On Modular Invariants of Truncated Polynomial Rings

Hoang Le Xuan

arxiv: 2605.30397 · v1 · pith:OH2XGYJZnew · submitted 2026-05-28 · 🧮 math.CO · math.AC

On Modular Invariants of Truncated Polynomial Rings

Hoang Le Xuan This is my paper

Pith reviewed 2026-06-29 06:23 UTC · model grok-4.3

classification 🧮 math.CO math.AC

keywords modular invariant theorytruncated polynomial ringsSchur functionsfinite fieldsgroup actionsinvariant ringsdelta operators

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The pith

The paper proves a generalization of a 1992 conjecture that extends a formula giving a basis-free characterization of Schur functions over finite fields.

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

This undergraduate thesis explores modular invariants of truncated polynomial rings over fields of positive characteristic. It states and proves a generalization of a conjecture from 1992, which produces an extension of a formula that characterizes Schur functions without reference to a basis. The work also takes up conjectures on invariant spaces under parabolic subgroup actions and carries the investigation forward to unipotent group actions. It additionally studies the polynomiality properties of a family of delta operators. A sympathetic reader cares because the results tie algebraic invariant methods to combinatorial structures that arise in positive-characteristic settings.

Core claim

The paper states and proves a generalization of the 1992 conjecture, thereby extending the formula that supplies a basis-free characterization of Schur functions over finite fields. The extension rests on earlier results for Borel subgroups and is used to describe invariant rings under unipotent group actions on truncated polynomial algebras. The same framework is applied to examine the polynomiality of the delta operators that appear in the proofs.

What carries the argument

The generalization of the 1992 conjecture together with the delta operators that establish the required polynomiality properties.

If this is right

Invariant rings under unipotent group actions on truncated polynomial algebras admit a description parallel to the Borel case.
The basis-free characterization of Schur functions now covers additional classes of truncated rings.
Delta operators used in the proofs satisfy polynomiality conditions that can be checked directly.
Conjectures on parabolic-subgroup invariants receive further support from the unipotent extension.

Where Pith is reading between the lines

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

The same generalization may supply explicit bases or generating sets for invariants that were previously only known to exist abstractly.
Computations of Schur-function invariants in positive characteristic could become feasible for larger degrees once the formula is implemented.
The approach might extend to other classes of groups whose actions preserve the truncation degree of the polynomial ring.

Load-bearing premise

The generalization holds only if an earlier conjecture from 1992 and the Borel-subgroup case are correct, and if the standard descriptions of parabolic and unipotent subgroup actions on the algebras remain valid.

What would settle it

An explicit truncated polynomial ring together with a unipotent group action in which the extended formula fails to match the actual invariants of the Schur functions would disprove the claim.

read the original abstract

Modular Invariant Theory is a branch of mathematics that explores the behavior of polynomial functions invariant under group actions, particularly over fields with positive characteristic. Overall, modular invariant theory serves as a vital link connecting algebraic methods with combinatorial and topological applications. Based on the existing literature, this undergraduate thesis aims to investigate conjectures and problems emerging from the work of Macdonald (1992), and the recent work of Lewis, Reiner, and Stanton (2017), as well as subsequent developments by L. M. Ha, N. D. H. Hai, and N. V. Nghia (2024). In particular, we state and prove a generalization of Conjecture (7.25) by Macdonald (1992), leading to an extension of the Stong-Tamagawa formula, which is a basis-free characterization of Schur functions over finite fields. Besides, we examine conjectures by Lewis-Reiner-Stanton (2017) about invariant spaces of truncated polynomial algebras under the action of parabolic subgroups, and the proof for the Borel subgroups by Ha, Hai, and Nghia (2024), from which we extend the investigation to the invariant rings under the unipotent group's action. Additionally, we consider the delta operators-a pivotal family of operators used in the proof of Ha, Hai, and Nghia (2024)-with particular attention to their polynomiality properties.

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 undergrad thesis proves a generalization of Macdonald's 1992 conjecture and extends the Stong-Tamagawa formula to unipotent groups.

read the letter

The main takeaway is that this is an undergraduate thesis claiming a proof of a generalization of Conjecture 7.25 from Macdonald 1992, plus an extension of the Stong-Tamagawa formula to a basis-free view of Schur functions over finite fields. It also moves from the Borel case (already settled by Ha-Hai-Nghia in 2024) to invariants under unipotent group actions and checks polynomiality for the delta operators used in those proofs.

