Blockwise Galois Alperin Weight Conjecture

• BGAWC is a conjecture unifying modular representation theory with local block structure through the incorporation of Galois automorphisms.
• It asserts an equality between the dimension of Galois-fixed Grothendieck groups and local summations over block weights, establishing deep categorical links.
• Verification relies on Puig’s reduction to almost-simple k*-extensions and functorial approaches that transform complex character correspondences into computationally feasible checks.

The Blockwise Galois Alperin Weight Conjecture (BGAWC) unifies modular representation theory, local block structure, and Galois actions, extending Alperin's classical weight conjecture to a setting incorporating Galois automorphisms and blockwise correspondences. Formulated independently by Geoffrey Robinson and others in the 1980s and subsequently given various equivariant and functorial reformulations, BGAWC postulates that blockwise character counts are governed locally by weights and globally by Galois symmetries. This conjecture is central to the modern understanding of modular representations of finite groups in nontrivial Galois extensions and has deeply influenced the categorical framework of block theory.

1. Conjecture Statement and Formulations

Let $p$ be a prime and $k$ an algebraically closed field of characteristic $p$. Given a finite group $G$, a $p$-block $b$ is a primitive idempotent in the center $Z(\mathcal O G)$, where $\mathcal O$ is a complete DVR with residue field $k$, and $K$ is a fraction field containing $\Q_p^{\rm unram}$.

A $b$–weight is a pair $(Q, \varphi)$, with $Q \leq G$ a $p$-subgroup and $\varphi$ an irreducible Brauer character of $N_G(Q)/Q$ of defect zero such that the corresponding Brauer pair $(Q, f)$ lies under $b$. The set of $G$-conjugacy classes of $b$-weights indexes local data relevant to block $b$.

Define $\mathcal G_k(G, b) := \bigoplus_{\chi \in \Irr_k(G, b)} \mathcal O \chi$, the Grothendieck group of $kG$-modules in block $b$ extended to an $\mathcal O$-lattice. Let $\Gamma = \text{Gal}(K / \Q_p^{\rm unram})$ and, for any cyclic $C \leq \Gamma$, denote by $\mathcal G_k(G, b)^C$ the subspace of $C$-fixed points.

The BGAWC asserts: $$\dim_K \mathcal G_k(G, b)^C = \sum_{(Q, \varphi)} \dim_K \mathcal G_k(N_G(Q)/Q, \varphi)^{C_{(Q, \varphi)}}$$ where $C_{(Q,\varphi)}$ denotes the stabilizer in $C$ of the class $(Q,\varphi)$, and the sum runs over representatives of conjugacy classes of $b$–weights (Puig, 2012). Equivalently, there exists a $K\Gamma$-module isomorphism

$$\mathcal G_k(G, b) \cong \bigoplus_{(Q, \varphi)} \mathcal G_k(N_G(Q)/Q, \varphi)$$

stable under Galois action.

Alternative formulations include:

• Character-counting form: For every subgroup $T \leq \Gamma_b$,

$$|(G, b)^T| = \sum_{[P]} |\mathcal W(N_G(P)/P, \bar b_P)^T|$$

with notation as above (Huang et al., 8 Dec 2025).

• Alternating sum (Knörr–Robinson style): $$\sum_{\tau \in G \backslash S_G} (-1)^{|\tau|} |(G_\tau, b_\tau)^T| = 0$$ for chains $\tau = (1 = P_0 < P_1 < \cdots < P_n)$ of $p$–subgroups, and $T \leq \Gamma_b$ (Huang et al., 8 Dec 2025).
• Functorial Grothendieck group formulation: $$\sum_{\tau \in G \backslash S_G} (-1)^{|\tau|} FProj(-, kG_\tau b_\tau)^T = 0$$ in $K_0(F^\Delta_{Fpp_k})$ for each $T$-stable block-functor (Huang et al., 8 Dec 2025).

2. Central $k^*$–Extensions in Block Theory

A central $k^*$–extension of a finite group $\bar G$ is a group $G$ fitting into an exact sequence

$$1 \longrightarrow k^* \longrightarrow G \longrightarrow \bar G \longrightarrow 1$$

with $k^*$ equal to the center of $G$. Puig's local theory lifts every $N_G(Q)/Q$ to a central $k^*$–extension

$$1 \longrightarrow k^* \longrightarrow F(b, G)(Q) \longrightarrow N_G(Q)/Q \longrightarrow 1$$

to retain block structure and Galois action (Puig, 2012). The group of $k^*$–automorphisms of $F(b, G)(Q)$, $\Aut_{k^*}(F(b,G)(Q))$, organizes both the block-theoretic and Galois data. The action of $\Gamma$ (Galois automorphisms) extends to all $F(b,G)(Q)$–modules, inducing corresponding actions on local Grothendieck groups.

