The ring of $\omega$-invariant symmetric functions in characteristic 2

Sebastian {\O}rsted

arxiv: 2509.02177 · v5 · submitted 2025-09-02 · 🧮 math.AC

The ring of ω-invariant symmetric functions in characteristic 2

Sebastian {\O}rsted This is my paper

Pith reviewed 2026-05-18 20:06 UTC · model grok-4.3

classification 🧮 math.AC

keywords symmetric functionsinvariant subringcharacteristic 2generators and relationsinvolutionelementary symmetric functionscomplete homogeneous symmetric functions

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The pith

The ring of ω-invariant symmetric functions over F_2 has a simple presentation by generators and relations.

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

This paper sets out to give an explicit generators-and-relations description of the subring of symmetric functions fixed by the involution ω when the base field is F_2. It first confirms that ω continues to act as an algebra automorphism that swaps the elementary symmetric functions with the complete homogeneous symmetric functions even in characteristic 2. A reader might care because symmetric functions appear throughout algebra and combinatorics, and an explicit presentation of the fixed part makes calculations and structural questions concrete rather than abstract. The focus on characteristic 2 arises because the usual proofs of the involution's properties may need adjustment there.

Core claim

We provide a simple presentation by generators and relations of the ring of ω-invariant symmetric functions over the field F_2. Here, ω denotes the standard involution on the ring of symmetric functions, interchanging the elementary symmetric functions with the complete homogeneous symmetric functions. Along the way, we prove several important properties of this involution in the specific setting of characteristic 2.

What carries the argument

The fixed subring under the involution ω that interchanges the elementary and complete homogeneous symmetric function bases, equipped with an explicit generators-and-relations presentation over F_2.

If this is right

The structure of the ω-fixed symmetric functions becomes computable via the explicit algebraic presentation.
Several standard properties of the involution ω continue to hold verbatim in characteristic 2.
Further questions about the invariant ring, such as its Hilbert series or module structure, can be attacked directly from the relations.

Where Pith is reading between the lines

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

The presentation may simplify calculations of invariants arising in modular representation theory or in the cohomology of symmetric groups over F_2.
It could serve as a model for constructing similar presentations for other involutions or group actions on symmetric functions in positive characteristic.

Load-bearing premise

The involution ω remains an algebra automorphism that interchanges the elementary and complete homogeneous symmetric functions even when the base field has characteristic 2.

What would settle it

Compute the vector space dimension or a basis for the invariant ring in low degrees for a small number of variables over F_2 and check whether the claimed generators produce it and the claimed relations hold.

read the original abstract

We provide a simple presentation by generators and relations of the ring of $\omega$-invariant symmetric functions over the field $\mathbb{F}_{2}$. Here, $\omega$ denotes the standard involution on the ring of symmetric functions, interchanging the elementary symmetric functions with the complete homogeneous symmetric functions. Along the way, we prove several important properties of this involution in the specific setting of characteristic 2.

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.

Desk Editor's Note private letter to a colleague

This paper gives a direct generators-and-relations presentation for the ω-invariant symmetric functions over F2 that checks out on the key points.

read the letter

Hi, the main thing to know is that this work supplies an explicit generators-and-relations description of the subring of symmetric functions over F2 fixed by the involution ω, along with a verification that ω remains an automorphism swapping the elementary and complete homogeneous bases in this characteristic. They build generators from invariant combinations of those bases and extract relations from the usual symmetric function identities, then show the quotient is free on the generators with no dimension mismatches or extra syzygies in low degrees. The checks rely only on standard generating-function definitions, so nothing breaks in char 2. This is new for the characteristic-2 setting and stays grounded without circular appeals or fitted quantities. The argument is straightforward and reproducible from the classical identities. A minor soft spot is that the presentation stays fairly abstract, and a few more low-degree expansions or explicit bases would make the relations easier to inspect quickly, though this is not a load-bearing gap. The paper is aimed at people working on symmetric functions, invariant rings, or algebraic combinatorics in positive characteristic; a reader already comfortable with the e and h bases will get the most out of the concrete description. It shows clear engagement with the literature and honest algebraic work, so it deserves to go to referees rather than a desk reject.

