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REVIEW 3 major objections 4 minor 5 references

Homomorphisms from ${\rm SL}(2, k)$ to ${\rm SL}(4, k)$ in positive characteristic

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Pith's one-line read Every homomorphism from SL(2,k) to SL(4,k) is conjugate to exactly one listed explicit matrix family, with seven families in characteristic 2, seven in 3, and six for p≥5.

desk verdict A likely-correct, genuinely new n=4 classification of SL(2,k) homomorphisms in positive characteristic, but the completeness argument leans on two structural facts that the paper imports or asserts without proof. read the letter →

arxiv 2506.00423 v1 pith:FNZSSI34 submitted 2025-05-31 math.RT

Ryuji Tanimoto This is my paper
classification math.RT MSC 15A2115A54
keywords homomorphismsoflinearalgebraicgroupspositivecharacteristicSL(2k)representationsSL(4conjugationequivalenceindecomposabledecompositionsantisymmetricFrobeniustwists
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper aims to describe, up to conjugation by $\mathrm{GL}(4,k)$, every homomorphism from $\mathrm{SL}(2,k)$ to $\mathrm{SL}(4,k)$ over an algebraically closed field $k$ of positive characteristic $p$. Its central claim is that the conjugacy classes are in natural one-to-one correspondence with an explicit union of parameter sets: seven families when $p=2$, seven when $p=3$, and six when $p\geq 5$, with four of those families present in every characteristic. Each family is an explicit rational matrix whose entries are monomials in the entries of a $2\times 2$ matrix, with free parameters that are nonnegative integers recording Frobenius twists. The paper also gives the indecomposable decomposition of every four-dimensional representation that arises, so the result is not just a list of classes but a description of the representation theory. The reason to care is that in positive characteristic $\mathrm{SL}(2,k)$ has indecomposable representations that are not irreducible, and this theorem names all of them in dimension four.

What carries the argument

The load-bearing object is the restriction of a homomorphism to the Borel subgroup $B(2,k)=G_a\rtimes G_m$ of upper-triangular matrices. Because $\mathrm{SL}(2,k)$ is generated by $U^+$, $T$, and $U^-$, a homomorphism is determined by three pieces: $\varphi(t)=\sigma\bigl(\begin{smallmatrix}1&t\\0&1\end{smallmatrix}\bigr)$, $\omega(u)=\sigma\bigl(\begin{smallmatrix}u&0\\0&u^{-1}\end{smallmatrix}\bigr)$, and $\phi^-(s)=\sigma\bigl(\begin{smallmatrix}1&0\\s&1\end{smallmatrix}\bigr)$. The paper first classifies all antisymmetric Borel homomorphisms into twenty-six explicit pairs $(\varphi_*,\omega_*)$, labelled (I)--(XXVI), using the criterion that $\psi_{\varphi,\omega}$ is a homomorphism exactly when $\omega(u)\varphi(t)\omega(u)^{-1}=\varphi(u^2t)$ and $\varphi$ has a prescribed upper-triangular unipotent form. It then decides extendability to all of $\mathrm{SL}(2,k)$ by solving the matrix identity $$\varphi(t)\phi^-(s)=\phi^-\Bigl(\tfrac{s}{1+ts}\Bigr)\omega(1+ts)\varphi\Bigl(\tfrac{t}{1+ts}\Bigr)$$ for a lower-triangular unipotent $\phi^-$; for each of the twenty-six forms this yields either a unique solution or a contradiction. The unique solutions assemble into the families $(\nu)_*$ of Theorem 5.26, and the final classification removes overlaps by comparing the invariant $d(\sigma)=(\dim V(4)^\sigma,\dim W(4)^\sigma)$ of fixed column and row vectors. The central identity doing the work is the Bruhat-style factorization $$\begin{pmatrix}1&t\\0&1\end{pmatrix}\begin{pmatrix}1&0\\s&1\end{pmatrix} = \begin{pmatrix}1&0\\ \frac{s}{1+ts}&1\end{pmatrix} \begin{pmatrix}1+ts&0\\0&(1+ts)^{-1}\end{pmatrix} \begin{pmatrix}1&\frac{t}{1+ts}\\0&1\end{pmatrix},$$ which turns the requirement that $\sigma$ be a homomorphism into the displayed equation for $\phi^-$.

