Eaton–Moretó Conjecture: Equality of minimal positive heights for a block and its defect group

Establish that for any finite group G, any prime p, and any Brauer p-block B of G with non-abelian defect group D, the minimal positive height among irreducible characters in B equals the minimal positive height determined by irreducible characters of D; that is, prove mh(B) = mh(D), where mh(B) denotes the minimum of the positive heights h(χ) (defined by χ(1)_p = p^{a−d+h(χ)} for χ ∈ Irr(B), with d the defect of B) and mh(D) denotes the smallest integer t ≥ 1 such that D has an irreducible character of degree p^t.

Background

The paper discusses the Eaton–Moretó conjecture, an extension of Brauer’s Height Zero conjecture. For a p-block B of a finite group G with defect group D, each χ ∈ Irr(B) has a height h(χ) determined by the p-part of χ(1). Let mh(B) denote the minimal positive height among characters in Irr(B).

The conjecture asserts that mh(B) equals a quantity determined by the defect group D, typically interpreted as the minimal exponent t such that D has an irreducible character of degree pt. The author proves this equality for principal p-blocks of p-solvable groups, but the general conjecture remains of broad interest beyond this case and has been studied in various settings.

References

If mh(B) is the minimum of the non-zero heights of Irr(B), then the Eaton–Moretó conjecture proposes that mh(B) = mh(D).

— The Eaton-Moreto Conjecture and p-Solvable Groups  (2409.01634 - Navarro, 2024) in Section 1. Introduction

Suppose now that $P$ is nonabelian. In , C. Eaton and A. Moretó conjectured that $m=h$, where $pm$ is the smallest degree greater than $1$ of an irreducible character of $P$, and $h$ is the smallest positive height among the irreducible characters in $B$.

— On The Eaton-Moretó Conjecture for Principal Blocks of Finite Groups  (2608.16398 - Arranz et al., 17 Aug 2026) in Section 1, Introduction

Eaton and Moretó showed in that Dade's Projective Conjecture implies the inequality $m \leq h$. Since Dade's conjecture is widely believed to hold, attention has therefore focused on proving the reverse inequality $h \leq m$.

— On The Eaton-Moretó Conjecture for Principal Blocks of Finite Groups  (2608.16398 - Arranz et al., 17 Aug 2026) in Section 1, Introduction