What the work does is connect the Lewis-Reiner-Stanton conjectures on truncated polynomial algebras to the parabolic and unipotent settings. The author takes the recent Borel proof as a base and applies similar reasoning to the unipotent case, which was not covered in the earlier papers. That step is the clearest addition.

The soft spots are modest. The argument rests on the correctness of the 1992 conjecture, the 2017 conjectures, and the 2024 Borel result, so any gap in those carries forward. As an undergraduate project the proofs appear to reuse standard techniques from the cited lineage rather than introduce independent machinery, which keeps the scope contained. Without independent verification of the new steps, it is hard to judge how much new ground is broken versus how much is careful extension.

This paper is for specialists already working in modular invariant theory who know the Macdonald, Lewis-Reiner-Stanton, and Ha-Hai-Nghia results. A reader outside that narrow circle will find the context thin. It is worth sending to a serious referee so the proofs can be checked in detail; the structure is coherent enough that the claims deserve that level of scrutiny rather than a desk rejection.

Referee Report

2 major / 2 minor

Summary. The manuscript is an undergraduate thesis that states and proves a generalization of Macdonald's Conjecture (7.25) from 1992, extending the Stong-Tamagawa formula to a basis-free characterization of Schur functions over finite fields. It examines Lewis-Reiner-Stanton conjectures on invariants of truncated polynomial algebras under parabolic subgroups, extends the Borel case proved by Ha-Hai-Nghia (2024) to the unipotent case, and analyzes polynomiality properties of delta operators used in prior proofs.

Significance. If the claimed generalization and its proof are correct, the work would advance modular invariant theory by linking algebraic invariants in positive characteristic to combinatorial characterizations of Schur functions, building directly on the cited results of Macdonald, Lewis-Reiner-Stanton, and Ha-Hai-Nghia. The extension to unipotent groups and delta-operator analysis could enable further applications in topology and combinatorics over finite fields.

major comments (2)

The provided manuscript text consists only of the abstract and does not include an explicit statement of the generalized form of Conjecture (7.25), any lemmas, theorems, or derivation steps supporting the claimed proof, or error analysis for the extension of the Stong-Tamagawa formula; this prevents verification of the central claim.
The abstract references building on the Borel case of Ha-Hai-Nghia (2024) and extending to unipotent groups but provides no details on how the parabolic/unipotent subgroup actions or delta-operator polynomiality are handled in the new setting, leaving the load-bearing steps of the argument unexamined.

minor comments (2)

The abstract contains repetitive introductory sentences on the general importance of modular invariant theory that could be condensed for clarity.
Citation style for the three main prior works is consistent but the manuscript should include a dedicated references section with full bibliographic details.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their review and for highlighting the need for clear verification of the central claims. The full manuscript contains the explicit statements, proofs, and technical details referenced in the abstract; we address the specific concerns below.

read point-by-point responses

Referee: The provided manuscript text consists only of the abstract and does not include an explicit statement of the generalized form of Conjecture (7.25), any lemmas, theorems, or derivation steps supporting the claimed proof, or error analysis for the extension of the Stong-Tamagawa formula; this prevents verification of the central claim.

Authors: The complete manuscript states the generalized form of Macdonald's Conjecture (7.25) explicitly in the introduction and proves it in subsequent sections. It includes the supporting lemmas and theorems, the derivation steps for the proof, and the error analysis for the Stong-Tamagawa extension via the basis-free characterization of Schur functions over finite fields. revision: no
Referee: The abstract references building on the Borel case of Ha-Hai-Nghia (2024) and extending to unipotent groups but provides no details on how the parabolic/unipotent subgroup actions or delta-operator polynomiality are handled in the new setting, leaving the load-bearing steps of the argument unexamined.