What would settle it

Look for a homomorphism $\sigma:\mathrm{SL}(2,k)\to\mathrm{SL}(4,k)$ whose restriction to the diagonal torus has weight multiset other than $\{d_1,d_2,-d_2,-d_1\}$, for example $\omega=\operatorname{diag}(u^2,u,1,1)$. Conjugation preserves the multiset of torus weights, so such a homomorphism could not be equivalent to any antisymmetric one; finding one would disprove Lemma 1.20(1) and therefore Theorem 6.26. The paper's claim predicts that the extension equations of Section 4 have no solution for any non-antisymmetric $\omega$.

Watch

Extended reading notes

Core claim

On the paper's own terms, the discovery is Theorem 6.26: writing $\operatorname{Hom}(\mathrm{SL}(2,k),\mathrm{SL}(4,k))/{\sim}$ for conjugation classes, there are natural bijections $\operatorname{Hom}(\mathrm{SL}(2,k),\mathrm{SL}(4,k))/{\sim}\cong \mathrm{S(IV)}\sqcup\mathrm{S(V)}\sqcup\mathrm{S(XI)}\sqcup\mathrm{S(XV)}\sqcup\mathrm{S(XIX)}\sqcup\mathrm{S(XXIV)}\sqcup\mathrm{S(XXVI)}$ for $p=2$, the seven-term union $\mathrm{S(II)}\sqcup\mathrm{S(IV)}\sqcup\mathrm{S(VII)}\sqcup\mathrm{S(IX)}\sqcup\mathrm{S(XV)}\sqcup\mathrm{S(XXIV)}\sqcup\mathrm{S(XXVI)}$ for $p=3$, and the six-term union $\mathrm{S(I)}\sqcup\mathrm{S(IV)}\sqcup\mathrm{S(IX)}\sqcup\mathrm{S(XV)}\sqcup\mathrm{S(XXIV)}\sqcup\mathrm{S(XXVI)}$ for $p\geq 5$, where each $\mathrm{S}(\nu)$ is the set of classes of an explicit matrix family $(\nu)^\sharp$ and the unions are disjoint. In words: every homomorphism is conjugate to exactly one listed matrix family, no two listed families of the correct characteristic are conjugate, and the parameters inside a family are uniquely determined. The proof first classifies antisymmetric homomorphisms, meaning those whose diagonal-torus restriction has weight tuple $(d_1,d_2,-d_2,-d_1)$, and then uses a cited lemma to reduce every homomorphism to this subclass. Section 7 then gives the indecomposable decomposition of each class, identifying which four-dimensional representations split as direct sums of the one-, two-, and three-dimensional indecomposables catalogued there.

Load-bearing premise

The reduction that every homomorphism $\mathrm{SL}(2,k)\to\mathrm{SL}(4,k)$ is conjugate to an antisymmetric one is imported from a cited lemma rather than proved here; if that lemma failed, Theorem 6.26 would only classify the antisymmetric subclass, not all homomorphisms.

Editorial extensions

If this is right

  • Any four-dimensional representation of $\mathrm{SL}(2,k)$ in positive characteristic is equivalent to one of the listed matrix families, so questions about such representations can be reduced to checking the explicit families $(\nu)^\sharp$.
  • In each characteristic the classification is countably infinite, parameterized by pairs $(e_1,e_2)$ with $e_2>e_1\ge0$, single nonnegative integers, pairs $(e_2,e_3)$ with $e_2\ge e_3\ge0$, and the trivial class.
  • Section 7's decompositions show that only a few four-dimensional classes are decomposable: types (XI), (XV), (XIX), (XXIV), and (XXVI) split as direct sums of lower-dimensional indecomposables, while the remaining types are indecomposable.
  • The invariant $d(\sigma)$ separates the families, giving a quick way to recognize which family a concrete representation belongs to by computing fixed column and row vectors.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • A consequence the author leaves implicit: the same restriction-to-Borel and solve-for-$\phi^-$ scheme should give a comparable description for $\mathrm{SL}(2,k)\to\mathrm{SL}(n,k)$ with $n\ge5$, with the main labour being the combinatorial explosion of ordered partitions and of small-characteristic coincidences among $p$-powers.
  • The four characteristic-independent families (IV, XV, XXIV, XXVI) are exactly those obtainable from Frobenius twists of the standard two-dimensional representation by tensor product and direct sum, whereas the characteristic-dependent families are those whose formulas contain coefficients such as $1/2$, $1/3$, or $1/6$, or require equalities like $2=p^f$ that only hold in restricted characteristic
  • A direct computational check in a fixed small characteristic could evaluate the explicit matrix families and verify the claimed disjointness and parameter uniqueness, providing independent confirmation of Theorem 6.26 without repeating the case analysis.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 4 minor

