Alperins Weight Conjecture for Modular Representations of Finite Groups

Alperins Weight Conjecture for Modular Representations of Finite Groups

OpenAI's claim

OpenAI presents a proof 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

• OpenAI, The Blockwise Alperin Weight Conjecture, 2026-09-23; citation key OAI:The-Blockwise-Alperin-Weight-Conjecture-September-23-2026. PDF (GitHub); PDF (direct); Manuscript page and citation; Latest version; Statement source: thm:main; Statement source: cor:block-consequences.
• OpenAI catalogue entry.

Alperins Weight Conjecture for Modular Representations of Finite Groups

Definitions and mathematical statement

Let $B$ be a $p$-block of a finite group $G$ over a splitting field of characteristic $p$, and let $\ell(B)$ be the number of isomorphism classes of its simple modules. A weight is a pair $(Q,\varphi)$ where $Q$ is a $p$-subgroup and $\varphi$ is an irreducible ordinary character of $N_G(Q)/Q$ of $p$-defect zero, meaning its degree has the full $p$-part of the group order. It is a $B$-weight when the block containing the inflated character corresponds to $B$ under the block-induction convention in the cited source. The assertion is \[ \ell(B)=\#\{G\text{-conjugacy classes of }B\text{-weights}\}. \] The local block-association condition is essential; counting all weights of $G$ instead would discard the selected blockwise statement. The conjecture asks for equality of finite numbers, not a specified canonical bijection.

Formulation and concept references: [S1].

Short English statement

Does each block have exactly as many simple modular representations as conjugacy classes of its associated local weights?

Research attempt: explicit blockwise counts and their limitations

Status and source check. The general equality in the statement is not proved here. A very recent primary manuscript must be distinguished from an unchanged literature frontier: Zhang's arXiv submission dated 21 September 2026 [E1] claims the inductive blockwise Alperin weight condition for simple groups of types $\mathsf B$ and $\mathsf C$, and treats remaining sporadic cases. Its abstract gives consequences for groups with the specified nonabelian simple sections. It does not announce the selected conjecture for every finite group and every block. This attempt neither verifies that long manuscript nor extends its asserted scope.

We work over an algebraically closed field $k$ of characteristic $p$, which is a splitting field for the finite groups used below. The arguments are unchanged after choosing a sufficiently large splitting field. Ordinary characters in a weight are characteristic-zero characters; they are not the simple $kG$-modules counted by $\ell(B)$. A defect-zero ordinary irreducible character $\varphi$ of a group $L$ must satisfy \[ \varphi(1)_p=|L|_p, \] where the subscript denotes the $p$-part. This degree condition is especially effective in the examples below.

Lemma 342.1 (normal $p$-subgroups on simple modules). If $R\lhd G$ is a $p$-subgroup and $M$ is a nonzero simple $kG$-module, then $R$ acts trivially on $M$.

Proof. First every nonzero finite-dimensional module for a finite $p$-group has a nonzero fixed vector. Induct on its order. A nontrivial $p$-group has a central element $z$ of order $p$. The operator $z-1$ satisfies $(z-1)^p=z^p-1=0$, so its kernel is nonzero and is stable under the whole group. On this kernel the action factors through the smaller group obtained by quotienting by $\langle z\rangle$, to which induction applies. Thus $M^R\ne0$. Normality makes $M^R$ a $G$-submodule: for $r\in R$, $g\in G$, and $v\in M^R$, one has $r(gv)=g(g^{-1}rg)v=gv$. Simplicity now gives $M^R=M$. $\square$

In particular a $p$-group $P$ has just one simple module, the trivial module. Since each block of a finite-dimensional algebra contains a simple module, $kP$ has a single block and its value of $\ell$ is one.

Lemma 342.2 (the complete $p$-group case). If $G=P$ is a finite $p$-group, its only weight up to conjugacy is $(P,1)$.

Proof. Suppose $Q<P$. Let $Q$ act by left multiplication on $P/Q$. Nontrivial orbits have cardinality divisible by $p$. Since $|P:Q|$ is divisible by $p$, the number of fixed cosets is also divisible by $p$. It is positive because $Q$ itself is fixed, and the fixed cosets are exactly $N_P(Q)/Q$. Hence this quotient is a nontrivial $p$-group. Any of its ordinary irreducible degrees $d$ satisfies $d^2\leq|N_P(Q)/Q|$, by the sum-of-squares formula for character degrees. Defect zero would require the $p$-part of $d$ to equal the whole order of this nontrivial $p$-group, forcing $d\geq|N_P(Q)/Q|$, a contradiction. Thus $Q<P$ gives no weight. For $Q=P$ the quotient is trivial and has its unique defect-zero character. The single block has one simple module and one associated weight. $\square$

This proof uses the normalizer quotient, rather than merely counting $p$-subgroups. A group can have many proper $p$-subgroups while all of them are excluded by the degree condition.

