Brauer height-zero conjecture

2020 Mathematics Subject Classification: Primary: 20C20 Secondary: 20C33 [MSN][ZBL]

For notation and definitions, see also Brauer first main theorem.

Let $\chi$ be an irreducible character in a block $B$ of a group $G$ with defect group $D$ (cf. also Defect group of a block). Let $\nu$ be the discrete valuation defined on the integers with $\def\a{\alpha}\nu(np^\a)=\a$ whenever $n$ is prime to $p$. By a theorem of Brauer, $\nu(\chi(1)\ge \nu(|G:D|)$. The height of $\chi$ is defined to be

$$\nu(\chi(1))-\nu(|G:D|).$$ Every block contains an irreducible character of height zero. Brauer's height-zero conjecture is the assertion that every irreducible character in $B$ has height zero if and only if $D$ is Abelian (cf. also Abelian group).

That every irreducible character in $B$ has height zero when $D$ is Abelian was proved for $p$-solvable groups (cf. also $\pi$-solvable group) by P. Fong (see [Fe], X.4). The converse for $p$-solvable groups was proved by D. Gluck and T. Wolf [GlWo], using the classification of finite simple groups. The "if" direction has been reduced to the consideration of quasi-simple groups by T.R. Berger and R. Knörr [BeKn]. The task of checking this half of the conjecture for the quasisimple groups was completed in 2011 by R. Kessar and G. Malle [KeMa], hence completing the proof of the "if" direction. The evidence for the "only if" direction is more slender.

References

[BeKn]	T.R. Berger, R. Knörr, "On Brauer's height $0$ conjecture" Nagoya Math. J., 109 (1988) pp. 109–116 MR0931954 Zbl 0637.20006
[Fe]	W. Feit, "The representation theory of finite groups", North-Holland (1982) MR0661045 Zbl 0493.20007
[GlWo]	D. Gluck, T.R. Wolf, "Brauer's height conjecture for $p$-solvable groups" Trans. Amer. Math. Soc., 282 : 1 (1984) pp. 137–152 MR0728707 Zbl 0543.20007
[KeMa]	R. Kessar, G. Malle, "Quasi-isolated blocks and Brauer's height conjecture" arXiv:1112.2642