Inductive Feit Condition

• The inductive Feit condition is a local-to-global principle that inductively determines character invariants in finite groups by analyzing suitable local subgroups.
• It establishes an equivariant bijection between irreducible characters of the full group and manageable local subgroups, ensuring compatibility of automorphic and Galois actions.
• Applied to alternating groups, Lie type groups, and covering groups, it underpins advances in modular representation theory and block classification.

The inductive Feit condition is a central concept in the reduction of Walter Feit’s conjecture on character-theoretic invariants—such as fields of values, conductors, and isomorphism classes of sources—for finite groups. It formalizes a powerful philosophy: that global properties of irreducible characters in complicated finite groups can be derived inductively from their behavior in simpler "local" subgroups, provided these local representations satisfy certain compatibilities. The inductive Feit condition enables a local-global principle, allowing for deep structural analyses in modular representation theory and character theory, and serves as a conduit to ultimately proving Feit’s conjecture for all finite groups by verifying it in the field of finite nonabelian simple groups.

1. Definition and Formulation of the Inductive Feit Condition

The inductive Feit condition is a local-to-global criterion imposed on finite (simple or nearly simple) groups and their irreducible characters. Precisely, for a universal covering group $X$ of a finite nonabelian simple group $S$ and a prime $\ell$, the inductive Feit condition demands the existence of:

• A proper ("local") subgroup $U \le X$ with suitably manageable structure (often a normalizer of a torus, Sylow, or self-normalizing subgroup),
• An explicit $\Gamma \times \mathcal{H}$-equivariant bijection

$$\Omega: \operatorname{Irr}_{\ell'}(X) \to \operatorname{Irr}_{\ell'}(U),$$

where $\operatorname{Irr}_{\ell'}(G)$ is the set of irreducible characters of $G$ with degree not divisible by $\ell$, and where $\Gamma$ and $\mathcal{H}$ are subgroups of the automorphism group and cyclotomic Galois group (often determined by character field and centralizer data),

• A matching of central ordering conditions, i.e., the automorphism and Galois stabilizers for the character $\chi \in \operatorname{Irr}_{\ell'}(X)$ and its image $\Omega(\chi) \in \operatorname{Irr}_{\ell'}(U)$ are related so that the relevant groups $(X \rtimes \Gamma_\chi)_{\mathcal{H}}$ and $(U \rtimes \Gamma_{\Omega(\chi)})_{\mathcal{H}}$ satisfy an inclusion or order equivalence.

This structure ensures that critical invariants—such as conductors (minimal cyclotomic fields containing the character values), extension properties, and stabilizers—can all be "read off" from local data in $U$.

2. Motivation and Historical Context

Feit’s conjecture, originally posed in 1980, concerns the possible isomorphism classes of vertex–source pairs (and related invariants) attached to simple modules of group algebras, and their dependence solely on the local subgroup structure of the group. The ultimate aim is to show that for any irreducible character of a finite group, the key invariants (e.g., the field of values; the minimal $n$ such that $\mathbb{Q}(\chi) \subset \mathbb{Q}_n$; the source structure of modules) are determined in a manner that can be reconstructed inductively from local subgroups.

The inductive Feit condition arises in the context of strategies that aim to reduce the proof of Feit’s conjecture to its verification for all finite nonabelian simple groups. This mirrors the approach found in other major conjectures (such as the McKay and Alperin weight conjectures) which depend upon analogous inductive conditions on simple groups (Tapp, 29 Jul 2025). The verification of the inductive Feit condition completes the program of reducing these global conjectures to the so-called "building blocks" of the classification of finite simple groups (Boltje et al., 25 Jul 2025).

3. Methodology: Local-Global Correspondence, Equivariant Bijections, and Central Order Matching

The implementation of the inductive Feit condition involves several technical steps:

1. Selection of Local Subgroups: For each relevant simple group $S$ (or its universal cover $X$), a proper subgroup $U$ is chosen to be, for example, the normalizer of a maximal torus, a Sylow subgroup, or a Hall subgroup. Frequently $U$ is self-normalizing, intravariant, and supports manageable character theory (Tapp, 29 Jul 2025).
2. Construction of Equivariant Bijections: A bijection $\Omega$ is constructed between $\operatorname{Irr}_{\ell'}(X)$ and $\operatorname{Irr}_{\ell'}(U)$, often by explicit combinatorial means. Equivariance under the action of subgroups of the automorphism group and the cyclotomic Galois group is required. When $X$ and $U$ are embedded in larger groups (such as symplectic groups), characters can sometimes be extended via known theorems (e.g., Gallagher's theorem), facilitating the bijection and its properties.
3. Verification of Central Ordering/Eigenvalue Conditions: For each paired character $(\chi, \Omega(\chi))$, one must verify that automorphism and Galois stabilizers are suitably matched—this often amounts to showing inclusion or equivalence of stabilizer subgroups and matching conductors or fields of values.

