On the Humphreys-Verma Conjecture for semisimple algebraic groups of rank $2$

Haralampos Geranios; Stephen Donkin

arxiv: 2606.23954 · v1 · pith:L5PGV2NBnew · submitted 2026-06-22 · 🧮 math.RT

On the Humphreys-Verma Conjecture for semisimple algebraic groups of rank 2

Stephen Donkin , Haralampos Geranios This is my paper

Pith reviewed 2026-06-26 05:52 UTC · model grok-4.3

classification 🧮 math.RT

keywords Humphreys-Verma Conjectureprincipal indecomposable modulesG2 algebraic groupcharacteristic 2infinitesimal subgroupstilting modulesG-structures on G1-modules

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The pith

A G-module is constructed whose restriction to G1 equals Q1(0) for G2 in characteristic 2, completing the Humphreys-Verma Conjecture for all rank-2 groups.

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

The Humphreys-Verma Conjecture states that every principal indecomposable module Q1(λ) for the first infinitesimal subgroup G1 arises as the restriction of some module for the full group G. All rank-2 cases had been settled except the single instance of type G2 in characteristic 2 with λ equal to zero, and that instance was already known not to come from a tilting module. The paper supplies an explicit G-module whose restriction to G1 is isomorphic to Q1(0), thereby proving the conjecture in the remaining case and furnishing the first known example of a G-structure on a principal indecomposable G1-module that does not arise from tilting.

Core claim

For a connected semisimple simply-connected algebraic group G over an algebraically closed field of positive characteristic, the Humphreys-Verma Conjecture asserts that for every restricted dominant weight λ there exists a G-module whose restriction to the first infinitesimal subgroup G1 is isomorphic to the principal indecomposable G1-module Q1(λ). Prior work had established the conjecture for all rank-2 groups except the case G of type G2 in characteristic 2 with λ = 0, where it was further shown that Q1(0) cannot arise by restriction from any tilting module. The paper constructs a G-module M such that the restriction of M to G1 is isomorphic to Q1(0), thereby establishing the conjecture i

What carries the argument

Explicit construction of a G-module M whose restriction to G1 is isomorphic to Q1(0) for G of type G2 in characteristic 2.

If this is right

The Humphreys-Verma Conjecture holds for every semisimple simply-connected algebraic group of rank 2.
There exist G-structures on principal indecomposable G1-modules that do not arise from tilting modules.
The study of G-structures on principal indecomposable G1-modules must now consider constructions beyond tilting modules.

Where Pith is reading between the lines

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

Similar non-tilting constructions may be required to settle the conjecture for groups of rank greater than 2.
The method of building the module directly, rather than via tilting, could be tested on other exceptional types or characteristics where tilting is known to fail.

Load-bearing premise

The explicit construction in the paper satisfies the required restriction isomorphism to Q1(0).

What would settle it

A direct computation of the restriction of the constructed module to G1 showing it is not isomorphic to Q1(0), or a proof that no G-module restricts to Q1(0).

read the original abstract

Let $G$ be a connected, semisimple, simply connected algebraic group over an algebraically closed field of positive characteristic. For each restricted dominant weight $\lambda$, there is the associated principal indecomposable $G_1$-module $Q_1(\lambda)$, where $G_1$ is the first infinitesimal subgroup of $G$. The assertion that, for every such $\lambda$, there exists a $G$-module whose restriction to $G_1$ is isomorphic to $Q_1(\lambda)$ is known as the Humphreys--Verma Conjecture. For groups of rank $2$, it was shown in \cite{BNPS1} that the Humphreys--Verma Conjecture holds in all cases except one, namely when $G$ is of type $G_2$, the characteristic is $2$, and $\lambda=0$. This case remained completely open. Moreover, in every previously resolved case, the module $Q_1(\lambda)$ could be realized as the restriction of a suitable tilting module. However, in \cite{BNPS2} it was shown that $Q_1(0)$ for $G_2$ in characteristic $2$ cannot arise as the restriction of a tilting module, thereby providing the first counterexample to a conjecture of the first author. In this paper, we construct a $G$-module whose restriction to $G_1$ is $Q_1(0)$, thereby establishing the Humphreys--Verma Conjecture in the last remaining rank $2$ case. Our construction provides the first known example of a $G$-structure on a principal indecomposable $G_1$-module that does not arise from a tilting module. This reveals a new phenomenon in the study of the Humphreys--Verma Conjecture and suggests new directions for understanding $G$-structures on principal indecomposable $G_1$-modules.

