Brauer's Height Zero Conjecture

• Brauer's Height Zero Conjecture is a key proposition in modular representation theory asserting that a p-block has all height zero characters if and only if its defect group is abelian.
• It has driven advances in block theory by inspiring generalizations such as projective and Galois-invariant versions, thus deepening our understanding of p-block structures.
• Recent progress confirms the conjecture across diverse group classes, underpinning modern local-global approaches in the study of finite group representations.

Brauer's Height Zero Conjecture is a cornerstone of modular representation theory, characterizing the relationship between the structure of defect groups of $p$-blocks of finite groups and the $p$-part of the degrees of their irreducible complex characters. The conjecture asserts that all irreducible ordinary characters of a $p$-block of a finite group have height zero if and only if its defect group is abelian. Over the decades, the conjecture motivated profound developments both in the structure theory of blocks and in local-global conjectures. Multiple generalizations and refinements exist, including projective and Galois-theoretic versions, as well as natural extensions to the case of non-abelian defect groups.

1. Foundations and Formal Statement

Consider a finite group $G$ and a prime $p$. The group algebra (over a field of characteristic $0$ or $p$) splits into a direct sum of indecomposable two-sided ideals called $p$-blocks. Each block $B$ has an associated $p$-subgroup $D$ (a defect group), which measures the "distance" from being of defect $0$ (i.e., semisimple in characteristic $p$). For any $\chi\in\Irr(B)$ (the set of ordinary irreducible characters belonging to $B$), Brauer defined the $p$-height $h(\chi)$ by the relation

$\chi(1)_p = |G:D|_p\, p^{h(\chi)},$

where $\chi(1)_p$ denotes the $p$-part of the character degree and $|G:D|_p$ the $p$-part of the index. Characters with $h(\chi) = 0$ are called height zero.

Brauer's Height Zero Conjecture: For every $p$-block $B$ of $G$ with defect group $D$,

$\forall\,\chi\in\Irr(B),\ h(\chi)=0\quad\Longleftrightarrow\quad D\ \text{is abelian}.$

Equivalently, all irreducible characters in $B$ have degree prime to $p$ times $|G:D|_p$ if and only if $D$ is abelian (Malle et al., 2022, Kessar et al., 2015, Sambale, 2018, Malle et al., 2021).

2. Historical Development and Key Partial Results

The conjecture dates to 1955 and quickly became central to block theory. Its "if"-direction ("abelian defect group $\Rightarrow$ all heights zero") was proved successively for special classes of groups:

• $p$-solvable groups (Gluck–Wolf),
• maximal-defect 2-blocks (Malle et al., 2022),
• quasi-simple and simple groups of Lie type (Deligne–Lusztig theory, Kessar–Malle, Navarro–Tiep, Enguehard, Bonnafé–Rouquier, et al.),
• all quasi-simple groups (Kessar et al., 2015, Kessar et al., 2011).

The "only if"-direction ("all heights zero implies abelian defect") was resolved in steps, culminating in the classification-based proof for all primes $p$ and all blocks by Malle, Navarro, Schaeffer Fry, and Tiep (Malle et al., 2022, Malle et al., 2021). For principal blocks, the implication was settled earlier (Malle et al., 2021).

3. Structural and Reduction Techniques

3.1 Berger–Knörr and Related Reductions

A major simplification is the reduction to the case of quasi-simple groups (Berger–Knörr [BK88]): if Brauer's Height Zero Conjecture holds for all quasi–simple groups, it holds in general (Kessar et al., 2015, Kessar et al., 2011). The proof relies on Clifford-theoretic control of defect groups under extension and induction–restriction arguments, allowing a minimal counterexample to be lifted to a quasi-simple group.

3.2 Techniques for Quasi-Simple and Lie Type Groups

For quasi-simple groups (especially of Lie type), one employs:

• Lusztig theory and $e$-Harish-Chandra theory for the parametrization of characters and blocks (Kessar et al., 2011),
• explicit determination of defect groups and their abelianity,
• analysis of unipotent blocks, quasi-isolated blocks, and principal series representations,
• block extension and Morita equivalence (Bonnafé–Rouquier).

In these settings, height-formulas often reduce to explicit combinatorial or character-theoretic computations, frequently using relative Weyl groups or explicit tables of character degrees.

4. Extensions and Generalizations

4.1 Projective Height Zero Conjecture

Malle–Navarro introduced a projective version (Malle et al., 2017, Sambale, 2018) involving central $p$-subgroups $Z\le G$ and characters $\lambda\in\Irr(Z)$. The Projective Height Zero Conjecture states: $\forall\,\chi\in\Irr(B\mid\lambda),\, h(\chi)=0\quad\Longleftrightarrow\quad D/Z\ \text{abelian and}\ \lambda\ \text{extends to}\ D,$ where $\Irr(B\mid\lambda)$ is the set of irreducibles covering $\lambda$.

Sambale (Sambale, 2018) proved that Brauer's original conjecture implies the projective version.

4.2 Galois and Local-Global Analogues

Recent work explores Galois-invariant versions (Malle et al., 13 Feb 2024, Malle et al., 2022, Moretó et al., 23 Nov 2025), where one restricts to irreducible characters fixed by specific Galois automorphisms (for instance, automorphisms of order $p$ fixing $p$-power roots of unity). The strengthened conjecture asserts that if all such Galois-fixed irreducibles in the principal block have degree prime to $p$, then the Sylow $p$-subgroup is abelian (Moretó et al., 23 Nov 2025). Results include:

• Complete proofs for $p=2$ (Malle et al., 2022),
• Proofs for all $p$ when $G$ avoids certain low-rank Lie-type composition factors (Malle et al., 13 Feb 2024),
• Full generality for principal blocks in the Galois framework (Moretó et al., 23 Nov 2025).