The local-global correspondence thus established ensures that all relevant data attached to characters (notably, the "conductor" or field of values) in $X$ is mirrored exactly for their images in $U$ (Boltje et al., 25 Jul 2025, Tapp, 29 Jul 2025).

4. Verification for Specific Families: Application to Simple and Covering Groups

The inductive Feit condition has been verified for a range of important simple group families.

(a) Alternating Groups and Double Covers: For alternating groups and their double covers, one can select, for each non-linear irreducible character $\chi$, a maximal self-normalizing subgroup $U$ and a corresponding character $\mu$ such that the field of values, conductor, and (automorphism × Galois group) stabilizer are precisely matched. This approach extends to spin (faithful) characters using explicit combinatorial and modular character-theoretic constructions, including Young diagrams and Schur's Q-functions (Boltje et al., 25 Jul 2025).

(b) Groups of Lie Type (Small Rank): For $\operatorname{PSL}_2(q)$ and the Suzuki groups ${}^{2}B_2(2^{2n+1})$, the inductive Feit condition is realized by selecting $U$ as the normalizer of suitable cyclic or Sylow subgroups with well-understood character sets. The equivariant bijection $\Omega$ is constructed with explicit assignments for each family of $\ell'$-irreducible characters, and central order conditions are checked using extension arguments, sometimes involving Gauss sum calculations or established properties of cyclotomic fields (Tapp, 29 Jul 2025). The central role of the dihedral or dicyclic structure of $U$ is notable.

5. Structural and Theoretical Consequences

Once the inductive Feit condition is established for a given family of simple groups, the following consequences hold:

• Reduction to Simple Groups: If all composition factors (nonabelian simple groups) involved in a finite group $G$ satisfy the inductive Feit condition, then Feit's conjecture holds for $G$ (Boltje et al., 25 Jul 2025).
• Perfect Isometry Conjecture: For double covers of symmetric and alternating groups, the inductive Feit condition implies the perfect isometry conjecture of Livesey, providing a natural isometry (compatible with Galois actions and block structures) between character sets of $G$ and of suitable local subgroups (Boltje et al., 25 Jul 2025).
• Block Theory and Modular Representation: Finiteness results for vertex–source pairs and the visibility of conductors and related invariants in local data underpin completeness and uniqueness proofs in block theory, facilitating inductive classification schemes.

6. Context in Broader Local-Global Strategies and Open Directions

The inductive Feit condition is emblematic of a range of local-global reduction strategies in modular and ordinary character theory, including those employed in the proof and reduction of the McKay, Alperin, and Dade conjectures. These methods rely on verifying tailored inductive conditions for each family of finite simple groups, then "lifting" the validity to all finite groups via composition factor analysis.

Current research focuses on extending the verification of the inductive Feit condition to additional classes of finite simple groups, particularly those of Lie type in higher rank or with more intricate automorphism and representation structures. This approach may ultimately resolve longstanding open conjectures (notably Brauer’s Problem 41, as observed in (Boltje et al., 25 Jul 2025)) and solidify local-global correspondences as a foundational methodology in the field.

A plausible implication is that further paper of the character-theoretic stabilization under automorphic and Galois actions in the context of covering groups and groups with nontrivial Schur multipliers will reveal new phenomena and potential generalizations of the inductive Feit condition. This suggests productive avenues for future research both in explicit computational character theory and in the categorical or diagrammatic formulation of representation-theoretic statements.

7. Summary Table: Key Components of the Inductive Feit Condition

Component	Description	Papers
Local subgroup $U$	Maximally self-normalizing, supports manageable character theory	(Boltje et al., 25 Jul 2025, Tapp, 29 Jul 2025)
Equivariant bijection $\Omega$	Matching of irreducible characters, Galois/automorphism equivariant	(Boltje et al., 25 Jul 2025, Tapp, 29 Jul 2025)
Central order matching	Stabilizer and field of values compatibility	(Boltje et al., 25 Jul 2025, Tapp, 29 Jul 2025)
Applicability	Verified for alternating groups, double covers, $\operatorname{PSL}_2(q)$, Suzuki groups	(Boltje et al., 25 Jul 2025, Tapp, 29 Jul 2025)

The central importance of the inductive Feit condition lies in its ability to localize global character-theoretic invariants, thereby enabling the transfer of deep properties from simple building blocks to arbitrary finite groups. Verification for all finite simple groups would affirm Feit’s conjecture and resolve related open problems in block theory and modular representation.