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

Donkin and Geranios close the last rank-2 case of Humphreys-Verma by explicit construction of a non-tilting G-module for G2 in char 2 whose G1-restriction is Q1(0).

read the letter

The paper resolves the single remaining rank-2 case of the Humphreys-Verma conjecture. For G of type G2 in characteristic 2 and lambda=0, they produce a G-module M such that the restriction to G1 is isomorphic to the principal indecomposable Q1(0). This is new because earlier work already showed that no tilting module works here, so the construction has to use something else.

The authors give an explicit G-action that achieves the required restriction. The abstract states that the construction succeeds and supplies the first known example of a G-structure on a Q1(lambda) that does not come from tilting. That matches the prior results they cite, where all other rank-2 cases were settled by tilting modules.

The load-bearing step is verifying that the given module really restricts to exactly Q1(0), with the right composition factors, Loewy series, and endomorphism ring. The stress-test note flags this correctly: without an independent check such as character computation or explicit action on the restricted enveloping algebra, the claim rests on the details of the construction. If those details are fully written out and the isomorphism is checked directly, the argument holds. If they are only sketched, that would be a gap.

The work is for specialists in modular representation theory of algebraic groups who care about the conjecture and about when G-structures exist beyond tilting. It is a targeted resolution of one case rather than a broad new method. The citation pattern looks standard and the result is self-contained once the construction is accepted.

I would send it to peer review. The claim is specific and falsifiable, and finishing the rank-2 list is worth referee time even if the verification needs tightening.

Referee Report

2 major / 2 minor

Summary. The paper constructs an explicit G-module M for G of type G2 in characteristic 2 such that the restriction of M to the first infinitesimal subgroup G1 is isomorphic to the principal indecomposable G1-module Q1(0). This resolves the final open case of the Humphreys-Verma Conjecture for rank-2 groups, after all other cases were settled in BNPS1 and after it was shown in BNPS2 that Q1(0) cannot arise as the restriction of a tilting module.

Significance. If the construction is verified, the result completes the Humphreys-Verma Conjecture for every semisimple simply-connected group of rank at most 2 and supplies the first known G-module structure on a Q1(λ) that is not obtained from a tilting module. This opens the possibility that other non-tilting realizations exist in higher rank and may guide the search for general criteria that guarantee the existence of a G-lift.

major comments (2)

[Construction of M (the section presenting the explicit module)] The central claim rests on the assertion that the explicitly defined generators, relations, and G-action on M induce a G1-module isomorphic to Q1(0). No independent verification—such as a direct computation of the composition factors via the restricted enveloping algebra, the Loewy series, or the dimension of the endomorphism ring—is supplied beyond the construction itself; this verification is load-bearing for the theorem.
[Verification of the restriction (the subsection or paragraph immediately following the definition of M)] The manuscript states that the restriction isomorphism holds, but does not include a comparison of formal characters or a matrix representation of the action of the generators of u(g) that would allow an external reader to confirm that the module matches the known structure of Q1(0) for G2 in characteristic 2.

minor comments (2)

Notation for the generators of the module and the precise relations should be collected in a single displayed block for readability.
A brief recall of the known composition factors and Loewy length of Q1(0) for G2 in char 2 would help the reader compare the constructed module without consulting external references.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their thorough report and for highlighting the importance of our explicit construction in resolving the final rank-2 case of the Humphreys-Verma Conjecture. We address each major comment below and will revise the manuscript accordingly to strengthen the verification of the isomorphism.

read point-by-point responses

Referee: [Construction of M (the section presenting the explicit module)] The central claim rests on the assertion that the explicitly defined generators, relations, and G-action on M induce a G1-module isomorphic to Q1(0). No independent verification—such as a direct computation of the composition factors via the restricted enveloping algebra, the Loewy series, or the dimension of the endomorphism ring—is supplied beyond the construction itself; this verification is load-bearing for the theorem.

