Minimal heights and defect groups with two character degrees.(English)Zbl 1540.20028
The paper under review proves a particular case of a global-local conjecture in the modular representation theory of finite groups.C. Eaton andA. Moretó [Int. Math. Res. Not. 2014, No. 20, 5581–5601 (2014;Zbl 1348.20009)] conjectured that the minimal positive height \(\operatorname{mh}(B)\) of the irreducible characters in a \(p\)-block \(B\) equals the minimal positive height \(\operatorname{mh}(D)\) of the irreducible characters of its defect group \(D\). This is an extension of Brauer’s height zero conjecture (which was proved recently [the first author et al., “Brauer’s height zero conjecture”, Preprint,arXiv:2209.04736]). Eaton and Moretó [loc. cit] showed that the inequality \(\operatorname{mh}(D) \leq \operatorname{mh}(B)\) follows from Dade’s projective conjecture. In the paper under review, the other inequality \(\operatorname{mh}(B) \leq \operatorname{mh}(D)\) is shown for principal blocks \(B\) whose defect group \(D\) has exactly two character degrees. The proof uses the classification of finite simple groups and also the classification of finite linear groups in characteristic \(p\) and order divisible by \(p\), but with no orbits of size divisible by \(p\) [M. Giudici et al., Trans. Am. Math. Soc. 368, No. 4, 2415–2467 (2016;Zbl 1382.20003)].
Reviewer: Frieder Ladisch (Rostock)
MSC:
| 20C20 | Modular representations and characters |
| 20C15 | Ordinary representations and characters |
| 20C33 | Representations of finite groups of Lie type |
Software:
GAPCite
References:
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