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arxiv: 2606.02395 · v1 · pith:3NAV4VKPnew · submitted 2026-06-01 · 🧮 math.CO

Shape changing identities for permuted-basement nonsymmetric Macdonald polynomials

Guilherme Zeus Dantas e Moura , Olya Mandelshtam This is my paper

Pith reviewed 2026-06-28 13:47 UTC · model grok-4.3

classification 🧮 math.CO
keywords permuted-basement Macdonald polynomialsnonsymmetric Macdonald polynomialsnon-attacking fillingsbijective proofsshape changing identitiesbasement permutationscombinatorial identitiesstraightening rules
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The pith

Bijective proofs establish identities for permuted-basement nonsymmetric Macdonald polynomials under adjacent basement and shape swaps.

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

The paper shows that identities relating the polynomials E_α^σ, E_α^{σ s_i}, E_{s_i α}^σ, and E_{s_i α}^{σ s_i} admit direct bijective proofs. These identities arise from two explicit operations on the non-attacking fillings that generate the polynomials: swapping adjacent basement entries or swapping adjacent parts of the shape. A reader would care because the bijections turn algebraic relations into combinatorial correspondences that hold for any fixed basement permutation σ. The proofs generalize an earlier basement-swap result and supply a straightening rule that expands one family of polynomials in another.

Core claim

Permuted-basement Macdonald polynomials admit combinatorial formulas as generating functions over composition-shaped non-attacking fillings. The paper supplies explicit bijections between the relevant sets of fillings that prove the four-term identities obtained by applying an adjacent transposition either to the basement permutation or to the shape composition. These bijections preserve the weight of each filling and therefore establish the algebraic identities directly.

What carries the argument

Two combinatorial operations on non-attacking fillings: (1) swapping adjacent entries in the basement and (2) swapping adjacent parts in the shape.

If this is right

  • The basement-swap bijection generalizes Alexandersson's 2019 result to arbitrary fixed σ.
  • The shape-swap bijection yields a straightening rule that expands any E_α^σ in the basis of polynomials with modified shapes.
  • Repeated application of the two operations relates any two permuted-basement versions whose basements and shapes differ by a sequence of adjacent transpositions.
  • The identities hold in the polynomial ring Q(q,t)[x] for each fixed basement permutation σ.

Where Pith is reading between the lines

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

  • The same bijections may supply recursive algorithms for computing the polynomials by reducing to simpler basements or shapes.
  • The straightening rule could be used to obtain positivity or other coefficient properties by induction on the number of shape swaps.
  • The methods might adapt to other families of polynomials that possess similar non-attacking filling models.

Load-bearing premise

The combinatorial formulas that express the polynomials as generating functions over non-attacking fillings are valid and the two described operations on fillings are weight-preserving bijections between the relevant sets.

What would settle it

A single explicit non-attacking filling of a given shape and basement for which one of the two swap operations produces a filling whose total weight differs from the weight predicted by the corresponding polynomial identity.

Figures

Figures reproduced from arXiv: 2606.02395 by Guilherme Zeus Dantas e Moura, Olya Mandelshtam.

Figure 1
Figure 1. Figure 1: A non-attacking augmented filling T of shape α = 2201, basement σ = [3, 1, 2, 4], and content(T) = 1202. The descents in T are shaded, with legs leg(1, 2) = leg(2, 2) = 0 and leg(2, 1) = 1. Thus, maj(T) = 1 + 1 + 2 = 4. We define the inversion and coinversion statistics, following [HHL08, Section 3]. Given a, b ∈ [n], let χab = 1 if a > b, and χab = 0 otherwise. Given a, b, c ∈ [n], define χabc = χab + χbc… view at source ↗
Figure 2
Figure 2. Figure 2: Configurations in which (u, v, w) forms a triple. A triple (u, v, w) is an inversion (resp. coinversion) triple if χT(u)T(v)T(w) = 1 (resp. 0), and inv(T) (resp. coinv(T)) denotes the number of such triples in T. Theorem 2.1 ([Fer11, Definition 4.4.2]). The permuted-basement Macdonald polynomial is given by E σ α (x; q, t) = X T ∈NAF(α,σ) wt(T), where wt(T) = x T q maj(T) t coinv(T) Y u∈dg(α) T(u)̸=T(d(u))… view at source ↗
Figure 3
Figure 3. Figure 3: For α = 2201, σ = [3, 1, 2, 4], β = 1202, we show Ti ∈ NAF(α, σ, β) and Uj ∈ NAF(α, σs1, β) and their q, t-weights. The label of an arrow from A to B is prob1 (A, B). The fact that the maps form a probabilistic bijection can be checked by computing that, for each T ∈ {T1, T2, T3} and each U ∈ {U1, U2}, we have prob1 (T, U)wtq,t(T) = prob1 (U, T)wtq,t(U). Moreover, using the fact that there exists a probabi… view at source ↗
Figure 4
Figure 4. Figure 4: Columns i and i+1 of a filling T with three blocks. . . . e f a b c d g h [PITH_FULL_IMAGE:figures/full_fig_p007_4.png] view at source ↗
Figure 6
Figure 6. Figure 6: Columns i, i + 1 of the fillings φ(T) and ψ(T) derived from T in [PITH_FULL_IMAGE:figures/full_fig_p008_6.png] view at source ↗
read the original abstract

