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 p^t. 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.