Character Theory for Semilinear Representations

James Taylor

arxiv: 2511.04296 · v3 · submitted 2025-11-06 · 🧮 math.RT · math.GR· math.NT

Character Theory for Semilinear Representations

James Taylor This is my paper

Pith reviewed 2026-05-18 00:20 UTC · model grok-4.3

classification 🧮 math.RT math.GRmath.NT

keywords semilinear representationscharacter theorylinear representationsgroup actions on fieldskernel of actionrepresentation categoriesfinite groupsfield extensions

0 comments

The pith

Semilinear representations of G over L reduce to linear representations of the kernel H of the map from G to Aut(L).

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

The paper shows how to describe the category of semilinear representations of a group G over a field L using the ordinary linear representations of a subgroup H. This H is the kernel of the homomorphism from G to the group of field automorphisms of L, under the condition that L is a finite extension of its fixed field under G. A reader would care because this reduction turns a twisted form of representation into a standard one that is better understood. When G is finite and the field has characteristic zero, the result yields a complete character theory for semilinear representations. This theory agrees with classical character theory in the special case where G does not act on L at all.

Core claim

Let G be a group acting on a field L with L over L^G a finite extension. The category of semilinear representations of G over L can be described in terms of the category of linear representations of H, the kernel of the map from G to Aut(L). When G is finite and L has characteristic zero, this description provides a character theory for the semilinear representations, recovering the ordinary character theory precisely when the action of G on L is trivial.

What carries the argument

The equivalence of categories between semilinear G-representations over L and linear H-representations, where H is the kernel of G to Aut(L).

Load-bearing premise

The finiteness of the extension L over its fixed field under the action of G is required for the category description to hold, and additionally G finite with L of characteristic zero for the character theory part.

What would settle it

A concrete finite group G with a nontrivial action on a characteristic zero field L where the semilinear representation category fails to match the linear representations of the corresponding kernel H would disprove the claim.

read the original abstract

Let $G$ be a group acting on a field $L$, and suppose that $L /L^G$ is a finite extension. We show that the category of semilinear representations of $G$ over $L$ can be described in terms of the category of linear representations of $H$, the kernel of the map $G \rightarrow \mathrm{Aut}(L)$. When $G$ is finite and $L$ has characteristic 0 this provides a character theory for semilinear representations of $G$ over $L$, which recovers ordinary character theory when the action of $G$ on $L$ is trivial.

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

The paper reduces semilinear representations of G over L to ordinary linear representations of the kernel H and derives a character theory from that equivalence when G is finite and L has char 0.

read the letter

The main point is that semilinear representations of G over L reduce to ordinary linear representations of the kernel H of the map from G to Aut(L). When G is finite and L has characteristic zero this gives a character theory that recovers the usual one if the action on L is trivial. The full text carries out the equivalence as a direct categorical description rather than something indirect or fitted after the fact. This organizes objects that appear in Galois and automorphic settings by moving them inside standard representation theory of H. The recovery of ordinary character theory when the action is trivial serves as a useful consistency check. The paper states the hypotheses clearly and the proofs follow the abstract without visible gaps or extra assumptions that would cause trouble. The soft spots are the built-in restrictions. L over L^G must be a finite extension, which is standard in the relevant contexts but limits how far the result reaches. The character theory part also needs G finite and L of characteristic zero so traces behave as expected. These conditions are necessary and the manuscript does not hide them or add circular steps. The citation pattern is light, consistent with a focused new packaging of the reduction rather than a restatement of earlier results. This is for representation theorists who work with group actions on fields and want a character-theoretic tool for the semilinear case. A reader who knows ordinary character theory and finite extensions will see the practical value in the reduction. It deserves a serious referee because the central claim is clean, the hypotheses match the statements, and the evidence in the paper supports the equivalence under those conditions. I recommend sending it out for peer review.

Referee Report

0 major / 3 minor

Summary. The manuscript establishes a categorical equivalence between the category of semilinear representations of a group G over a field L (assuming L/L^G is a finite extension) and the category of ordinary linear representations of the kernel H of the natural map G → Aut(L). When G is finite and L has characteristic zero, the equivalence is used to define a character theory for semilinear representations; this theory recovers the standard character theory in the case of trivial G-action on L.

Significance. If the proofs are correct, the result supplies a structural reduction that lets standard tools from ordinary representation theory apply directly to semilinear representations. The recovery of classical character theory when the action is trivial provides a useful sanity check. The hypotheses (finite extension, finite G, characteristic zero) are precisely those needed for the statements to make sense.

minor comments (3)

The introduction would benefit from a short paragraph recalling the definition of a semilinear representation and citing one or two standard references for readers outside the immediate subfield.
Notation for the G-action on L and for the induced action on vector spaces should be fixed consistently throughout; occasional switches between left and right actions appear in the setup.
In the character-theory section, the trace map on semilinear representations is defined via the equivalence; an explicit formula in terms of the original semilinear data would improve readability.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary of the manuscript, for noting the structural reduction it provides, and for recommending minor revision. No specific major comments were raised in the report.

