Alperin's Weight Conjecture

• Alperin's Weight Conjecture is a local-global principle in modular representation theory establishing a bijection between simple modules and G-conjugacy classes of weights.
• The conjecture employs radical p-subgroups and canonical bijections, using techniques like polarizations, Fitting sequences, and interior algebra embeddings.
• Its natural, automorphism-equivariant framework has led to deep insights and reductions, especially for p-solvable and odd-order groups.

Alperin's Weight Conjecture is a central local-global statement in modular representation theory of finite groups, first formulated by Jon Alperin. It predicts that the number of simple modules of a finite group algebra in characteristic $p$ coincides with the number of $G$-conjugacy classes of certain weighted local data called “weights.” This conjecture encapsulates the philosophy that global representation-theoretic information is entirely determined by local subgroup structure. Over the past decades, Alperin's conjecture has spurred extensive research, producing a highly developed reduction theory, categorical reformulations, and connections to several other deep conjectures in block theory and representation theory of finite and algebraic groups.

1. Precise Statement and Parameterization of Weights

For a finite group $G$, an algebraically closed field $k$ of characteristic $p$, and the group algebra $kG$, Alperin's Weight Conjecture asserts: $\#\{\text{isomorphism classes of simple } kG\text{-modules}\} = \#\{\text{%%%%6%%%%-conjugacy classes of weights}\}$ A "weight" is defined as a pair $(R, Y)$, where:

• $R \subset G$ is a $p$-subgroup that is radical (i.e., $R=O_p(N_G(R))$), often realized as a vertex in the sense of Green theory, or more precisely, as a radical pointed group.
• $Y$ is an isomorphism class of simple projective modules over a twisted group algebra $k_{+}(N_G(R)/R)$, or, more precisely, over a central $k^*$-extension of $N_G(R)$ canonically determined up to a choice of "polarization" (a system of splittings compatible with automorphisms).

Puig establishes in (1005.3748) that for $p$-solvable groups:

• The set of simple $kG$-modules can be parameterized by weights $(R, Y)$.
• Starting with an indecomposable module $M$ with vertex $R$ and source $E$, one constructs a multiplicity module $V$ (a simple projective module for a twisted group algebra associated to the stabilizer of $E$), and after fixing a polarization, a one-to-one correspondence $M \mapsto (R, Y)$ is achieved, where $Y$ is the isomorphism class of the restriction of $V$ to $N_G(R)$.

2. Existence and Natural Structure of the Weight–Module Bijection

A critical structural claim is that the correspondence between simple modules and weights is canonical and natural:

• The bijection is constructed to be equivariant with respect to the outer automorphism group $\operatorname{Out}(G)$.
• The method first establishes bijections at the level of blocks by using interior algebra embeddings, then “glues” these via Fitting block sequences, yielding an overall bijection: $\Gamma : \Irr_k(G) \longrightarrow \text{Wgt}_k(G)$
• This bijection is independent of arbitrary choices (the polarization removes the ambiguity in central $k^*$-extensions) and is compatible with all group isomorphisms and automorphisms, as established in [(1005.3748), Proposition 5.2] and the commutative diagrams in Section 5.

When $G$ has odd order (invoking the Feit–Thompson Theorem), the unique “odd polarizations” ensure a unique, tensor-compatible system of splittings, and the correspondence reduces to that defined by Navarro, as shown in the Appendix of (1005.3748).

3. Relationship Between Multiplicity Modules and Weights

A nuanced point concerns the explicit relationship between the “multiplicity module” $V$ and the weight module $Y$:

• Given an indecomposable $kG$-module $M$ with associated vertex $R$ and source $E$, and the stabilizer $N$ (satisfying $R \subset N \subset N_G(R)_E$), there exists a simple $kN$-module $W$ with canonical isomorphisms

$U \cong \Ind_{N_G(R)_E}^{N_G(R)} (W), \qquad V \cong \Ind_{N_G(R)_E}^{N_G(R)}(W_E)$

where $U$ is the simple projective module with vertex $R$ appearing in the weight, and $W_E$ is a module associated with $W$ (see formula (1.6.1) and Section 3.3 in (1005.3748)). Thus the local multiplicity data are induced from the stabilizer, and the parameterization is controlled by canonical induction.

