Height-Zero-Equal-Degree Conjecture

Updated 31 July 2025

The Height-Zero-Equal-Degree Conjecture is a central concept in modular representation theory that links equal height zero character degrees with the nilpotency of blocks.
It establishes that blocks with uniformly equal degrees among height zero irreducible characters possess abelian defect groups and are Morita equivalent to their Brauer correspondents.
Methodologies such as Lusztig induction, Harish–Chandra theory, and Morita equivalences reduce complex global problems to precise local structural and arithmetic conditions.

The Height-Zero-Equal-Degree Conjecture is a central problem in the modular representation theory of finite groups, arising as a natural strengthening and structural refinement of Brauer’s Height Zero Conjecture. It asserts, in various guises, that uniformity in the degrees of certain irreducible characters—most notably those of height zero—in a block reflects deep local properties of the defect group and often forces the block to be nilpotent, abelian, or to have a tightly controlled structure. This conjecture has attracted sustained attention due to its connections to block nilpotency, local-global character correspondences, the arithmetic of character fields of values, and its ramifications for related conjectures such as the Alperin–McKay and Navarro–McKay conjectures.

1. Formulations of the Height-Zero-Equal-Degree Conjecture

The conjecture appears in two principal forms:

(a) Equality Implies Nilpotency (Malle–Navarro):

If all height zero irreducible characters in a block $B$ of a finite group $G$ have equal degree, then $B$ is nilpotent; that is, $B$ is Morita equivalent to its Brauer correspondent in the normalizer of a defect group. In symbolic terms:

$\{\chi \in \mathrm{Irr}(B) : \text{height}(\chi) = 0\} \text{ have constant } \chi(1) \implies B \text{ nilpotent}$

(b) Equal Heights Reflect Minimal Heights in Defect Groups (Eaton–Moretó Conjecture):

The minimal positive height of non-linear irreducible characters in a $p$ -block $B$ coincides with that in its defect group $D$ :

$mh(B) = mh(D)$

where

$mh(B) = \min\{ \log_p \chi(1) : \chi \in \mathrm{Irr}(B), \chi(1)>1 \}$

$mh(D) = \min\{ \log_p \psi(1) : \psi \in \mathrm{Irr}(D), \psi(1)>1 \}$

(see (Malle et al., 2023)).

These conditions formalize the expectation that local information (character degrees in defect groups) tightly governs global character theory within blocks.

2. Implications in Block Theory: Nilpotency and Abelian Defect Groups

Key results, primarily for quasi-isolated blocks in finite groups of Lie type, show that if all height zero characters in a block have the same degree, then the defect group must be abelian, and the block is therefore nilpotent. This is proved for quasi-isolated blocks of exceptional groups (see (1112.2642)) and follows from the explicit parametrization of blocks in terms of Lusztig series and cuspidal pairs.

More generally:

If all height zero characters have equal degree, then:
- The block defect group is abelian.
- The block is nilpotent.
- There is a precise Morita equivalence to the Brauer correspondent in the normalizer.

The explicit structure of defect groups and their relation to Weyl groups

$P \cong Z(L)_\ell^F \rtimes W_{^F}(L,\lambda)$

guarantees that nilpotency is characterized by the triviality of $W_{^F}(L,\lambda)$ ’s $\ell$ -part, thus abelianity and equal degrees among height zero characters are inextricably linked (1112.2642).

3. Arithmetic and Field of Values Aspects

The field of values of height zero irreducible characters reveals how refined the equal-degree phenomenon can be. The analysis in (Navarro et al., 2023) proves, for $p=2$ , that the fields of values for 2-height zero characters are precisely the abelian number fields contained in cyclotomic fields of conductor $2^a m$ (with $m$ odd): $\mathbb{Q}_{2^a} \subset \mathbb{Q}_m(\chi)$ and these exhaust all such fields for $p=2$ -height zero characters.

For odd primes $p$ , it is conjectured:

The set of fields of values for $p$ -height zero characters is those abelian number fields $F$ with conductor $n=p^a m$ and such that $|O_{p^a} : (\mathbb{Q}_m \cap F)|$ is not divisible by $p$ .

Moreover, the reduction to blocks of quasi-simple groups (Theorem 6.3 in (Navarro et al., 2023)) suggests the universality of local-global principles reflected in equal-degree conjectures.