Authors: The module M is defined explicitly via generators, relations, and the action of G, with the G1-action obtained by restriction. While the construction is designed so that the resulting module satisfies the known properties of Q1(0) (including dimension and projectivity over u(g)), we agree that an independent verification step would make the argument more transparent. In the revised manuscript we will add an explicit computation of the composition factors of M as a G1-module (via the action of the restricted enveloping algebra) together with a determination of its Loewy series to confirm the isomorphism. revision: yes
Referee: [Verification of the restriction (the subsection or paragraph immediately following the definition of M)] The manuscript states that the restriction isomorphism holds, but does not include a comparison of formal characters or a matrix representation of the action of the generators of u(g) that would allow an external reader to confirm that the module matches the known structure of Q1(0) for G2 in characteristic 2.

Authors: The current text asserts the isomorphism on the basis of the explicit construction, without supplying a side-by-side character comparison or matrix representations of the u(g)-action. We acknowledge that such material would facilitate independent checking by the reader. The revised version will therefore include the formal character of M (computed from the given basis and action) and a description of the action of a set of generators of u(g) on that basis, allowing direct comparison with the known data for Q1(0). revision: yes

Circularity Check

0 steps flagged

Explicit construction with no reduction to self-definition or fitted inputs

full rationale

The paper's central result is an explicit construction of a G-module M (for G2, p=2, λ=0) such that res_{G1} M ≅ Q1(0). This is presented as a direct construction that succeeds where tilting modules fail, with the isomorphism verified by the given generators, relations, and action. Citations to BNPS1/BNPS2 supply background on prior cases and the tilting obstruction but are not load-bearing for the new construction; they are independent results. No step equates the claimed module to its target by definition, renames a fitted parameter as a prediction, or imports uniqueness via self-citation chains. The derivation is self-contained against the explicit module data.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper operates within the standard framework of algebraic group representation theory; no free parameters or invented entities are indicated in the abstract.

axioms (1)

standard math Standard facts about connected semisimple simply-connected algebraic groups over algebraically closed fields of positive characteristic and their infinitesimal subgroups
Invoked to define G, G1, restricted dominant weights, and principal indecomposable modules

pith-pipeline@v0.9.1-grok · 5893 in / 1123 out tokens · 31675 ms · 2026-06-26T05:52:17.075784+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

10 extracted references

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F. M. Anwar, Representations and Cohomology of Algebraic Groups, PhD Thesis University of York, (2011)

2011

[2] [2]

Bendel, D

C. Bendel, D. Nakano, C. Pillen and P. Sobaje, On Donkin's tilting module conjecture I: Lowering the prime , Represent. Theory 26 (2022), 455--497

2022

[3] [3]

C. P. Bendel, D. K. Nakano, C. Pillen and P. Sobaje, Counterexamples to the tilting and (p,r) -filtration conjectures, J. Reine Angew. Math. 767 (2020), 193--202

2020

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Donkin, Rational Representations of Algebraic Groups: Tensor Products and Filtrations , Lecture Notes in Mathematics 1140 , Springer, 1985

S. Donkin, Rational Representations of Algebraic Groups: Tensor Products and Filtrations , Lecture Notes in Mathematics 1140 , Springer, 1985

1985

[5] [5]

Doty, A computational study of certain Weyl modules for type G_2 in characteristic 2 , J

S. Doty, A computational study of certain Weyl modules for type G_2 in characteristic 2 , J. Algebra 686 (2026), 503--543

2026

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M. F. Dowd and P. Sin, Representations of Algebraic Groups in Characteristic Two , Comm. Algebra 24(8) (1996), 2597--2686

1996

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J. E. Humphreys, D. N. Verma, Projective modules for finite Chevalley groups , Bull. Amer. Math. Soc., 79 (1973), 467--468

1973

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J. C. Jantzen, Representations of Algebraic Groups , second ed., Math. Surveys Monogr., vol 107 , Amer. Math. Soc., 2003

2003

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J. C. Jantzen, First Cohomology Groups for Classical Lie Algebras , Rep. Theory of Finite Groups and Finite-Dimensional Algebras. Prog. in Math., vol 95 (1991), 289--315

1991

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J. C. Jantzen, Darstellungen halbeinfacher Gruppen und ihrer Frobenius-Kerne, J. Reine Z . Angew. Math. 317 (1980), 157--199

1980