3. Puig’s Reduction and the Role of Almost-Simple $k^*$–Groups

Puig's main theorem, under the standard solvability-of-outer-automorphism-groups-of-simples (SOSFG) hypothesis (Puig, 2012), establishes that to prove BGAWC for all finite groups, it suffices to verify the conjecture for all central $k^*$–extensions of finite almost-simple groups. An almost-simple $k^*$–group $H$ is characterized by: $1 \to k^* \to H \to H/S \text{ (with %%%%58%%%% simple, %%%%59%%%% cyclic of order prime to %%%%60%%%%, %%%%61%%%%)}$ If the BGAWC fixed-point dimension formula (and equivariant module correspondence) holds for every block $c$ of each such $H$, then BGAWC is valid for all finite $k^*$–extensions. Globally, for all $(G, b)$,

$$\mathcal G_k(G, b) \cong \varprojlim_{q} \mathcal G_k(F(b,G)(q))$$

with compatible $\Gamma$ action, and

$$\dim_K \mathcal G_k(G, b)^C = \sum_{(Q, f)} \dim_K \mathcal G_k(F(b,G)(Q), 0(Q, f))^{C_{(Q, f)}}$$

4. Equivariant and Blockwise Local Conditions

The verification on almost-simple $k^*$–groups requires three main conditions:

1. Equivariant Weight Counting: For any cyclic $C \subset \Gamma$,

$$\dim_K \mathcal G_k(H, c)^C = \sum_{(Q, f)} \dim_K \mathcal G_k(F(c, H)(Q), 0(Q, f))^{C_{(Q, f)}}$$

with $(Q, f)$ running over $H$–conjugacy classes of selfcentralizing Brauer pairs (Puig, 2012).

1. Genuine Blockwise Correspondence: A $K \mathrm{Out}_{k^*}(H)_c$–module isomorphism must exist: $$\mathcal G_k(H, c) \cong \bigoplus_{(Q, f)} \mathcal G_k(F(c, H)(Q), 0(Q, f))$$ enforcing Robinson’s "equivariant condition (E)" (Puig, 2012).
2. Almost-Necessary Kernel Condition: For Clifford-theoretic descendants, the kernels of induced $C$-actions on local summands must coincide: $$\ker(C \,\text{action}\, \mathcal G_k(H_\psi | \psi)) = \ker(C \,\text{action}\, \mathcal G_k(N_H(R, g) | \widehat{g}))$$ with notation as above (Puig, 2012).

For blocks with abelian defect, (Huang et al., 8 Dec 2025) shows that if the block and its Brauer correspondent are functorially equivalent over the minimal field, BGAWC holds for such blocks.

5. Proof Strategies, Functorial Approaches, and Alternating Sums

Inductive, categorical, and functorial methods have been developed for BGAWC:

• Induction: The proof proceeds by induction on $|G|$, reducing via Brauer chains and Frobenius-localizer techniques to smaller groups and local block data (Puig, 2012).
• Deformation to Frobenius Categories: All local data reside in a folded Frobenius $P$–category $(P, \mathcal F, \text{aut}^{\rm sc})$, with Grothendieck groups described by inverse limits over selfcentralizing chains (Puig, 2012).
• Alternating Sum Formulation: Knörr–Robinson-style alternating sums over chains of $p$-subgroups, and their categorical analogues, provide reformulations equivalent to BGAWC (Huang et al., 8 Dec 2025).
• Functorial Equivalences: The Grothendieck group of diagonal $p$-permutation functors and block-functor equivalence criteria translate blockwise character correspondences into categorical isomorphisms (Huang et al., 8 Dec 2025).
• Cohomological Vanishing: Thévenaz’s radical function arguments and involution machinery show that combinatorial cancellation leaves only radical chains contributing to the sum (Puig, 2012).

6. Key Formulas and Character Theoretic Correspondences

Important formulas central to BGAWC include:

• Fong-Reynolds block induction/restriction: $kH c \cong \mathrm{Ind}_{N_H(c)}^H kN_H(c) c$
• Inverse-limit global description: $$\mathcal G_k(G, b) \cong \varprojlim_{q \in \text{ch}^*(\mathcal F)} \mathcal G_k(\mathcal F(q))$$ with $q$ running over selfcentralizing chains (Puig, 2012).
• Alternating-sum (chain-based) in the character-counting context: $$\sum_{\tau \in G \backslash S_G} (-1)^{|\tau|} |(G_\tau, b_\tau)^T| = 0$$ or, when including defect-zero,

$$\sum_{\tau \in G \backslash N_G} (-1)^{|\tau|} |(G_\tau, b_\tau)^T| = |\mathcal W(G, b)^T|$$

(Huang et al., 8 Dec 2025).

• Equivariant bijection via functorial equivalence: For $k$–pairs $(G, b)$ and $(H, c)$ functorially equivalent over $R$ with the same minimal field,

$(G, b) \longleftrightarrow (H, c)$

of irreducible characters compatible with Galois action (Huang et al., 8 Dec 2025).

7. Implications, Verification, and Manageability

Puig’s reduction confines proof of BGAWC to finitely many almost-simple $k^*$–extensions: groups of Lie type (in non-defining characteristic), alternating groups, and sporadic groups, each with finitely many blocks (Puig, 2012). Verification for each block of every such group, via explicit check of conditions (A)–(C), implies the full conjecture for all finite groups.

Concrete implications and practical considerations include:

• The case-based approach, informed by CFSG, enables possible algorithmic or computational verification for all families.
• Each group family presents distinct local block-theoretic and cohomological complexities, but always within the almost-simple framework.
• Functorial equivalences guarantee BGAWC for blocks with abelian defect groups and ensure that Galois-conjugate blocks are functorially equivalent over an algebraically closed field of characteristic zero (Huang et al., 8 Dec 2025).

A plausible implication is the feasibility of verifying BGAWC globally via computational enumeration and local checks on finite data sets. The categorical framework developed—particularly the functorial, Grothendieck group, and chain-sum viewpoints—clarifies the conceptual grounding and the reduction of BGAWC to local structure and Galois fixed points.