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Math-2344 · mathematics · Algebra

Alperin Weight Conjecture and Perfect Localities

Status Partially solved · unsolvedmath.comLLM status Open unverified

Statement

Definitions and mathematical statement

Shared notation.
$\mathbb N$, $\mathbb Z$, $\mathbb Q$, $\mathbb R$, and $\mathbb C$ denote the natural numbers, integers, rationals, reals, and complex numbers, respectively. All additional parameters and hypotheses are specified in the question.
Statement recorded in the supplied source.
In modular representation theory, understand and prove Alperin's Weight Conjecture, including the more general form suggested by Frobenius categories; also determine whether a perfect locality (centric linking system) associated with a Frobenius category exists and is unique.

Formulation references: [S1], [S2].

Short English statement

In modular representation theory, understand and prove Alperin's Weight Conjecture, including the more general form suggested by Frobenius categories; also determine whether a perfect locality (centric linking system) associated with a Frobenius category exists and is unique.

Sources

Catalogue entry

Statement licence: CC-BY-SA-4.0 · Source file

Contributions 1

Math-2344-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-2344

Theorem 1.1 proves the numerical blockwise Alperin weight equality for every block. It does not verify the additional automorphism-equivariant inductive condition, Alperin–McKay equality, or independent locality clauses requested in the original bundled question. This entry records partial support; complete coverage of the question is not established by this catalogue.

Alperin Weight Conjecture and Perfect Localities

OpenAI's claim

OpenAI presents a mathematical result relevant to this question in the selected manuscript. For each prime and each block of a finite group, the claimed count of simple modular characters equals the count of block weights up to conjugacy.

Mathematical statement of the claimed result

Result 1.
For a finite group $G$, prime $p$, and $p$-block $B$, let $l(B)$ count its irreducible Brauer characters and $W_p(B)$ its conjugacy classes of block weights $(Q,\phi)$, with $Q$ a $p$-subgroup and $\phi$ a defect-zero character of $N_G(Q)/Q$ inducing the block $B$. The claimed identity is \[ l(B)=|W_p(B)|. \] If $D$ is a defect group and $b$ its Brauer-correspondent block in $N_G(D)$, the corollary gives $l(B)\ge l(b)$, with equality when $D$ is abelian. For $P\in\operatorname{Syl}_p(G)$, it gives $|\operatorname{IBr}_p(G)|\ge|\operatorname{IBr}_p(N_G(P))|$. Writing $B_0$ for the principal block and $k_p(G)$ for the number of $p$-power-order classes including the identity, it also gives \[ k_p(G)=3\Rightarrow l(B_0)\ge(p-1)/2,\qquad p\mid|G|\Rightarrow l(B_0)\ge2\sqrt{p-1}+1-k_p(G). \]

Outline of the argument

The proof organizes alternating cancellations along subgroup chains and detects defect-zero contributions by central traces. A marked genus-one covering construction, valued lifting, and descent provide the uniform arithmetic input needed for the final chain-orbit cancellation.

Formal verification

Family 202 has no Lean scope page in this release. Its selected result is not presented as formalized. This site has not executed the Lean code. The scope descriptions do not certify the remaining manuscript arguments.

Status caveat

OpenAI attributes these manuscripts to work by an internal model. Its release cautions that arguments without formalization can contain mistakes. This catalogue has not independently checked the proofs; the labels report the mathematical claims attributed to OpenAI.

Sources

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