On Loewy lengths of blocks

Abstract

We give a lower bound on the Loewy length of a p-block of a finite group in terms of its defect. We then discuss blocks with small Loewy length. Since blocks with Loewy length at most 3 are known, we focus on blocks of Loewy length 4 and provide a relatively short list of possible defect groups. It turns out that p-solvable groups can only admit blocks of Loewy length 4 if p = 2. However, we find (principal) blocks of simple groups with Loewy length 4 and defect 1 for all p ≡ 1 (mod 3). We also consider sporadic, symmetric and simple groups of Lie type in defining characteristic. Finally, we give stronger conditions on the Loewy length of a block with cyclic defect group in terms of its Brauer tree.

The Projective Cover of the Trivial Representation for a Finite Group of Lie Type in Defining Characteristic

“…This sheds new light on a couple of our earlier results in [24]. Firstly, we get the following, providing an alternative proof of [24,Proposition 4.12]:…”

“…Then we have O p ′ (H) = H, and H ≤ G being a characteristic subgroup, from O p ′ (G) = {1} we infer O p ′ (H) = {1}.By [26, Lemma 4.1] we have LL(B 0 (H)) = 4, where B 0 (H) is the principal block algebra of kH. Hence we have LL(P (k H )) ≤ 4, and thus from Proposition 3.2 we get that the heart S := H(k H ) of P (k H ) is simple; see also[24, Proposition 4.6]. Moreover, if S ∼ = k H , then from O p ′ (H) = {1} we conclude that H ∼ = C 3 , hence LL(B 0 (H)) = LL(P (k H )) = 3, a contradiction.…”

“…Moreover, this is also related to the recent result [29,Theorem 7.1] saying that, if G is a classical simple finite group of Lie type, and l is an odd prime such that the Sylow l-subgroups are non-cyclic, then the l-modular Cartan invariant c (l) 11 (G) ≥ 3. But in the proof given there the defining characteristic case l = p is erroneously attributed to [24,Proposition 4.12], which only states that the Loewy length of the block algebra B 0 (G) as a whole is LL(B 0 (G)) = 4, but nothing comprehensive is said about P (k G ) or c 11 (G). Actually, when we were preparing the present manuscript, we noticed the above-mentioned gap, and informed the authors of [29] about it, and after this manuscript was completed we learned that they have been able to close the gap in [30].…”

Corrections to: “A Murnaghan–Nakayama rule for values of unipotent characters in classical groups”

“…But contrary to what was claimed in the first line of the proof of [5,Thm. 7.1], the same assertion for the case of defining characteristic does not follow from the work of Koshitani, Külshammer, and Sambale [2]. We thank Shigeo Koshitani and Jürgen Müller for pointing this out to us; see [3].…”

Finite groups whose first Cartan invariants over a field of characteristic two are two

“…Finite groups G such that c 11 (G) = 2 when k has an odd prime characteristic p, are discussed in a recent paper [21].…”