Spectral Separation and Eigenvalue Labelling for Polynomial Tensor Representations of General Linear Groups

Dang Vo Phuc

arxiv: 2512.00263 · v5 · submitted 2025-11-29 · 🧮 math.RT · math.GR· math.NT

Spectral Separation and Eigenvalue Labelling for Polynomial Tensor Representations of General Linear Groups

Dang Vo Phuc This is my paper

Pith reviewed 2026-05-17 04:07 UTC · model grok-4.3

classification 🧮 math.RT math.GRmath.NT

keywords spectral separationeigenvalue labellingpolynomial tensor representationsSinger cyclegeneral linear groupsbase-q injectivityfinite field representations

0 comments

The pith

Distinct weights produce distinct Singer cycle eigenvalues on bounded-degree polynomial tensor representations of GL_d(q).

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

The paper shows that if a module W for a subgroup containing a genuine Singer cycle has its scalar extension equal to an untwisted polynomial tensor representation of total degree K less than q minus one, then distinct weights correspond to distinct eigenvalues of the cycle on the extended module. This separation matters because it turns spectral data into a label for the weights and opens a route to recovering the original module action from eigenvalue information. The argument rests on an elementary lemma that vectors with digits bounded by the degree give unique residues modulo q^d minus one. The authors also supply a formula for shifted cases and illustrate a rewriting approach that reduces action reconstruction to an inversion step, backed by explicit computations on multiplicity-free examples.

Core claim

If the total polynomial degree satisfies K < q-1, distinct weights give distinct eigenvalues of s on W ⊗_{F_q} F_{q^d} for a genuine Singer cycle s. The proof relies on an elementary base-q injectivity lemma: bounded digit vectors determine distinct residues modulo q^d-1. When the tensor product is multiplicity-free for the diagonal torus, the Singer cycle has a simple spectrum. A shifted exponent formula separates distinct shifted digit vectors under the same bound.

What carries the argument

base-q injectivity lemma: bounded digit vectors determine distinct residues modulo q^d-1

If this is right

When the tensor product is multiplicity-free for the diagonal torus, the Singer cycle has a simple spectrum.
A shifted exponent formula separates distinct shifted digit vectors under the same bound K < q-1.
The results supply a uniform spectral explanation for eigenvalue separation in bounded-degree polynomial tensor representations.
A conditional rewriting framework uses compatible base-q eigenvalue labelling to reduce reconstruction of the natural action to a functor-specific inversion problem.

Where Pith is reading between the lines

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

The same injectivity idea might allow eigenvalue labelling to continue working for selected families of representations whose total degree exceeds q-1.
Explicit computational reconstruction of the natural action from a multiplicity-free genuine tensor product suggests the spectral data can serve as a practical starting point for identifying modules in larger examples.
The approach connects to broader questions of when character or trace information on a cyclic subgroup determines the full module structure for linear groups over finite fields.

Load-bearing premise

The scalar extension of W restricts to an untwisted polynomial tensor representation of the algebraic group GL_d.

What would settle it

Two distinct weights producing the same eigenvalue for the Singer cycle s on such a W when total degree K is strictly less than q-1, or two different bounded base-q digit vectors yielding the same residue modulo q^d-1.

read the original abstract

Let $q=p^f$ be a prime power, $H \leq \mathrm{GL}_d(q)$ a subgroup containing a genuine Singer cycle $s$ of order $q^d-1$, and $W$ an $\mathbb{F}_q H$-module whose scalar extension restricts to an untwisted polynomial tensor representation $\bigotimes L(\lambda^{(t)})$ of the algebraic group $\mathrm{GL}_d$. If the total polynomial degree satisfies $K < q-1$, we prove that distinct weights give distinct eigenvalues of $s$ on $W \otimes_{\mathbb{F}_q} \mathbb{F}_{q^d}$. The proof relies on an elementary base-$q$ injectivity lemma: bounded digit vectors determine distinct residues modulo $q^d-1$. Consequently, when the tensor product is multiplicity-free for the diagonal torus, the Singer cycle has a simple spectrum. We also provide a shifted exponent formula for situations where Singer eigenvalue data undergo $q$-Frobenius shifts, proving separation of distinct shifted digit vectors under the same bound $K<q-1$. These results provide a uniform spectral explanation for eigenvalue separation in bounded-degree polynomial tensor representations. Motivated by this, we formulate a conditional rewriting framework that uses compatible base-$q$ eigenvalue labelling to reduce the reconstruction of the natural action to a functor-specific inversion problem. Finally, the viability of this framework is explicitly demonstrated through computational experiments, prominently featuring a non-trivial, full algebraic reconstruction of the natural action from a strictly multiplicity-free, genuine tensor product representation.