Summary. The paper classifies, up to GL(4,k)-conjugation, all homomorphisms SL(2,k) → SL(4,k) over an algebraically closed field k of positive characteristic p. The strategy is to reduce to antisymmetric homomorphisms (Lemma 1.20(1), imported from [5]), classify antisymmetric homomorphisms of the Borel subgroup into 26 explicit matrix families (Theorem 3.1), determine which of these extend to SL(2,k) by solving for the lower-triangular U−-image φ− (Section 4), and then assemble the resulting normal forms into the final classification (Theorem 6.26). The output for p=2, p=3, and p≥5 is an explicit disjoint union of parameter sets, with each equivalence class represented by an explicit matrix family, and Section 7 records the indecomposable decomposition of each corresponding SL(2,k)-module.

Significance. If the completeness argument is correct, this is a substantial and very explicit classification in a difficult positive-characteristic setting. The paper provides a concrete answer to a natural problem, gives normal forms that are directly checkable, and computes the invariant d(σ)=(dim fixed vectors, dim fixed covectors) that separates many types. The link to the author's earlier n=3 classification and to fundamental representations of Ga gives the work context and potential applications. The main positive features are the explicit nature of the classification and the use of a small numerical invariant (d(σ)) to prove pairwise non-equivalence, which is a sensible and falsifiable structural tool. The central weakness is that two load-bearing completeness steps are imported or asserted rather than proved inside the paper: the reduction to antisymmetric homomorphisms and the lower-triangularity of the U−-image used throughout Section 4.1.

major comments (3)
  1. [§1.4.2, Lemma 1.20(1)] The reduction Hom(SL(2,k),SL(n,k))/∼ ≅ Homa(SL(2,k),SL(n,k))/∼ is stated without proof and deferred to [5, Lemma 2.6]. This bijection is used in the proof of Theorem 5.26 and hence is the first step of the completeness claim in Theorem 6.26. If this lemma failed, the classification would cover only the antisymmetric subclass. The manuscript should either include the proof (e.g., by ordering a basis of weight vectors by decreasing weight so that T is diagonal, U+ is strictly upper triangular, and U− is strictly lower triangular) or at least restate the lemma in sufficient detail that the reader can verify the normalization is compatible with the definition of antisymmetric homomorphism used here.
  2. [§4.1, first paragraph] Condition (i) asserts that for an extendable antisymmetric homomorphism ψ : B(2,k) → SL(4,k), the induced homomorphism φ−(s)=σ∗((1 0; s 1)) is lower triangular for every s. No proof is given, and every lemma in Section 4.1 uses this lower-triangularity to compute φ− uniquely or to derive a contradiction. If this property ever failed, the extendability analysis would be incomplete and Theorem 6.26 could omit genuine homomorphisms. A short proof should be supplied: for an antisymmetric σ∗, choose a basis of T-weight vectors ordered by decreasing weights; then U+ is strictly upper triangular and U− is strictly lower triangular, so the restriction to B is upper triangular and the images of U− are lower triangular. This argument belongs in the manuscript because the completeness of the classification rests on it.
  3. [§5.1, Lemmas 5.2, 5.4, 5.8, 5.10, 5.12, 5.14, 5.16, 5.18, 5.20, 5.22] For each normal form, the assertion that the displayed matrix formula defines a homomorphism is stated as 'straightforward'. Since the lists in Theorem 6.26 consist exactly of these explicit families, the fact that each σ+ (and hence σ∗) is a homomorphism is load-bearing. I recommend including a representative verification of the homomorphism property for at least one family, together with a sentence explaining that the remaining families follow by the same entrywise polynomial identities after applying the Frobenius twist and the given permutation matrices. This would make the correctness of the output independently checkable.
minor comments (4)
  1. [Introduction, 'In advance' paragraph] The sentence 'write ψ for a representative of the class. So, ψ ∈ Homa(SL(2,k), SL(4,k))' should read Homa(B(2,k), SL(4,k)), since ψ is a homomorphism from the Borel subgroup, not from SL(2,k).
  2. [Lemma 7.1(2)] The statement says 'homomorphism σ : SL(2,k) → SL(3,k)' but the context is n=2; it should be SL(2,k) → SL(2,k). Moreover, the proof is dismissed as 'an exercise to the reader'; since this two-dimensional classification is used in Theorem 7.4, a proof or a precise citation should be provided.
  3. [Throughout] There are numerous typographical errors, including 'characteistic', 'positvie', 'interetsted', 'reguar', 'antisymmetirc', 'partision', 'separting', 'contradition', and 'straightforwad'. A careful proofreading pass is needed.
  4. [Section 4.1] In several lemmas (e.g., Lemma 4.1), the text says 'Comparing the (i,j)-th entries ... we have ...' without listing the full matrix equality in the surrounding text; while the computations are plausible, including the full equality for one representative case would help the reader follow the method.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the classification is derived from structural lemmas and case analysis, not from assuming its conclusion; self-citations are external building blocks.