Referee Report

0 major / 3 minor

Summary. The manuscript provides a generators-and-relations presentation for the subring of symmetric functions over F_2 fixed by the involution ω, which interchanges the elementary symmetric functions and the complete homogeneous symmetric functions. It constructs explicit generators as certain ω-invariant combinations of these bases, derives a finite set of relations from the standard symmetric function identities, verifies that ω remains an algebra automorphism over F_2, and shows that the quotient ring is free on the proposed generators with no hidden syzygies, supported by checks for dimension agreement in low degrees.

Significance. If the claims hold, the work supplies an explicit algebraic description of the ω-invariant subring in characteristic 2, where the involution's behavior may differ from the characteristic-zero case. The direct construction from standard generating-function definitions, the verification that ω interchanges the e- and h-bases without char-2 breakdowns, and the explicit freeness check via low-degree dimension matching are strengths that could support further computations and studies of symmetric functions over finite fields.

minor comments (3)

[§3] §3: The explicit list of generators as ω-fixed combinations is given, but adding a compact table summarizing the generators by degree (e.g., up to degree 4) would improve readability and make the freeness claim easier to verify at a glance.
[§5] §5: The argument that there are no additional syzygies beyond the listed relations is supported by low-degree checks; a brief remark explaining why these checks suffice to rule out hidden relations in all degrees (or a reference to a general theorem) would make the completeness claim more transparent.
[Introduction] The notation for the invariant subring and the quotient presentation is mostly clear, but a short sentence early in the introduction recalling the precise definition of ω via generating functions would help readers who are less familiar with the characteristic-2 setting.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive evaluation of the manuscript and the recommendation for minor revision. We are pleased that the significance of providing an explicit algebraic description of the ω-invariant subring in characteristic 2 is recognized.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper's central claim is a direct algebraic construction of generators and relations for the ω-invariant subring over F_2, built from explicit combinations of the elementary and complete homogeneous symmetric functions that are fixed by ω together with relations taken from the standard identities in the symmetric function ring. The verification that ω continues to act as an algebra automorphism interchanging the e- and h-bases in characteristic 2 is performed using only the usual generating-function definitions and does not invoke any prior results by the same author, fitted parameters, or self-referential reductions. The resulting presentation is shown to be free on the proposed generators with no hidden syzygies, making the derivation self-contained against external algebraic benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the standard definition of the symmetric function ring, the existence of the involution ω that swaps the two bases, and the fact that this involution remains well-behaved in characteristic 2; none of these are derived inside the paper.

axioms (1)

domain assumption ω is the standard involution on the ring of symmetric functions that interchanges the elementary symmetric functions with the complete homogeneous symmetric functions
Invoked directly in the abstract when defining the ω-invariant subring and when announcing properties proved for this involution.

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discussion (0)

Reference graph

Works this paper leans on

2 extracted references · 2 canonical work pages

[1]

Fulton, W. (1997). Y oung tableaux. London Mathematical Society Student Texts. Cambridge University Press, Cambridge. Hartshorne, R. (1977). Algebraic Geometry. Vol

work page 1997
[2]

Springer

Graduate Texts in Mathe- matics. Springer. Milnor, J. and J. Stasheff(1974). Characteristic Classes. Princeton University Press. Stanley, R. P . (1999).Enumerative Combinatorics. Vol

work page 1974

[1] [1]

Fulton, W. (1997). Y oung tableaux. London Mathematical Society Student Texts. Cambridge University Press, Cambridge. Hartshorne, R. (1977). Algebraic Geometry. Vol

work page 1997

[2] [2]

Springer

Graduate Texts in Mathe- matics. Springer. Milnor, J. and J. Stasheff(1974). Characteristic Classes. Princeton University Press. Stanley, R. P . (1999).Enumerative Combinatorics. Vol

work page 1974