Principal blocks with 5 irreducible characters

Principal Blocks for Different Primes, II

“…With these constraints, we see that S is not of Suzuki or Ree type, that p and q are at most 7, and that d is a regular number in the sense of Springer [34] (see also [33,Definition 2.5]). Hence we see that the principal blocks B p (S) and B q (S) are the unique blocks of S containing p -, respectively, q -degree unipotent characters (see, e.g., [31,Lemma 3.6]). Under these conditions, we see by observing the explicit list of unipotent character degrees in [5, Section 13.9] that there exists a unipotent character χ satisfying either χ ∈ Irr p (B p (S)) and q | χ(1) or χ ∈ Irr q (B q (S)) and p | χ (1).…”

“…(Indeed, that s ∈ G = O r 0 ( G) implies that the corresponding semisimple character χ s is trivial on Z( G) and that sz is not conjugate to s for nontrivial z ∈ Z( G) implies that χ s is irreducible on restriction to G using e.g. [32,Proposition 2.7] and [31,Lemma 1.4]; the remaining conditions χ s ∈ Irr(B p ( G)) using [12,Corollary 3.4] and χ s (1) is p but divisible by q since χ s (1) = [ G : C G (s)] r 0 .) Now, if 2 ∈ {p, q}, we see using the results of Weir [35] that P and Q are naturally isomorphic to the corresponding Sylow subgroups of GL we (r), embedded naturally into GL we (r) × GL b (r) ≤ GL n (r) where n = we + b with 0 ≤ b < e. The results of Carter-Fong [6] yield the same when p or q is 2, except that if (e, b) = (2, 1), then |GL b (r)| is divisible by 2 exactly once, and the Sylow 2-subgroup of GL n (r) in this case is that of GL we (r) × GL b (r).…”

Height zero characters in principal blocks

“…Let p be a prime. Recall that the principal p-block of a finite group G is the one containing the principal character 1 G and that its defect groups are the Sylow p-subgroups of G. Thanks to the recent work of Koshitani-Sakurai [KS21] and of Rizo-Schaeffer Fry-Vallejo [RSV21], those Sylow p-subgroups of finite groups with principal p-blocks having up to five irreducible characters have been completely determined. This in turn has contributed to the solution of the principal-block case [HS21] of Héthelyi-Külshammer's conjecture [HK00] and to obtaining a p-local lower bound for the number of height-zero characters in principal blocks [HSV23].…”

The blocks with four irreducible characters

“…It is well known that if 𝑛 = 1 or 2, then 𝑘(𝐵) = 𝑛 if and only if |𝐷| = 𝑛 (see [15,Theorem 3.18] and [1]); for 𝑛 = 3, this is known to be a consequence of the Alperin-McKay conjecture, but no proof is yet available. Although the cases where 𝐵 is a principal block and 𝑘(𝐵) = 4 or 5 have been recently solved in [9] and [19], the non-principal block cases remain open. It is well known that many blocks with 𝑘(𝐵) = 4 have defect groups with |𝐷| = 4 or 5 (for instance 2.𝖠 5 for 𝑝 = 5, or 2.𝖲 5 for 𝑝 = 2), but it is not known if these are the only possibilities, even assuming the Alperin-McKay conjecture.…”

“…It is well known that if $$n=1$$ or $$\hskip.001pt 2$$, then $$k(B)=n$$ if and only if $$|D|=n$$ (see [15, Theorem 3.18] and [1]); for $$n=3$$, this is known to be a consequence of the Alperin–McKay conjecture, but no proof is yet available. Although the cases where $$B$$ is a principal block and $$k(B)=4$$ or 5 have been recently solved in [9] and [19], the non‐principal block cases remain open. It is well known that many blocks with $$k(B)=4$$ have defect groups with $$|D|=4$$ or $$\hskip.001pt 5$$ (for instance $$2.$…”