On H\'{e}thelyi-K\"{u}lshammer's conjecture for principal blocks

We prove that the number of irreducible ordinary characters in the principal p-block of a finite groupG of order divisible by p is always at least 2 √ p− 1. This confirms a conjecture of Héthelyi and Külshammer [HK00] for principal blocks and provides an affirmative answer to Brauer’s Problem 21 [Bra63] for principal blocks of bounded defect. Our proof relies on recent works of Maróti [Mar16] and Malle-Maróti [MM16] on bounding the conjugacy class number and the number of p-degree irreducible characters of finite groups, earlier works of Broué-Malle-Michel [BMM93] and Cabanes-Enguehard [CE94, CE04] on the distribution of characters into unipotent blocks and e-Harish-Chandra series of finite reductive groups, and known cases of the Alperin-McKay conjecture.

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