A block-by-block calculation for all finite abelian groups. Let $G=P\times A$, where $P$ is an abelian $p$-group and $A$ has order prime to $p$. Every irreducible character $\chi$ of $A$ has degree one. Over a splitting field, the central idempotents \[ e_\chi=\frac1{|A|}\sum_{a\in A}\chi(a)^{-1}a \] are pairwise orthogonal and sum to one. Their orthogonality follows by summing a nontrivial character over $A$: if its value at $b$ is not one, multiplying the sum by that value leaves the sum unchanged and forces it to vanish. Consequently \[ kG\cong\prod_{\chi\in\widehat A}kP e_\chi \cong\prod_{\chi\in\widehat A}kP. \tag{342.1} \] Each factor is a block with one simple module.

All $p$-subgroups $Q$ of $G$ lie in $P$, and $N_G(Q)=G$. The ordinary irreducible characters of \[ G/Q=(P/Q)\times A \] have degree one. Such a character has defect zero precisely when $P/Q=1$. Thus the weights are $(P,\chi)$, indexed by $\widehat A$. Because the normalizer is the whole group, the block associated with this weight is exactly the block selected by $e_\chi$. There is one weight for each block, agreeing with $342.1$.

This example makes the word "blockwise" concrete. For $G=C_3\times C_2$ at $p=3$, there are two weights in total and two blocks, each with one simple module. Comparing the total of two weights with the simple-module count of either single block would produce a false apparent counterexample. The block association cannot be discarded.

A slight extension is also transparent. If $P$ is any finite $p$-group and $A$ is any $p'$-group, then the splitting semisimple algebra $kA$ is a product of matrix algebras. Accordingly the blocks of $k(P\times A)$ are matrix algebras over $kP$, each with one simple module. A weight with $Q<P$ is again excluded by the nontrivial normalizer $p$-quotient, using the same degree estimate for its $p$-group character factor; weights with $Q=P$ are indexed by the ordinary irreducible characters of $A$. The matching of these factors gives the equality for each block of this direct product.

A nonabelian example where normal-subgroup reduction can mislead. Take $G=S_3$ and $p=3$, with $R=A_3\cong C_3$. Lemma 342.1 says every simple module factors through $G/R\cong C_2$. Thus there are two simple modules, the trivial and sign modules. Nevertheless these do not lie in two separate blocks of $kS_3$.

To see this directly, consider the two-dimensional augmentation submodule \[ W=\{(x_1,x_2,x_3)\in k^3:x_1+x_2+x_3=0\} \] of the permutation module. In characteristic three the line $T=k(1,1,1)$ lies in $W$ and is trivial. On $W/T$ a transposition acts by $-1$, so the quotient is the sign module. The resulting exact sequence \[ 0\longrightarrow T\longrightarrow W\longrightarrow\operatorname{sgn} \longrightarrow0 \tag{342.2} \] does not split. Indeed, the $R$-fixed vectors of $k^3$ are exactly the constant vectors, so $W^R=T$. A splitting would make $R$ act trivially on both summands and hence on all of $W$, contradicting this one-dimensional fixed space.

Modules belonging to different blocks have no nontrivial extensions between them: the orthogonal central block idempotents decompose every module, and would split $342.2$. Therefore both simple modules belong to the same block. These are all the simple modules, so $kS_3$ has one block, with $\ell(B)=2$.

For its weights, there are only two conjugacy classes of $3$-subgroups, represented by $1$ and $R$. At $Q=1$, the ordinary character degrees $1,1,2$ have $3$-part one, whereas $|S_3|_3=3$, so there is no weight. At $Q=R$, the normalizer quotient is $C_2$, with two defect-zero characters. They both belong to the unique block. The two counts therefore agree.

The point is not merely that this small group satisfies the conjecture. It disproves a tempting reduction step: although a normal $p$-subgroup acts trivially on every simple module, the blocks of the original group algebra need not be in bijection with the blocks of the quotient algebra. Here $kC_2$ has two blocks while $kS_3$ has one. Simple-module descent alone cannot transport the entire blockwise weight problem.

What the finite calculations test. The accompanying exact script constructs the characteristic-three permutation action on $W$, lists its one-dimensional subspaces, and checks that the only $R$-fixed line is $T$. It verifies the nonsplit extension used above, as well as the ordinary degree and normalizer counts in these examples. The computations do not classify radical subgroups of arbitrary groups, construct all block correspondents, or implement the inductive condition appearing in [E1].

Outcome and first gap. The equality has been proved here for $p$-groups, finite abelian groups, the indicated direct products, and the unique $3$-block of $S_3$. The examples show why normalizer quotients, character defect, and block association are all necessary. What is missing is a general mechanism matching modular simple modules with locally defined weights while respecting the individual block. The recent claims in [E1] address major families through substantially deeper local and character-theoretic structure; neither those restricted claims nor the elementary reductions above supply that mechanism for all finite groups. The selected conjecture remains unresolved by this attempt.

Sources

• S1 Radical subgroups and the inductive blockwise Alperin weight conditions for PSp4(q), Introduction, printed page 1181, lines 20-24. https://par.nsf.gov/servlets/purl/10275354. Formulation source.
• E1 Baoyu Zhang, On the inductive blockwise Alperin weight condition for type $\mathsf B$ and type $\mathsf C$, arXiv:2609.24774, submitted 21 September 2026. https://arxiv.org/abs/2609.24774. Primary manuscript consulted 27 September 2026; its stated simple-group families and corresponding section hypotheses are not replaced by a claim for all finite groups.