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arxiv: 2605.26775 · v2 · pith:ORQRQUSZnew · submitted 2026-05-26 · 🧮 math.CO · math.NT· math.RA

The V/L recursion for Macdonald's 7th Variation Schur polynomials

Darij Grinberg This is my paper

Pith reviewed 2026-06-29 17:33 UTC · model grok-4.3

classification 🧮 math.CO math.NTmath.RA
keywords Schur polynomialsMacdonald variationsfinite fieldsFrobenius maprecursionvector spacescombinatoricssymmetric functions
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The pith

Macdonald's seventh variation Schur polynomials satisfy the conjectured V/L recursion.

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

The paper generalizes and proves the recursive relation S_λ(V) = sum over lines L in V of S_λ(V//L) for Macdonald's seventh variation of the Schur polynomials. These polynomials are constructed over finite fields using powers of the Frobenius map to mimic the behavior of ordinary Schur polynomials, including both straight and skew cases. The relation decomposes the polynomial on a vector space by summing its values on the quotients obtained by factoring out each one-dimensional subspace. A reader would care because the recursion mirrors a key structural property of classical Schur functions, allowing the same kinds of recursive computations and combinatorial arguments to carry over to this finite-field setting.

Core claim

The recursive relation S_λ(V) = ∑_{L⊆V line} S_λ(V//L) holds for the seventh variation Schur polynomials S_λ defined via the Frobenius endomorphism over finite fields, proving the conjecture of Macdonald.

What carries the argument

The V/L recursion, which expresses the polynomial on V as a sum of the same polynomial on the quotients V//L for each line L contained in V.

If this is right

  • The recursion holds for both straight and skew versions of these polynomials.
  • It supplies a recursive method for evaluating or defining the polynomials on vector spaces.
  • The same decomposition can be used to derive further identities that hold for ordinary Schur polynomials.

Where Pith is reading between the lines

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

  • The recursion may extend to other symmetric-function generalizations built from the Frobenius.
  • It could allow transfer of combinatorial interpretations of Schur polynomials to the finite-field setting.
  • The proof technique might apply to recursions involving higher-dimensional subspaces.

Load-bearing premise

The seventh variation polynomials are constructed using powers of the Frobenius so that they inherit the line-decomposition property of ordinary Schur polynomials.

What would settle it

Direct computation of both sides of the recursion for partition (2) and vector space F_2^3 would show equality or a mismatch.

read the original abstract

We generalize and prove the recursive relation \[ S_{\lambda}(V) = \sum_{L\subseteq V\text{ line}} S_{\lambda}(V \mathbin{/\mkern-5mu/} L) \] conjectured by I. G. Macdonald for his "7th variation" of the Schur functions. This variation is a family of polynomials over a finite field that mimic the (straight and skew) Schur polynomials using powers of the Frobenius.

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 generalizes and proves the recursive relation S_λ(V) = ∑_{L ⊆ V line} S_λ(V // L) conjectured by I. G. Macdonald for the 7th variation Schur polynomials, which are defined over finite fields via powers of the Frobenius endomorphism to mimic straight and skew Schur polynomials.

Significance. If the proof holds, the result confirms an important conjecture and supplies a recursive decomposition that parallels the standard Schur case, potentially enabling inductive arguments or explicit computations for these variation polynomials in algebraic combinatorics.

minor comments (1)
  1. [Abstract] The abstract introduces the notation V // L and the summation over lines without a brief parenthetical gloss; a short clarification would improve immediate readability.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their summary of the manuscript, which correctly identifies the main result as a proof of Macdonald's conjectured V/L recursion for the 7th variation Schur polynomials. No specific major comments were provided in the report, so we have no individual points to address. The proof is given in full in the manuscript, and we believe it is complete.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper proves an externally conjectured recursion for 7th-variation Schur polynomials (defined independently via Frobenius powers over finite fields) rather than deriving any claimed result from quantities defined in terms of the target recursion itself. No self-citations, fitted parameters renamed as predictions, or ansatzes smuggled via prior work by the same authors appear in the provided abstract or claim description. The derivation chain is therefore self-contained against the external conjecture and definition.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Based solely on the abstract; the proof is expected to rest on standard combinatorial identities and finite-field properties without introducing new fitted parameters or invented entities.

axioms (2)
  • standard math Standard properties of Schur polynomials and their variations over finite fields
    The work extends established theory of symmetric functions and Frobenius endomorphisms.
  • domain assumption The 7th variation mimics straight and skew Schur polynomials via Frobenius powers
    Invoked in the abstract to justify the recursion applicability.

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discussion (0)

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Reference graph

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