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arxiv: 2605.11988 · v2 · pith:PE4226REnew · submitted 2026-05-12 · 🧮 math.RT · math.GR

Alperin's Main Problem of Block Theory

Alexander Moret\'o This is my paper

Pith reviewed 2026-05-22 10:02 UTC · model grok-4.3

classification 🧮 math.RT math.GR
keywords Alperin conjectureblock theorycharacter theoryMcKay conjectureSylow subgroupssubnormalizernonvanishing characters
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The pith

The right local objects for Alperin's Main Problem are sets of characters non-vanishing at a given element together with its subnormalizer subgroup.

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

This paper proposes a conjectural framework for Alperin's Main Problem of block theory. The character sets considered are defined by nonvanishing at given elements rather than only by degree conditions. The guiding idea is that the right local objects governing character values are the sets Irr^x(G) of irreducible characters not vanishing at a given element x together with the subnormalizer subgroup Sub_G(x). McKay's conjecture is recovered as a first degree-level consequence. The author states the basic conjectures, proposes stronger versions, and verifies the main conjectures in several families including the simple groups with TI Sylow p-subgroups.

Core claim

The right local objects governing character values are not, in general, the sets Irr_{p'}(G) and the normalizers of Sylow p-subgroups, but rather the sets Irr^x(G) of irreducible characters not vanishing at a given element x, together with the subnormalizer subgroup Sub_G(x).

What carries the argument

The sets Irr^x(G) of irreducible characters not vanishing at a given element x, together with the subnormalizer subgroup Sub_G(x).

If this is right

  • McKay's conjecture follows as a first degree-level consequence of the new framework.
  • The main conjectures hold for all simple groups with TI Sylow p-subgroups.
  • This perspective reorganizes several classical questions in character theory.
  • Stronger versions of the conjectures can be stated and checked in the same families.

Where Pith is reading between the lines

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

  • The approach may extend naturally to other local-global principles in representation theory that involve non-vanishing conditions.
  • Explicit computation of Sub_G(x) for small groups could produce new examples that distinguish this framework from the classical one.
  • If the conjectures hold more generally they could suggest a way to reorganize the search for bijections in other block-theoretic problems.

Load-bearing premise

That the proposed sets Irr^x(G) and subgroups Sub_G(x) correctly capture the local data needed to resolve Alperin's Main Problem as opposed to the classical p'-degree and Sylow normalizer approach.

What would settle it

A finite group outside the verified families, such as a solvable group or a quasisimple group with non-TI Sylow p-subgroup, where no bijection exists between the Irr^x(G) sets and the corresponding Irr^y(N) sets for y in the subnormalizer.

read the original abstract

This paper proposes a conjectural framework for Alperin's Main Problem of Block Theory from 1976. The character sets considered here are defined by nonvanishing at given elements, not only by degree conditions. From this point of view, McKay's conjecture is usually recovered as a first degree-level consequence. The guiding idea is that the right local objects governing character values are not, in general, the sets ${\rm Irr}_{p'}(G)$ and the normalizers of Sylow $p$-subgroups, but rather the sets ${\rm Irr}^x(G)$ of irreducible characters not vanishing at a given element $x$, together with the subnormalizer subgroup ${\rm Sub}_G(x)$. I state the basic conjectures of this theory, propose stronger versions, and verify the main conjectures in several families, including the simple groups with TI Sylow $p$-subgroups. I also show how this perspective reorganizes several classical questions in character theory.

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.

Referee Report

1 major / 2 minor

Summary. The manuscript proposes a conjectural reformulation of Alperin's Main Problem of block theory. It posits that the sets Irr^x(G) of irreducible characters non-vanishing at a given element x, together with the subnormalizer Sub_G(x), supply the correct local data governing character values, in place of the classical Irr_{p'}(G) and normalizers of Sylow p-subgroups. Basic and stronger conjectures are stated; the main conjectures are verified for simple groups with TI Sylow p-subgroups; and the perspective is shown to reorganize several classical questions, with McKay's conjecture recovered as a degree-level consequence.