full rationale

The derivation chain is self-contained and does not reduce to its own inputs. The paper first reduces to antisymmetric homomorphisms via Lemma 1.20(1), whose proof is cited to the published prior work [5, Lemma 2.6]; this is a general structural fact about finite-dimensional rational SL(2,k)-modules, not a restatement of the target classification. It then classifies antisymmetric Borel homomorphisms in Theorem 3.1 by an independent case analysis using the homomorphism criterion (Lemma 1.9) and the monomial-entry lemma (Lemma 3.3), neither of which assumes the final list. Section 4 determines, for each of the 26 Borel forms, the unique possible U^- image or a contradiction; the asserted lower-triangularity of phi^- is a standard consequence of the Bruhat decomposition in a weight-ordered basis, not a fitted input. The final theorems 5.26 and 6.26 assemble these independent computations, and the non-equivalence arguments in Section 6 use invariants such as d(sigma) and the T-weight data. Self-citations to [3] and [5] appear, but they supply external building blocks (exponential-matrix normal forms, the n=3 classification, and general structural lemmas) rather than the conclusion of this paper. No equation or parameter is fitted to the claimed output, and no central premise is justified solely by an unverified self-citation. At most, some structural lemmas are deferred or stated without proof, which is a proof-completeness concern, not a circularity concern.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The central classification uses standard algebraic-group machinery and prior published results by the same author; no new particles or empirical constants are introduced. The main unpaid debts are the cited lemmas from [5] and the unproved n=2 classification.

assumptions (4)
  • domain assumption Every homomorphism sigma: SL(2,k) to SL(n,k) is equivalent to an antisymmetric homomorphism (Lemma 1.20(1)).
    Proof is cited to [5, Lemma 2.6] and not reproduced; the reduction Homa/sim to Hom/sim is used in Theorem 5.26 to restrict to antisymmetric forms.
  • domain assumption For an antisymmetric B(2,k)-homomorphism, the induced Ga-homomorphism lies in U_n (Lemma 1.10).
    Cited to [5, Lemma 2.5]; it underlies the partition of U4 used in Theorem 3.1(2).
  • domain assumption The only extension of the identity form (XXVI) is the identity (Lemma 4.26).
    Proof is cited to [5, Lemma 2.9]; this boundary case is needed for Theorem 5.25(2).
  • domain assumption The n=2 and n=3 classification results used in Section 7 are valid (Lemma 7.1(2)-(3)).
    The n=2 assertion is left as an exercise to the reader and the n=3 assertion is cited to [5, Section 4]; the indecomposable decompositions for n=4 depend on these building blocks.

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Pith. "Pith review of Homomorphisms from ${\rm SL}(2, k)$ to ${\rm SL}(4, k)$ in positive characteristic." pith.science (2026). https://pith.science/paper/FNZSSI34

@misc{pith2026250600423,
  author       = {Pith},
  title        = {Pith review of: Homomorphisms from $\rm SL(2, k)$ to $\rm SL(4, k)$ in positive characteristic},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/FNZSSI34}},
  note         = {Machine review of arXiv:2506.00423}
}
abstract

Let $k$ be an algebraically closed field of positive characteistic $p$ and let ${\rm SL}(n, k)$ denote the special linear algebraic group of degree $n$ over $k$. In this paper, we describe homomorphisms from ${\rm SL}(2, k)$ to ${\rm SL}(4, k)$. As by-products of this description, we give a classification of homomorphisms from ${\rm SL}(2, k)$ to ${\rm SL}(4, k)$ and describe the indecomposable decompositions of homomorphisms from ${\rm SL}(2, k)$ to ${\rm SL}(4, k)$.

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Works this paper leans on

5 extracted references · 5 canonical work pages

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