Permuted-basement Macdonald polynomials $E_\alpha^\sigma(\mathbf{x};q,t)$ are nonsymmetric generalizations of symmetric Macdonald polynomials that form a basis for the polynomial ring $\mathbb{Q}(q,t)[\mathbf{x}]$ for each fixed $\sigma$. There are combinatorial formulas for them as generating functions over composition-shaped non-attacking fillings. In this extended abstract, we bijectively prove identities for the relationship between $E_\alpha^\sigma$, $E_\alpha^{\sigma s_i}$, $E_{s_i\alpha}^\sigma$, and $E_{s_i\alpha}^{\sigma s_i}$. These identities correspond to two combinatorial operations on non-attacking fillings: (1) swapping adjacent entries in the basement, generalizing a result of Alexandersson (2019), and (2) swapping adjacent parts in the shape, which yields a straightening rule for expanding $E_\alpha^\sigma$ in the polynomials $\{E_{s_i\alpha}^\tau\}_\tau$.

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

0 major / 1 minor

Summary. The manuscript claims to establish bijective proofs for identities relating the four families of permuted-basement nonsymmetric Macdonald polynomials E_α^σ, E_α^{σ s_i}, E_{s_i α}^σ and E_{s_i α}^{σ s_i}. The proofs are realized by two explicit operations on non-attacking fillings of composition shapes: adjacent basement swaps (generalizing Alexandersson 2019) and adjacent shape-part swaps (providing a straightening rule), both asserted to preserve the non-attacking condition and the weight.

Significance. If the bijections are correctly verified, the work supplies combinatorial maps that directly relate the generating functions for these polynomial bases, extending prior combinatorial formulas for nonsymmetric Macdonald polynomials and furnishing an explicit straightening rule. The explicit case analysis on the operations constitutes a concrete combinatorial contribution.

minor comments (1)
  1. The manuscript is presented as an extended abstract; for journal publication the case analysis establishing that the two operations are weight-preserving bijections on the relevant sets of fillings should be expanded with all verification details included in the main text.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary, significance assessment, and recommendation of minor revision. No specific major comments were raised in the report.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper's central results are explicit bijective proofs establishing identities among four families of permuted-basement nonsymmetric Macdonald polynomials via two weight-preserving operations on non-attacking fillings (basement swap and adjacent shape-part swap). These bijections are constructed directly from the combinatorial definitions of the polynomials as generating functions; the underlying formulas are imported from prior literature without self-referential redefinition, and no parameters are fitted or renamed as predictions. No load-bearing self-citations, uniqueness theorems imported from the authors' prior work, or ansatzes smuggled via citation appear in the derivation chain. The argument is therefore self-contained against external combinatorial benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper works inside the established theory of Macdonald polynomials; the only background assumptions are the standard combinatorial formulas for the polynomials and the fact that they form bases.

axioms (2)
  • domain assumption Permuted-basement Macdonald polynomials form a basis for Q(q,t)[x] for each fixed σ
    Stated directly in the abstract as the starting point for the identities.
  • domain assumption The polynomials admit combinatorial formulas as generating functions over composition-shaped non-attacking fillings
    Invoked to define the objects on which the bijections act.

pith-pipeline@v0.9.1-grok · 5700 in / 1359 out tokens · 15432 ms · 2026-06-28T13:47:48.100028+00:00 · methodology

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Reference graph

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