These versions also lead to Galois-theoretic analogues of the Itô–Michler theorem (relation between the irreducible character degrees and the existence of normal abelian Sylow subgroups) (Moretó et al., 10 Jun 2024, Moretó et al., 23 Nov 2025).

4.3 Non-Abelian Defect Groups: Eaton–Moretó Conjecture

Eaton–Moretó conjectured that, for blocks with non-abelian defect group $D$,

$\min \{h(\chi)>0 : \chi\in\Irr(B)\} = \min \{\log_p\lambda(1) : \lambda\in\Irr(D),\ \lambda(1)>1\},$

i.e., the minimal positive character height in $B$ equals that of $D$ (Brunat et al., 2014, Malle et al., 2023). This strictly generalizes Brauer's height zero conjecture, and substantial progress has been made in the special cases of principal blocks, finite reductive groups in defining characteristic, and covering groups of symmetric/alternating groups (Brunat et al., 2014, Malle et al., 2023).

5. Explicit Verifications and Class Analyses

The conjecture has been verified explicitly for many group and block types:

• Quasi-simple and finite simple groups (Kessar et al., 2015, Brunat et al., 2014),
• Blocks with defect group $D_{2^n}\times C_{2^m}$ and $D_{2^n}*C_{2^m}$ (Sambale, 2011, Sambale, 2011),
• Metacyclic defect groups (Sambale, 2012),
• Principal blocks for all groups (Malle et al., 2021),
• Non-principal blocks in various controlled cases (Brunat et al., 2014), and
• For $p$-solvable and nilpotent blocks (Malle et al., 2017).

The proofs for blocks of finite groups with metacyclic, extraspecial, and dihedral-cyclic defect groups proceed by case-by-case analysis of fusion systems, decomposition numbers, and congruences, coupled with detailed enumeration of block invariants: $k_i(B)$ (number of irreducible characters of height $i$), $k(B)$ (number of irreducibles), $l(B)$ (number of irreducible Brauer characters), and Cartan invariants—see (Sambale, 2011, Sambale, 2011, Sambale, 2012).

6. Implications, Corollaries, and Local-Global Conjectures

6.1 Local-Global Conjectures

Brauer's Height Zero Conjecture is fundamental in the structure of blocks and connects deeply to other local-global conjectures:

• Alperin–McKay and Alperin weight conjectures,
• Dade's projective conjecture (which supplies partial implications for the minimal height conjecture (Malle et al., 2023)),
• Malle–Navarro's conjecture on the characterization of nilpotent blocks by degree patterns (Kessar et al., 2011),
• Extensions to multi-prime and normality detection (block-theoretic analogues of Itô–Michler) (Moretó et al., 10 Jun 2024).

Recent proofs show that local data (such as the vanishing of specific block character degrees modulo $p$) can detect normal or abelian Sylow subgroups and more general subgroup structure.

6.2 Structural and Computational Applications

The explicit determination of block invariants and the behavior of character heights is crucial for block classification, computational group theory (e.g., in GAP), and for understanding the distribution of character degrees in finite groups of Lie type and their covering groups.

The geometric techniques (Deligne–Lusztig theory, Harish–Chandra induction, Lusztig series) have become central tools in these applications, particularly for quasi-simple and algebraic groups over finite fields.

7. Open Problems and Future Directions

While the conjecture is now resolved for all finite groups (Malle et al., 2022), the following directions remain active:

• Uniform proofs of the Eaton–Moretó minimal height conjecture in cross-characteristic types, especially for classical groups (Brunat et al., 2014).
• Further Galois-theoretic refinements, extending "height zero" or other block invariants to sets of automorphism-fixed characters (Malle et al., 13 Feb 2024, Moretó et al., 23 Nov 2025).
• Explicit classification of all height spectra for blocks with "complex" defect groups, refining the local-global block theory.
• Deeper connections to the inductive Alperin–McKay and related conditions, which could target fine correspondences and bijections between blocks of related groups.

The interplay between block-theoretic invariants, character-theoretic properties, local subgroups, and Galois actions continues to shape the landscape of modular representation theory, largely inspired by the scope and resolution of Brauer's Height Zero Conjecture.

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References:

• (Sambale, 2011): "Blocks with defect group D_{2^n} x C_{2^m}"
• (Sambale, 2011): "Blocks with defect group D_{2^n} * C_{2^m}"
• (Kessar et al., 2011): "Quasi-isolated blocks and Brauer's height zero conjecture"
• (Sambale, 2012): "Brauer's Height Zero Conjecture for metacyclic defect groups"
• (Brunat et al., 2014): "Characters of positive height in blocks of finite quasi-simple groups"
• (Kessar et al., 2015): "Brauer's height zero conjecture for quasi-simple goups"
• (Malle et al., 2017): "The Projective Height Zero Conjecture"
• (Sambale, 2018): "On the projective height zero conjecture"
• (Malle et al., 2021): "Brauer's Height Zero Conjecture for Principal Blocks"
• (Malle et al., 2022): "Brauer's Height Zero Conjecture"
• (Malle et al., 2022): "Height Zero Conjecture with Galois Automorphisms"
• (Malle et al., 2023): "Minimal heights and defect groups with two character degrees"
• (Malle et al., 13 Feb 2024): "A Brauer--Galois height zero conjecture"
• (Moretó et al., 10 Jun 2024): "A normal version of Brauer's height zero conjecture"
• (Moretó et al., 23 Nov 2025): "Height zero characters and Galois automorphisms"