Significance. If the conjectures hold, the framework would supply a refined local-global principle in modular representation theory that potentially resolves or reorganizes Alperin's problem and related questions. The explicit verifications in the TI Sylow case and the reorganization of classical character-theoretic questions constitute concrete contributions even if the general conjectures remain open.

major comments (1)
  1. [Verifications for TI Sylow p-subgroups] Section on verifications for simple groups with TI Sylow p-subgroups: the argument that Irr^x(G) and Sub_G(x) supply the local data relies on the fact that distinct Sylow p-subgroups intersect only at the identity, which forces Sub_G(x) to coincide with N_G(<x>) for p-elements x in many cases and restricts the non-vanishing loci. This geometric simplification is special to the TI setting and does not obviously persist when non-trivial intersections occur; the manuscript therefore provides no direct evidence that the proposed objects capture the required local information in the general case, which is load-bearing for the central claim.
minor comments (2)
  1. [Definitions and notation] The definition of the subnormalizer Sub_G(x) is introduced without an explicit comparison to the normalizer N_G(<x>) or to existing notions of subnormalizers in the literature; a short paragraph clarifying the distinction would improve readability.
  2. [Statement of conjectures] In the statement of the stronger conjectures, the precise relationship between the proposed bijections and the classical Alperin weight conjecture is left implicit; an explicit remark on how the new objects reduce to or differ from the classical ones in the p'-degree case would help readers.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for the constructive observation regarding the scope of the verifications. We address the major comment below.

read point-by-point responses

  1. Referee: Section on verifications for simple groups with TI Sylow p-subgroups: the argument that Irr^x(G) and Sub_G(x) supply the local data relies on the fact that distinct Sylow p-subgroups intersect only at the identity, which forces Sub_G(x) to coincide with N_G(<x>) for p-elements x in many cases and restricts the non-vanishing loci. This geometric simplification is special to the TI setting and does not obviously persist when non-trivial intersections occur; the manuscript therefore provides no direct evidence that the proposed objects capture the required local information in the general case, which is load-bearing for the central claim.

    Authors: We agree that the verifications for simple groups with TI Sylow p-subgroups exploit the trivial intersection property, which causes Sub_G(x) to coincide with N_G(<x>) in many instances and restricts the loci where characters can be non-vanishing. This is a genuine geometric simplification special to the TI case. The conjectures themselves, however, are stated for arbitrary finite groups using the general definitions of Irr^x(G) and Sub_G(x). The TI verifications are presented as an accessible family in which the conjectures can be checked explicitly, not as a substitute for a general proof. We acknowledge that these checks do not furnish direct evidence for groups with non-trivial Sylow intersections. In the revised version we will insert a short paragraph in the introduction and a remark at the end of the verification section that explicitly notes the special character of the TI setting, clarifies the role of these verifications as initial supporting evidence rather than a general demonstration, and invites further checks in non-TI families. This revision makes the scope of the current evidence more transparent without altering the conjectural nature of the main claims. revision: partial

Circularity Check

0 steps flagged

No circularity: conjectural framework with independent verifications

full rationale

The paper states new conjectures replacing Irr_{p'}(G) and N_G(P) with Irr^x(G) and Sub_G(x) for Alperin's Main Problem, then verifies the conjectures in specific families (including simple groups with TI Sylow p-subgroups) and reorganizes classical questions. No load-bearing derivation, equation, or prediction reduces to its own inputs by construction; the central claims are explicit conjectures whose truth is checked externally in the cited families rather than forced by definition or self-referential fitting. Self-citations, if present, are not used to justify uniqueness or to smuggle ansatzes that close the argument.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 2 invented entities

The claims rest on standard background facts from character theory of finite groups together with the new definitions of Irr^x(G) and Sub_G(x) whose validity is conjectural.

axioms (1)
  • domain assumption Standard facts from the character theory of finite groups and block theory hold.
    The framework is built on existing results in representation theory.
invented entities (2)
  • Irr^x(G) no independent evidence
    purpose: Set of irreducible characters nonvanishing at a fixed element x
    Introduced to replace classical degree-based sets as the local object.
  • Sub_G(x) no independent evidence
    purpose: Subnormalizer subgroup associated to element x
    Proposed as the correct local subgroup replacing Sylow normalizers.

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Forward citations

Cited by 1 Pith paper

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

  1. Subnormalizers and character correspondences in $p$-solvable groups

    math.RT 2026-05 unverdicted novelty 7.0

    Establishes the strong subnormalizer conjecture for p-solvable groups with odd p under stated conditions and obtains new Glauberman correspondence results.

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43 extracted references · 43 canonical work pages · cited by 1 Pith paper

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