Eaton–Moretó conjecture for principal blocks

The Eaton–Moretó conjecture compares character degrees of a nonabelian Sylow $p$-subgroup with those of characters in the principal $p$-block. The general principal-block case is not settled.

Known results

• Finite $p$-solvable groups: the full principal-block equality was proved in 2024, conditional on the Dade–Robinson conjecture for the opposite inequality.
• Quasi-simple groups: substantial cases were proved, including all stated cases whose component is not of Lie type in non-defining characteristic; some cases for $p=2,3$ remain.
• Earlier work established the equality for further quasi-simple-group cases and related positive-height questions.

August 2026 unconditional inequality

Arranz, Gómez-Serrano, Navarro, and Schaeffer Fry prove that, for every finite group with nonabelian Sylow $p$-subgroup $P$, if the smallest nonlinear character degree of $P$ is $p^m$, then some principal-block character $\chi$ satisfies $1<\chi(1)_p\leq p^m$. They state that Dade’s Projective conjecture would supply the reverse inequality and hence the full conjecture. This is a preprint claim with no independent verification found.

Current status (as of August 2026): One inequality is proved unconditionally, and the full statement is known for finite $p$-solvable groups, but the general conjecture remains open pending Dade’s Projective conjecture.