4. The Role of Automorphisms, Blocks, and Fitting Sequences

Central to the construction is the compatibility of the bijection with group automorphisms:

• The bijections at the block level are compatible with the action of $\operatorname{Out}(G)$ and do not rely on the choice of a block isomorphism (since Fitting block sequences ensure the consistency across different blocks).
• This naturality extends to commutative algebraic diagrams (cf. (1005.3748), formulas (5.1.1), (5.3.4), (5.4.2), and diagram (5.8.2)), confirming that the module-weight correspondence commutes with the action of automorphisms and splitting of central extensions.

For each block $b$, there is a canonical bijection

$\overline{T}^{G,b} : \Irr_k(G,b) \leftrightarrow \Irr_k(\widehat{G}_G(O))$

with subsequent isomorphisms yielding

$\Irr_k(G) \simeq \text{Wgt}_k(G)$

valid after a coherent choice of polarization.

5. Specialization to p-Solvable and Odd-Order Groups

For $p$-solvable groups, the conjecture holds in full generality:

• Okuyama had earlier established the bijection in this setting, and Puig's construction is compatible with and extends this work.
• In the case where $p$ is odd and $G$ has odd order, after fixing the unique odd polarization, the bijection constructed by Puig coincides with Navarro’s bijection ((1005.3748), Appendix, Theorem A.3 and Section A.8).

This demonstrates that specific choices of splits (polarizations) can remove ambiguities that may arise in the general parameterization, particularly in the presence of automorphisms.

6. Conceptual and Technical Impact

The technical development in (1005.3748) anchors the validity of Alperin's Weight Conjecture for $p$-solvable groups by:

• Providing explicit structural descriptions and bijections for the weight–module correspondence.
• Exhibiting natural and automorphism-compatible parameterizations of weights through modules, vertices, sources, and twisted group algebras.
• Demonstrating that the local theory of vertices, sources, and multiplicity modules—as refined via Fitting sequences and Dade $p$-algebras—gives a complete local–global description of simple module counts.

This natural bijection, with its compatibility and structural depth, is both a combinatorial and categorical realization of Alperin's vision that the representation theory of finite groups is governed entirely by local subgroup data. In addition, the linkage to polarizations and block-isomorphism theory exemplifies the importance of canonical choices for constructing module- and automorphism-equivariant correspondences. The reduction to well-understood cases in $p$-solvable and odd-order groups further clarifies the conjecture's scope and the canonical nature of “weights.”

7. Summary Table: Key Elements in Weight Parameterization

Concept	Definition / Role	Key Technical Tool
Weight $(R, Y)$	$R$: radical $p$-subgroup; $Y$: simple projective over extension of $N_G(R)/R$	Vertex-source-multiplicity module decomposition
Multiplicity module $V$	Simple projective over twisted group algebra for source stabilizer	Canonical induction from $N_G(R)_E$
Polarization	Coherent splitting of central extensions to make the correspondence canonical	Removes ambiguity in central k^*-extensions
Automorphism equivariance	Bijections compatible with $\operatorname{Out}(G)$ and inner automorphisms	Gluing bijections via Fitting block sequences
Relationship to blocks	Natural bijections at block level, glued over all blocks	Interior algebra embeddings
Navarro correspondence	Coincides with Puig’s bijection for odd order using unique polarization	Reduction to known bijection in odd-order case

The synthesis in (1005.3748) thus establishes that for $p$-solvable groups, the module theory and local data (weights) are intricately and naturally parameterized, providing a canonical and automorphism-compatible realization of Alperin's Weight Conjecture.