Blocks with only one irreducible Brauer character orbit

Abstract: In this paper, we prove that a (p)-block with abelian defect group is inertial if it covers a (p)-block of a normal subgroup of (p)-power index having only one irreducible Brauer character orbit.

• The paper demonstrates that blocks with abelian defect groups and a single Brauer character orbit are inertial under group extensions, leveraging Puig's inertial framework.
• It applies a combination of Clifford theory, Fong-Reynolds correspondence, and Jordan decomposition to rigorously reduce the problem to the quasi-simple group case.
• The work resolves a gap in block theory by linking character orbit properties with Morita equivalence and inertial structure in modular representation theory.

Inertiality of Blocks with Only One Irreducible Brauer Character Orbit

Introduction and Problem Setting

This paper addresses a specific structural question in the modular representation theory of finite groups: given a $p$-block $b$ with abelian defect groups, if the set $\operatorname{IBr}(b)$ of irreducible Brauer characters forms a single orbit under the stabilizer of $b$ in $\operatorname{Aut}(G)$, under what conditions is $b$ inertial? In the modern language of block theory, a block is inertial if it is basically Morita equivalent to a block with a normal defect group (Puig's sense). The main result provides a positive answer for $\tilde{b}$, a block covering $b$ in a group extension of $p$-power index, given that $b$ has a single irreducible Brauer character orbit.

This fills a notable gap in the landscape, since nilpotent-covered blocks are both prototype inertial blocks and have this single-character orbit property, yet the implication is nontrivial beyond that. For nonabelian defect groups, explicit counterexamples show the phenomenon does not persist.

Structural Reductions and Technical Lemmas

The proof architecture combines deep group-theoretic and block-theoretic reductions. The approach leverages several layers:

• By careful analysis using Fong-Reynolds correspondents and Clifford-theoretic techniques, the orbit property of $b$0 percolates through to relevant subgroups, quotient blocks, and their automorphism actions.
• Via an intricate induction on the order of the group (minimizing the index of the normal $b$1-subgroup in the center), the question is reduced to the quasi-simple group case, using layer and Fitting subgroup arguments. Essential here are technical lemmas from Navarro, Linckelmann, and Puig, ensuring the stability of block-theoretic invariants under intricate group-theoretical operations.
• Lemmas on the solvability of outer automorphism groups, the structure of block stabilizers, and the actions across abelian defect classes undergird the main reduction (cf. Lemmas on the layer and generalized Fitting subgroup).

A crucial step is to show that for nilpotent blocks covered in some extension group, all relevant blocks in the chain are explicitly inertial. Uniqueness and basic Morita equivalence are inherited along the chain, allowing the main assertion to be reduced to the key quasi-simple cases.

Analysis for Quasi-simple and Classical Groups

Handling quasi-simple groups, especially groups of Lie type and their extensions, constitutes the core technical difficulty. The paper makes heavy use of advanced local-global theory for finite groups of Lie type, particularly:

• Lusztig's classification of irreducible characters via Lusztig series $b$2 and the parametrization of blocks by dual (semisimple) elements,
• The Bonnafé–Rouquier correspondence for blocks and their behavior under Levi subgroups,
• The theory of Jordan decomposition (Kessar–Malle framework), which allows block properties to be transferred from centralizers of semisimple elements back to the ambient group,
• The behavior of Brauer characters in (quasi-)isolated blocks and the effect of bad primes on defect structures.

Technically, remarkable use is made of basic sets (in sense of Geck–Malle), heights of irreducible Brauer characters, and the fine structure of blocks in exceptional and classical groups. For example, the cited result that all unipotent blocks with central defect groups in exceptional groups (types $b$3, $b$4, $b$5) must be nilpotent is pivotal.

Another critical ingredient is the classification of nilpotent, cyclic, and $b$6-length blocks for sporadic, alternating, and some of the small-rank groups of Lie type. This relies on the detailed tables and character data (e.g., Atlas, Brauer trees, etc.), reducing all exceptional or low-prime cases to ones covered by Puig's inertial results.

Main Theorem and Its Proof

Main Theorem:

Let $b$7 be a finite group with $b$8 of $b$9-power index, $\operatorname{IBr}(b)$0 a block of $\operatorname{IBr}(b)$1 with abelian defect group, and $\operatorname{IBr}(b)$2 a block of $\operatorname{IBr}(b)$3 covered by $\operatorname{IBr}(b)$4. If $\operatorname{IBr}(b)$5 is a single $\operatorname{IBr}(b)$6-orbit, then $\operatorname{IBr}(b)$7 is inertial.

The essence of the proof is as follows:

1. Reduce to the case in which $\operatorname{IBr}(b)$8 is quasi-simple and $\operatorname{IBr}(b)$9's layer is maximal and normal.
2. Employ Jordan decomposition and basic sets machinery to show that in all possible configurations, either the block in question is nilpotent (whence inertial), or the covering block in the extension group is nilpotent, or the problem further reduces by consideration to an inertial case.
3. For $b$0 and special cases of Lie type with exceptional central quotients or twisted types, careful case analysis using decomposition numbers, block distribution, and Clifford theory is performed to confirm the assertion.

A bold claim deduced is that for all blocks with abelian defect group covered in a $b$1-power index extension, the single orbit property on Brauer characters ensures inertiality of the cover. Furthermore, explicit exceptions are excluded in the reduction steps and verified using character tables.

Implications and Future Directions

Practically, this result identifies a broad, computable class of blocks whose local structure (i.e., inertiality) is controlled by a character-theoretic symmetry condition. This advances the correspondence between local module-theoretic behavior—Morita equivalence types—and global character orbit data, which is a persistent topic in block theory and modular representation.

Theoretically, this result fits into ongoing efforts to better organize the categorical structure of blocks (cf. basic Morita equivalence, source algebra methods) and to classify blocks by invariants visible in the character table or through automorphism actions.

Future research may seek to:

• Characterize which non-abelian defect blocks might exhibit related phenomena and whether stronger versions of the orbit property (e.g., orbits under larger automorphism groups) imply deeper structural consequences.
• Investigate the interaction of block inertiality with global derived categories and equivalences beyond Morita, perhaps following the template suggested by Broué's abelian defect group conjecture.
• Systematically study the failure of the implication in the nonabelian defect case, seeking new invariants or obstructions.

Conclusion

This paper rigorously demonstrates that, for blocks with abelian defect groups in a suitable group-theoretic covering framework, the single orbit condition on irreducible Brauer characters dictates inertiality. The proof synthesizes advanced techniques from modern block theory, representation theory of finite groups of Lie type, and automorphism group actions. The results provide both practical tools for block classification and theoretical support for the predictiveness of orbit-theoretic block properties in modular representation theory.