Corrections To: “a Murnaghan–nakayama Rule for Values of Unipotent Characters in Classical Groups”

We settle a missing case in the proof of one of the main applications of our results in [Frank Lübeck and Gunter Malle, A Murnaghan– Nakayama rule for values of unipotent characters in classical groups, Represent. Theory 20 (2016), 139–161, MR3466537]. In [5] we derived a Murnaghan–Nakayama formula for the values of unipotent characters of finite classical groups. As one application, we showed that the first Cartan invariant c ( ) 11 of a finite simple classical group G in non-defining characteristic is never equal to 2 if Sylow -subgroups of G are non-cyclic. 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]. Here we give a short argument for the missing case. The argument also covers the analogous statement in [4, Prop. 6.3] pertaining to simple groups of exceptional Lie type. The crucial observation is that Sylow p-subgroups in groups of Lie type of characteristic p are large. Except for a few small rank cases, we do not need any knowledge about the ordinary character table of the group G. Note that if c (p) 11 = 2, then there exists χ ∈ Irr(G) such that 1G+χ is the character of a projective module; in particular, χ(g) = −1 for all p-singular elements g ∈ G. Lemma 1. Let G be a finite group and p a prime. Assume that H ≤ G is a subgroup such that |G : H| is divisible by p, but |G : H| < |G|p. Then c 11 > 2. Proof. The permutation character π of G on the cosets of H has exactly one trivial constituent 1G. On the other hand, as |G : H| is divisible by p, the modular reduction of the permutation representation has (at least) two trivial modular constituents: one in the socle, one in the head. Thus, the p-modular reduction of one of the non-trivial ordinary constituents χ of π must contain the trivial p-modular character of G. By Brauer reciprocity, χ then occurs in the projective cover Ψ of the trivial character. If we had c (p) 11 = 2, then necessarily Ψ = 1G + χ. Since the degree of any projective character is divisible by |G|p, this is a contradiction to |G|p ≤ Ψ(1) = 1G(1) + χ(1) ≤ π(1) = |G : H|. Received by the editors October 25, 2016. 2010 Mathematics Subject Classification. Primary 20C20; Secondary 20C33.