Difference between revisions of "Brauer height-zero conjecture"
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− | + | {{MSC|20C20|20C33}} | |
+ | {{TEX|done}} | ||
− | + | For notation and definitions, see also | |
+ | [[Brauer first main theorem|Brauer first main theorem]]. | ||
− | + | Let $\chi$ be an irreducible character in a block $B$ of a | |
+ | [[Group|group]] $G$ with defect group $D$ (cf. also | ||
+ | [[Defect group of a block|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 | ||
− | Every block contains an irreducible character of height zero. Brauer's height-zero conjecture is the assertion that every irreducible character in | + | $$\nu(\chi(1))-\nu(|G:D|).$$ |
+ | 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|Abelian group]]). | ||
− | That every irreducible character in | + | That every irreducible character in $B$ has height zero when $D$ is Abelian was proved for $p$-solvable groups (cf. also |
+ | [[Pi-solvable group|$\pi$-solvable group]]) by P. Fong (see | ||
+ | {{Cite|Fe}}, X.4). The converse for $p$-solvable groups was proved by D. Gluck and T. Wolf | ||
+ | {{Cite|GlWo}}, using the | ||
+ | [[Simple finite group|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 | ||
+ | {{Cite|BeKn}}. The task of checking this half of the conjecture for the quasisimple groups was completed in 2011 by R. Kessar and G. Malle | ||
+ | {{Cite|KeMa}}, hence completing the proof of the "if" direction. The evidence for the "only if" direction is more slender. | ||
====References==== | ====References==== | ||
− | + | {| | |
+ | |- | ||
+ | |valign="top"|{{Ref|BeKn}}||valign="top"| T.R. Berger, R. Knörr, "On Brauer's height $0$ conjecture" ''Nagoya Math. J.'', '''109''' (1988) pp. 109–116 {{MR|0931954}} {{ZBL|0637.20006}} | ||
+ | |- | ||
+ | |valign="top"|{{Ref|Fe}}||valign="top"| W. Feit, "The representation theory of finite groups", North-Holland (1982) {{MR|0661045}} {{ZBL|0493.20007}} | ||
+ | |- | ||
+ | |valign="top"|{{Ref|GlWo}}||valign="top"| D. Gluck, T.R. Wolf, "Brauer's height conjecture for $p$-solvable groups" ''Trans. Amer. Math. Soc.'', '''282''' : 1 (1984) pp. 137–152 {{MR|0728707}} {{ZBL|0543.20007}} | ||
+ | |- | ||
+ | |valign="top"|{{Ref|KeMa}}||valign="top"| R. Kessar, G. Malle, "Quasi-isolated blocks and Brauer's height conjecture" arXiv:1112.2642 | ||
+ | |- | ||
+ | |} |
Latest revision as of 15:21, 13 April 2012
2010 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 |
Brauer height-zero conjecture. Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Brauer_height-zero_conjecture&oldid=24289