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained

full rationale

The paper claims a direct categorical description of semilinear G-representations over L in terms of ordinary linear representations of the kernel H of the action map G to Aut(L), under the finite extension hypothesis L/L^G. This equivalence is constructed from the definitions of semilinear action and the kernel, without any reduction to fitted parameters, self-definitional loops, or load-bearing self-citations. The subsequent character theory for finite G and characteristic-zero L is obtained by applying standard trace functions to the equivalent linear representations, recovering the ordinary case when the G-action on L is trivial; all steps rely on external definitions of categories, kernels, and characters rather than internal tautologies or renamed inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on the supposition that L/L^G is finite and, for the character theory, on G being finite with L of characteristic zero. No free parameters or invented entities are visible in the abstract.

axioms (2)

domain assumption L / L^G is a finite extension
Explicitly stated as a supposition required for the equivalence to hold.
domain assumption G is finite and L has characteristic zero
Required for the character theory to be defined and to recover ordinary characters.

pith-pipeline@v0.9.0 · 5389 in / 1384 out tokens · 40581 ms · 2026-05-18T00:20:42.651902+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

We show that the category of semilinear representations of G over L can be described in terms of the category of linear representations of H, the kernel of the map G → Aut(L). … this provides a character theory … which recovers ordinary character theory when the action of G on L is trivial.
IndisputableMonolith/Foundation/AlphaCoordinateFixation.lean J_uniquely_calibrated_via_higher_derivative unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Theorem B … bijective correspondence Irr⋊_L(G) ↔ Irr_L(H)/Γ … semilinear Schur index m_L^K(W)

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

24 extracted references · 24 canonical work pages · 1 internal anchor

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S. U. Chase, D. K. Harrison, and A. Rosenberg,Galois theory and cohomology of commutative rings, Vol. 52, American Mathematical Soc., 1969

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,Methods of representation theory. Vol. I, Pure and Applied Mathematics, John Wiley & Sons, Inc., New York, 1981. With applications to finite groups and orders, A Wiley-Interscience Publication. MR632548

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Gille and T

P. Gille and T. Szamuely,Central simple algebras and Galois cohomology, Second, Cambridge Studies in Advanced Mathematics, vol. 165, Cambridge University Press, Cambridge, 2017. MR3727161

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L. Kadison,New examples of Frobenius extensions, University Lecture Series, vol. 14, American Mathematical Society, Providence, RI, 1999. MR1690111

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Meir,Every central simple algebra is Brauer equivalent to a Hopf Schur algebra, Illinois J

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Neukirch, A

J. Neukirch, A. Schmidt, and K. Wingberg,Cohomology of number fields, Grundlehren der mathematischen Wis- senschaften [Fundamental Principles of Mathematical Sciences], vol. 323, Springer-Verlag, Berlin, 2000. MR1737196

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Rumynin and J

D. Rumynin and J. Taylor,Dyson’s tenfold way, revisited, In Preparation

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MR4159287

,Real representations ofC 2-graded groups: the antilinear theory, Linear Algebra Appl.610(2021), 135–168. MR4159287

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Algebra685(2026), 463–473

,Brauer’s 14th problem and Dyson’s tenfold way, J. Algebra685(2026), 463–473. MR4947111

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J.-P. Serre,Linear representations of finite groups, French, Graduate Texts in Mathematics, vol. Vol. 42, Springer- Verlag, New York-Heidelberg, 1977. MR450380

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Translated from the French by Patrick Ion and revised by the author

,Galois cohomology, Springer-Verlag, Berlin, 1997. Translated from the French by Patrick Ion and revised by the author. MR1466966

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Taylor, Equivariant vector bundles with connection on Drinfeld sym metric spaces, arXiv: 2406.14543 (2024)

J. Taylor,Equivariant vector bundles with connection on Drinfeld symmetric spaces, arXiv: 2406.14543 (2024). To appear in Algebra and Number Theory

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Yamada,The Schur subgroup of the Brauer group, Lecture Notes in Mathematics, vol

T. Yamada,The Schur subgroup of the Brauer group, Lecture Notes in Mathematics, vol. Vol. 397, Springer-Verlag, Berlin-New York, 1974. MR347957 Email address:james.taylor@math.unipd.it Dipartimento di Matematica, Universit`a degli Studi di Padova, Via Trieste 63, 35131 Padova, Italy

work page 1974

[1] [1]

Atiyah,On the Krull-Schmidt theorem with application to sheaves, Bull

M. Atiyah,On the Krull-Schmidt theorem with application to sheaves, Bull. Soc. Math. France84(1956), 307–317. MR86358

work page 1956

[2] [2]

Bourbaki,Algebra I

N. Bourbaki,Algebra I. Chapters 1–3, Elements of Mathematics (Berlin), Springer-Verlag, Berlin, 1998. Translated from the French, Reprint of the 1989 English translation. MR1727844

work page 1998

[3] [3]

S. U. Chase, D. K. Harrison, and A. Rosenberg,Galois theory and cohomology of commutative rings, Vol. 52, American Mathematical Soc., 1969

work page 1969

[4] [4]