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 proves that distinct weights give distinct Singer eigenvalues on low-degree untwisted tensor modules for GL_d(q) using an elementary base-q lemma, plus a conditional framework for action reconstruction with some computational checks.

read the letter

The main takeaway is that when the total degree K is less than q-1, different weights in an untwisted polynomial tensor representation lead to different eigenvalues for a genuine Singer cycle s after extending scalars to F_{q^d}. This gives a uniform way to see why the spectrum separates in these cases. The argument rests on showing that the difference of two such weight vectors produces a base-q number strictly between -(q^d-1) and q^d-1, so it can only be congruent to zero modulo q^d-1 if the vectors are identical. That step is direct and avoids heavier representation-theoretic machinery. The paper uses this to prove the separation and then extends it to shifted exponents under Frobenius. They also introduce a rewriting framework that reduces action reconstruction to an inversion problem once the eigenvalues are labeled this way. The computational experiments demonstrate this in practice, with one example achieving a complete reconstruction from a multiplicity-free tensor product. This approach avoids more advanced tools from algebraic group theory or character theory, sticking to direct linear algebra and number theory. The computational verification helps confirm the framework works in concrete cases. On the downside, the result is confined to K < q-1, which is a real restriction since higher degrees can have collisions. The assumption that the scalar extension is untwisted polynomial tensor is necessary for the weights to behave as expected, but it narrows the applicability. The framework is conditional, so it doesn't give unconditional reconstruction theorems. No major gaps in the logic appear from the setup, though checking the full derivations would be needed to confirm no hidden assumptions in the module restrictions. This paper targets researchers in modular representation theory of finite linear groups and those interested in explicit computations with Singer cycles. Someone studying eigenvalue problems or module reconstructions in this setting could pick up useful techniques here. It deserves peer review. The ideas are accessible, the lemma is elementary, and the examples provide concrete support. A referee could help tighten the scope and check the computational claims.

Referee Report

0 major / 3 minor

Summary. The paper claims that for a prime power q, a subgroup H ≤ GL_d(q) containing a genuine Singer cycle s of order q^d−1, and an F_q H-module W whose scalar extension restricts to an untwisted polynomial tensor representation ⊗ L(λ^{(t)}) of the algebraic group GL_d with total degree K < q−1, distinct weights yield distinct eigenvalues of s on W ⊗_{F_q} F_{q^d}. The proof rests on an elementary base-q injectivity lemma: for distinct weight vectors μ, ν the difference vector a satisfies ∑a_i=0 and |a_i|≤K<q−1, so that ∑a_i q^{i−1} lies strictly inside (−(q^d−1), q^d−1) and congruence to 0 mod (q^d−1) forces a=0. The manuscript also supplies a shifted-exponent formula for q-Frobenius shifts of digit vectors and formulates a conditional rewriting framework that reduces reconstruction of the natural action to a functor-specific inversion problem, with viability shown by computational experiments on multiplicity-free genuine tensor products.

Significance. If the central separation result holds, the work supplies a uniform, number-theoretic explanation for simple spectra of Singer cycles on bounded-degree polynomial tensor representations. This is potentially useful for the representation theory of finite groups of Lie type and for computational reconstruction of actions from eigenvalue data. The elementary character of the base-q lemma together with explicit computational verification of a non-trivial reconstruction constitute clear strengths.

minor comments (3)

The abstract and setup refer to an 'untwisted polynomial tensor representation'; a short clarifying sentence on how the untwisted hypothesis guarantees that the weights are the ordinary ones to which the base-q labelling applies would improve readability.
In the description of the computational experiments, the specific tensor products, the value of d, and the software or libraries employed are not detailed; adding these would strengthen the reproducibility claim.
The shifted-exponent formula is stated for q-Frobenius shifts; a brief remark on the range of shifts for which the same K < q−1 bound continues to guarantee separation would be helpful.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary and recommendation of minor revision. The report accurately captures the main results and strengths of the manuscript. Since no specific major comments or criticisms are raised, we have no points requiring rebuttal or clarification at this stage.