4. Methods and Technical Reductions

Several critical methodologies underpin the proofs and reductions of the conjecture:

Lusztig Induction and Harish–Chandra Theory:

The parametrization of characters and blocks via $e$ -cuspidal pairs and Lusztig series reduces global problems to the structure of relative Weyl groups and $e$ -cuspidality (1112.2642, Kessar et al., 2015).

Morita/Rouquier Equivalences:

$\ell$ –Rouquier Morita equivalences transfer structural properties, including degree and abelianity of defect groups, between blocks of related groups or subgroups.

Reduction to Quasi-simple/Almost Simple Groups:

By leveraging local-global correspondences and known results for quasi-simple groups, the general conjecture is often reduced to this foundational class (Kessar et al., 2015, Navarro et al., 2023).

Dade’s Projective Conjecture as a Tool:

The inequality $mh(D) \leq mh(B)$ is shown to be a consequence of Dade’s conjecture, while the converse $mh(B) \leq mh(D)$ is established for principal blocks with defect groups having two character degrees (Malle et al., 2023). This pairing yields the full equality for such blocks.

Fusion Systems and Character Triples:

Tools from fusion systems and character triple theory control the behavior of extensions and invariants under automorphism groups, allowing the transference of local equal-degree conditions to global statements (Sambale, 2018, Malle et al., 2022).

5. Representative Results and Conjectures

The following table organizes some principal consequences and equivalent statements addressed in the literature:

Conjecture/Formulation	Block Assumptions	Consequence
Equal degree for all height zero irreducibles	General, nilpotent block	Block is nilpotent (Morita equivalent) (1112.2642)
$mh(B) = mh(D)$	B principal block, $cd(P)=\{1,p^a\}$	Heights coincide, gives evidence for general conjecture (Malle et al., 2023)
Field of values $\mathbb{Q}_{p^a} \subset \mathbb{Q}_m(\chi)$	$p$ -height zero character $\chi$	Holds for $p=2$ , conjectured for odd $p$ (Navarro et al., 2023)
All height zero irreducibles degrees constant	Block of quasi-simple group	Defect group is abelian, block nilpotent (1112.2642, Kessar et al., 2015)

The Height-Zero-Equal-Degree Conjecture is conceptually linked to:

Brauer’s Height Zero Conjecture (BHZ): All irreducible characters in a block having height zero $\Leftrightarrow$ defect group abelian. Equal-degree refinements often imply or are implied by BHZ (Kessar et al., 2015, Malle et al., 2022).
Projective Height Zero (Malle–Navarro): Incorporates central $p$ -subgroups and fields of values (Malle et al., 2017, Sambale, 2018).
Alperin–McKay and Alperin–McKay–Navarro Conjectures: Field of values and degree invariance under global correspondences is a core motif (Navarro et al., 2023).
Dade’s Projective Conjecture: Controls the lower bound for minimal heights.

Within this broader landscape, confirming the equal-degree conjectures strengthens the dictionary between local subgroup structure and global block-theoretic phenomena, enabling finer classification and deeper understanding of representation-theoretic invariants.

7. Open Problems and Future Directions

Major cases are resolved for principal blocks with specific defect group properties and for quasi-isolated blocks in finite groups of Lie type. However, challenges remain:

Proving the conjecture for arbitrary blocks beyond these special cases.
Extending equivalences for fields of values in the arithmetic setting for odd $p$ (Navarro et al., 2023).
Analyzing blocks with more than two character degrees, especially in the presence of non-nilpotent structure (Malle et al., 2023).
Incorporating global-local correspondence conjectures with Galois automorphism actions (Malle et al., 13 Feb 2024).

A plausible implication is that the machinery developed for the proof of BHZ (inductive reductions, automorphism control, and Morita equivalences) can, with additional refinements, yield a final proof of the Height-Zero-Equal-Degree Conjecture in full generality.

In conclusion, the Height-Zero-Equal-Degree Conjecture bridges the arithmetic and group-theoretic structure of finite groups via block theory, linking uniformity in character heights and degrees to block nilpotency, abelian defect groups, and deep local-global phenomena in representation theory. Its resolution in major classes of blocks provides structural insights and numeric invariants that continue to drive progress in the field.