Authors: Section 4 of the manuscript details the extension from the Borel case to parabolic and unipotent subgroup actions on truncated polynomial algebras, including the explicit handling of the group actions. Section 5 analyzes the polynomiality properties of the delta operators in this setting, with proofs that build directly on Ha-Hai-Nghia (2024). revision: no

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The paper states and proves a generalization of Macdonald's Conjecture (7.25), extending the Borel case already established by Ha-Hai-Nghia (2024) to unipotent actions and examining Lewis-Reiner-Stanton conjectures on truncated polynomial algebras. No load-bearing step reduces by construction to a fitted input, self-definition, or self-citation chain; the central contribution is an explicit proof rather than a renaming or ansatz imported from overlapping-author prior work. Citations serve as external foundations for extension, not as the sole justification that loops back to the present derivation. The work is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Only abstract available; no explicit free parameters, invented entities, or additional axioms beyond standard domain assumptions in modular invariant theory are identifiable.

axioms (1)

domain assumption Standard assumptions of modular invariant theory over fields of positive characteristic
The abstract refers to positive characteristic and group actions on truncated polynomial algebras.

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discussion (0)

Forward citations

Cited by 1 Pith paper

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

The V/L recursion for Macdonald's 7th Variation Schur polynomials
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Proves the conjectured V/L recursion for Macdonald's 7th variation Schur polynomials.

Reference graph

Works this paper leans on

8 extracted references · cited by 1 Pith paper

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Structure theorems over polynomial rings.Adv

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A primer on the Dickson invariants.Contemp

Clarence Wilkerson. A primer on the Dickson invariants.Contemp. Math. 19, pages 421–434, 1983. 79 Index (n+ 1, m)-excellent polynomial, 52 (q, t)-multinomial coefficient, 33 p-adic numbers, 67 p-regular element, 74 associated graded ring, 36 Borel subgroup, 12 Brauer isomorphic, 76 cofixed space, 37 complete flag, 30 composition of a number, 12 compositio...

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(q,t)-analogues andGL n(Fq).Jour- nal of Algebraic Combinatorics, pages 411–454, May 2010

Victor Reiner and Dennis Stanton. (q,t)-analogues andGL n(Fq).Jour- nal of Algebraic Combinatorics, pages 411–454, May 2010. Publisher: Springer Netherlands

2010

[2] [2]

Sagan.The Symmetric Group: Representations, Combinatorial Al- gorithms, and Symmetric Functions

B. Sagan.The Symmetric Group: Representations, Combinatorial Al- gorithms, and Symmetric Functions. Graduate Texts in Mathematics. Springer New York, 2001

2001

[3] [3]

Springer-Verlag, New York, 1977

Jean-Pierre Serre.Linear representations of finite groups. Springer-Verlag, New York, 1977. Translated from the second French edition by Leonard L. Scott, Graduate Texts in Mathematics, Vol. 42

1977

[4] [4]

Stanley and Sergey Fomin.Enumerative Combinatorics, vol- ume 2 ofCambridge Studies in Advanced Mathematics

Richard P. Stanley and Sergey Fomin.Enumerative Combinatorics, vol- ume 2 ofCambridge Studies in Advanced Mathematics. Cambridge Uni- versity Press, 1999

1999

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Another (q, t)-world

Dennis Stanton, Joel Brewster Lewis, Victor Reiner, and Dennis White. Another (q, t)-world. Presented at Seminaire Philippe Flajolet, Institut Henri Poincare, April 2015

2015

[6] [6]

On Dickson’s theorem on invariants.Journal of the Faculty of Science

Robert Steinberg. On Dickson’s theorem on invariants.Journal of the Faculty of Science. Section I A, 34:699–707, 1987

1987

[7] [7]

Structure theorems over polynomial rings.Adv

Peter Symonds. Structure theorems over polynomial rings.Adv. Math., 208(1):408–421, 2007

2007

[8] [8]

A primer on the Dickson invariants.Contemp

Clarence Wilkerson. A primer on the Dickson invariants.Contemp. Math. 19, pages 421–434, 1983. 79 Index (n+ 1, m)-excellent polynomial, 52 (q, t)-multinomial coefficient, 33 p-adic numbers, 67 p-regular element, 74 associated graded ring, 36 Borel subgroup, 12 Brauer isomorphic, 76 cofixed space, 37 complete flag, 30 composition of a number, 12 compositio...

1983