Hung–Martínez–Navarro conjecture

The conjecture concerns a lower bound and equality criterion for sums of squares of character degrees of finite groups. An earlier paper treated the statement as a general conjecture, while a note dated 15 August 2026 claims an unconditional proof.

Known results

• The inequality was known for $p=2$ through work of Hung, Martínez and Navarro.
• The earlier paper proved the inequality and equality characterization for $p=2$, using the corresponding strengthening of the McKay conjecture.
• For general $p$, the earlier paper reduced the inequality to Giannelli’s strengthening of the McKay conjecture and proved the equality case assuming that strengthening.

15 August 2026 claimed proof

The note On a conjecture of Hung, Martínez and Navarro claims, for every prime $p$, that

$$\sum_{\chi\in\operatorname{Irr}_{p'}(G)}\chi(1)^2\geq |N_G(P):P'|,$$

with equality exactly when $N_G(P)$ has a normal complement in $G$. It says the proof uses the now-proven McKay conjecture and does not assume Giannelli’s stronger conjecture; the preprint is unrefereed and remains unconfirmed.

Current status (as of August 2026): The conjecture has a claimed proof for all primes, but its correctness has not been independently verified.