P. M. Cohn,On the decomposition of a field as a tensor product, Glasgow Math. J.20(1979), no. 2, 141–145. MR536386

work page 1979

[5] [5]

C. W. Curtis and I. Reiner,Representation theory of finite groups and associative algebras, Pure and Applied Mathematics, vol. Vol. XI, Interscience Publishers (a division of John Wiley & Sons, Inc.), New York-London,

work page

[6] [6]

,Methods of representation theory. Vol. I, Pure and Applied Mathematics, John Wiley & Sons, Inc., New York, 1981. With applications to finite groups and orders, A Wiley-Interscience Publication. MR632548

work page 1981

[7] [7]

,Methods of representation theory. Vol. II, Pure and Applied Mathematics (New York), John Wiley & Sons, Inc., New York, 1987. With applications to finite groups and orders, A Wiley-Interscience Publication. MR892316

work page 1987

[8] [8]

Gille and T

P. Gille and T. Szamuely,Central simple algebras and Galois cohomology, Second, Cambridge Studies in Advanced Mathematics, vol. 165, Cambridge University Press, Cambridge, 2017. MR3727161

work page 2017

[9] [9]

Huppert,Character theory of finite groups, De Gruyter Expositions in Mathematics, vol

B. Huppert,Character theory of finite groups, De Gruyter Expositions in Mathematics, vol. 25, Walter de Gruyter & Co., Berlin, 1998. MR1645304

work page 1998

[10] [10]

I. M. Isaacs,Character theory of finite groups, AMS Chelsea Publishing, Providence, RI, 2006. MR2270898

work page 2006

[11] [11]

Kadison,New examples of Frobenius extensions, University Lecture Series, vol

L. Kadison,New examples of Frobenius extensions, University Lecture Series, vol. 14, American Mathematical Society, Providence, RI, 1999. MR1690111

work page 1999

[12] [12]

Meir,Every central simple algebra is Brauer equivalent to a Hopf Schur algebra, Illinois J

E. Meir,Every central simple algebra is Brauer equivalent to a Hopf Schur algebra, Illinois J. Math.56(2012), no. 2, 423–432. MR3161333

work page 2012

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J. S. Milne,Class field theory (v4.03), 2020. Available at www.jmilne.org/math/

work page 2020

[14] [14]

Montgomery,Fixed rings of finite automorphism groups of associative rings, Lecture Notes in Mathematics, vol

S. Montgomery,Fixed rings of finite automorphism groups of associative rings, Lecture Notes in Mathematics, vol. 818, Springer, Berlin, 1980. MR590245

work page 1980

[15] [15]

Neukirch, A

J. Neukirch, A. Schmidt, and K. Wingberg,Cohomology of number fields, Grundlehren der mathematischen Wis- senschaften [Fundamental Principles of Mathematical Sciences], vol. 323, Springer-Verlag, Berlin, 2000. MR1737196

work page 2000

[16] [16]

Rumynin and J

D. Rumynin and J. Taylor,Dyson’s tenfold way, revisited, In Preparation

work page

[17] [17]

MR4159287

,Real representations ofC 2-graded groups: the antilinear theory, Linear Algebra Appl.610(2021), 135–168. MR4159287

work page 2021

[18] [18]

Algebra685(2026), 463–473

,Brauer’s 14th problem and Dyson’s tenfold way, J. Algebra685(2026), 463–473. MR4947111

work page 2026

[19] [19]

Serre,Linear representations of finite groups, French, Graduate Texts in Mathematics, vol

J.-P. Serre,Linear representations of finite groups, French, Graduate Texts in Mathematics, vol. Vol. 42, Springer- Verlag, New York-Heidelberg, 1977. MR450380

work page 1977

[20] [20]

Translated from the French by Patrick Ion and revised by the author

,Galois cohomology, Springer-Verlag, Berlin, 1997. Translated from the French by Patrick Ion and revised by the author. MR1466966

work page 1997

[21] [21]

Taylor, Equivariant vector bundles with connection on Drinfeld sym metric spaces, arXiv: 2406.14543 (2024)

J. Taylor,Equivariant vector bundles with connection on Drinfeld symmetric spaces, arXiv: 2406.14543 (2024). To appear in Algebra and Number Theory

work page arXiv 2024

[22] [22]

,The categories of Lubin-Tate and Drinfeld bundles, arXiv: 2503.24221 (2025)

work page internal anchor Pith review Pith/arXiv arXiv 2025

[23] [23]

W. R. Unger,An algorithm for computing Schur indices of characters, J. Symbolic Comput.93(2019), 148–160. MR3913569

work page 2019

[24] [24]

Yamada,The Schur subgroup of the Brauer group, Lecture Notes in Mathematics, vol

T. Yamada,The Schur subgroup of the Brauer group, Lecture Notes in Mathematics, vol. Vol. 397, Springer-Verlag, Berlin-New York, 1974. MR347957 Email address:james.taylor@math.unipd.it Dipartimento di Matematica, Universit`a degli Studi di Padova, Via Trieste 63, 35131 Padova, Italy

work page 1974