Circularity Check

0 steps flagged

No significant circularity; central claim rests on elementary injectivity lemma

full rationale

The paper establishes spectral separation for distinct weights under total degree K < q-1 by applying a standard base-q uniqueness argument to difference vectors a with sum zero and |a_i| < q-1, showing that the associated integer sum lies strictly inside (-(q^d-1), q^d-1) and hence vanishes modulo q^d-1 only when a = 0. This is an independent number-theoretic fact independent of the representation-theoretic setup. The untwisted polynomial tensor hypothesis simply ensures that the weights in question are the ordinary ones to which the lemma applies, without any self-definition, fitted parameter renamed as prediction, or load-bearing self-citation. The shifted-exponent formula and conditional rewriting framework are derived consequences rather than presuppositions. No equation or step reduces to its own input by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper relies on standard facts about algebraic groups, finite fields, and Singer cycles; the new content is the injectivity lemma and the labelling framework rather than additional free parameters or invented entities.

axioms (2)

domain assumption W is an F_q H-module whose scalar extension restricts to an untwisted polynomial tensor representation of GL_d
This is the explicit setup required for the eigenvalue separation statement.
domain assumption s is a genuine Singer cycle of order q^d - 1 inside H
The theorem is stated for subgroups containing such an element.

pith-pipeline@v0.9.0 · 5583 in / 1408 out tokens · 52362 ms · 2026-05-17T04:07:09.600996+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

7 extracted references · 7 canonical work pages · 1 internal anchor

[1]

P. A. Brooksbank,Constructive recognition of classical groups in their natural representation, J. Symbolic Comput.35(2003), 195–239

work page 2003
[2]

B. P. Corr,A Las Vegas rewriting algorithm for the symmetric square representation of classical groups, Preprint, 2015, Arxiv:1507.05671

work page internal anchor Pith review Pith/arXiv arXiv 2015
[3]

Dietrich, C

H. Dietrich, C. R. Leedham-Green and E. A. O’Brien,Effective black-box constructive recognition of classical groups, J. Algebra421(2015), 460–492

work page 2015
[4]

G¨ ul and N

K. G¨ ul and N. Ankaralıo˘ glu,On the twisted modules for finite matrix groups, Turkish J. Math.40(2016), 191–200

work page 2016
[5]

J. C. Jantzen,Representations of Algebraic Groups, 2nd ed., Mathematical Surveys and Monographs, Vol. 107, American Mathematical Society, Providence, RI, 2003

work page 2003
[6]

L¨ ubeck,Small degree representations of finite Chevalley groups in defining characteristic, LMS J

F. L¨ ubeck,Small degree representations of finite Chevalley groups in defining characteristic, LMS J. Comput. Math.4(2001), 135–169

work page 2001
[7]

Magaard, E

K. Magaard, E. A. O’Brien and ´A. Seress,Recognition of small dimensional representations of general linear groups, J. Aust. Math. Soc.85(2008), 229–250. Department of Mathematics, FPT University, Quy Nhon AI Campus, An Phu Thinh New Urban Area, Quy Nhon City, Binh Dinh, Vietnam Email address:phucdv14@fpt.edu.vn, dangphuc150488@gmail.com 23

work page 2008

[1] [1]

P. A. Brooksbank,Constructive recognition of classical groups in their natural representation, J. Symbolic Comput.35(2003), 195–239

work page 2003

[2] [2]

B. P. Corr,A Las Vegas rewriting algorithm for the symmetric square representation of classical groups, Preprint, 2015, Arxiv:1507.05671

work page internal anchor Pith review Pith/arXiv arXiv 2015

[3] [3]

Dietrich, C

H. Dietrich, C. R. Leedham-Green and E. A. O’Brien,Effective black-box constructive recognition of classical groups, J. Algebra421(2015), 460–492

work page 2015

[4] [4]

G¨ ul and N

K. G¨ ul and N. Ankaralıo˘ glu,On the twisted modules for finite matrix groups, Turkish J. Math.40(2016), 191–200

work page 2016

[5] [5]

J. C. Jantzen,Representations of Algebraic Groups, 2nd ed., Mathematical Surveys and Monographs, Vol. 107, American Mathematical Society, Providence, RI, 2003

work page 2003

[6] [6]

L¨ ubeck,Small degree representations of finite Chevalley groups in defining characteristic, LMS J

F. L¨ ubeck,Small degree representations of finite Chevalley groups in defining characteristic, LMS J. Comput. Math.4(2001), 135–169

work page 2001

[7] [7]

Magaard, E

K. Magaard, E. A. O’Brien and ´A. Seress,Recognition of small dimensional representations of general linear groups, J. Aust. Math. Soc.85(2008), 229–250. Department of Mathematics, FPT University, Quy Nhon AI Campus, An Phu Thinh New Urban Area, Quy Nhon City, Binh Dinh, Vietnam Email address:phucdv14@fpt.edu.vn, dangphuc150488@gmail.com 23

work page 2008