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Math-2371 · mathematics · Group theory

Inductive Blockwise Alperin Weight Condition

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.
Does the inductive blockwise Alperin weight condition hold for every finite simple group and every prime $p$, and hence imply the blockwise Alperin weight conjecture?

Formulation references: [S1], [S2].

Short English statement

Does the inductive blockwise Alperin weight condition hold for every finite simple group and every prime $p$, and hence imply the blockwise Alperin weight conjecture?

Sources

Catalogue entry

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

Contributions 1

Math-2371-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-2371

The global numerical blockwise conclusion in this two-part target is proved for all groups and primes, but the inductive condition for all finite simple groups is not established; the paper expressly avoids the inductive route. This entry records partial support; complete coverage of the question is not established by this catalogue.

Inductive Blockwise Alperin Weight Condition

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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