numbers with explicit diagram-based formulas, a strength that is already visible in the special-case treatments mentioned in the abstract.

major comments (2)
  1. [Abstract] Abstract (paragraph 2) and the statement of the central claim: the assertion that Young diagrams furnish a complete bookkeeping device for normal ordering of arbitrary words rests on the unproven assumption that the set of diagrams remains closed under the inhomogeneous commutation move XY - q YX = h Y^s Z_p when s > 1 or when multiple Z_p factors are present. No inductive argument or exhaustive check for general words is indicated in the provided text; without it the claimed combinatorial identities may miss contributions.
  2. [Abstract] The commutation relations are inhomogeneous. For the diagram encoding to enumerate all terms, every reordering path must produce only diagrams already included in the posited family. The abstract gives no explicit verification that this closure holds once the right-hand side contains Y^s with s > 1; this is load-bearing for the claim that the procedure works for arbitrary words.
minor comments (1)
  1. The abstract states that certain special cases are treated explicitly; the manuscript should clarify in the introduction which values of s and which monomial degrees are covered by the explicit identities versus which remain conjectural.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. The major comments correctly identify that the central claim regarding the Young-diagram encoding for arbitrary words requires explicit justification of closure under the inhomogeneous relations. We address each point below and will revise the manuscript to include the necessary argument.

read point-by-point responses

  1. Referee: [Abstract] Abstract (paragraph 2) and the statement of the central claim: the assertion that Young diagrams furnish a complete bookkeeping device for normal ordering of arbitrary words rests on the unproven assumption that the set of diagrams remains closed under the inhomogeneous commutation move XY - q YX = h Y^s Z_p when s > 1 or when multiple Z_p factors are present. No inductive argument or exhaustive check for general words is indicated in the provided text; without it the claimed combinatorial identities may miss contributions.

    Authors: We agree that the current manuscript does not contain an explicit inductive argument establishing closure of the diagram family for general s > 1 and multiple Z_p factors. While the framework and special cases are developed in detail, the general claim would be strengthened by such a verification. In the revised version we will add a new subsection providing an inductive proof that every term generated by the relation XY - q YX = h Y^s Z_p remains within the posited family of diagrams. revision: yes

  2. Referee: [Abstract] The commutation relations are inhomogeneous. For the diagram encoding to enumerate all terms, every reordering path must produce only diagrams already included in the posited family. The abstract gives no explicit verification that this closure holds once the right-hand side contains Y^s with s > 1; this is load-bearing for the claim that the procedure works for arbitrary words.

    Authors: We concur that explicit verification of closure is essential for the claim to hold for arbitrary words. The revised manuscript will incorporate an inductive argument demonstrating that repeated applications of the inhomogeneous relation, including cases with s > 1, produce only diagrams already accounted for in the encoding. This will confirm that no contributions are missed and support the combinatorial identities. revision: yes

Circularity Check

0 steps flagged

No circularity; derivation builds directly from commutation relations

full rationale

The paper introduces the algebra via explicit commutation relations XY - q YX = h Y^s Z_p, X Z_p = p Z_p X, Z_p Y = p Y Z_p and then constructs normal-ordering rules using Young diagrams as a bookkeeping device, deriving combinatorial identities (including specializations to generalized Stirling numbers) as consequences. No self-definitional loops, fitted inputs renamed as predictions, or load-bearing self-citations appear in the abstract or described framework. The approach is presented as an algebraic investigation that treats special cases explicitly from the given relations, remaining self-contained without reducing any claimed result to its own inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on the three given commutation relations (standard in deformed algebra literature) and on the unproven assertion that Young diagrams capture all reordering coefficients. No free parameters are fitted inside the paper; p, q, s are part of the algebra definition. No new entities are postulated.

axioms (2)
  • domain assumption The generators satisfy XY - q YX = h Y^s Z_p, X Z_p = p Z_p X, Z_p Y = p Y Z_p for s in non-negative integers.
    These relations are stated in the abstract as the definition of the algebra; they are taken as given rather than derived.
  • ad hoc to paper Young diagrams provide a complete and faithful encoding of the normal-ordering coefficients.
    The abstract asserts that normal ordering is investigated 'with the help of Young diagrams' without supplying a proof that the encoding is bijective or exhaustive.

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Forward citations

Cited by 1 Pith paper

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

  1. Normal ordering in the $(p,q)$-deformed generalized Weyl algebra. II: Interpretation in terms of rook placements

    math.CO 2026-07 unverdicted novelty 5.0

    Normal ordering in the (p,q)-deformed generalized Weyl algebra yields (p,q)-deformed s-rook numbers that give combinatorial interpretations of (p,q)-generalized Stirling numbers on staircase boards.

Reference graph

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