Namespaces
Variants
Actions

Difference between revisions of "Brauer height-zero conjecture"

From Encyclopedia of Mathematics
Jump to: navigation, search
(reference to simple finite group)
m (tex,msc,mr,zbl)
 
Line 1: Line 1:
For notation and definitions, see also [[Brauer first main theorem|Brauer first main theorem]].
+
{{MSC|20C20|20C33}}
 +
{{TEX|done}}
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120450/b1204501.png" /> be an irreducible character in a block <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120450/b1204502.png" /> of a [[Group|group]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120450/b1204503.png" /> with defect group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120450/b1204504.png" /> (cf. also [[Defect group of a block|Defect group of a block]]). Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120450/b1204505.png" /> be the discrete valuation defined on the integers with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120450/b1204506.png" /> whenever <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120450/b1204507.png" /> is prime to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120450/b1204508.png" />. By a theorem of Brauer, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120450/b1204509.png" />. The height of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120450/b12045010.png" /> is defined to be
+
For notation and definitions, see also
 +
[[Brauer first main theorem|Brauer first main theorem]].
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120450/b12045011.png" /></td> </tr></table>
+
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120450/b12045012.png" /> has height zero if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120450/b12045013.png" /> is Abelian (cf. also [[Abelian group|Abelian group]]).
+
$$\nu(\chi(1))-\nu(|G:D|).$$
 +
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 <img align="absmiddle" border="0"  src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120450/b12045012.png" /> has height zero when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120450/b12045013.png" /> is Abelian was proved for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120450/b12045014.png" />-solvable groups (cf. also [[Pi-solvable group|<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120450/b12045015.png" />-solvable group]]) by P. Fong (see [[#References|[a2]]], X.4). The converse for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120450/b12045014.png" />-solvable groups was proved by D. Gluck and T. Wolf [[#References|[a3]]], 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 [[#References|[a1]]]. The task of checking this half of the conjecture for the quasisimple groups was completed in 2011 by R. Kessar and G. Malle [[#References|[a4]]], hence completing the proof of the "if" direction. The evidence for the  "only if"  direction is more slender.
+
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====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> T.R. Berger,  R. Knörr,  "On Brauer's height <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120450/b12045016.png" /> conjecture"  ''Nagoya Math. J.'' , '''109'''  (1988)  pp. 109–116</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> W. Feit,  "The representation theory of finite groups" , North-Holland  (1982)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top"> D. Gluck,  T.R. Wolf,  "Brauer's height conjecture for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120450/b12045017.png" />-solvable groups"  ''Trans. Amer. Math. Soc.'' , '''282''' :  1  (1984)  pp. 137–152</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top"> R. Kessar, G. Malle,  "Quasi-isolated blocks and Brauer's height conjecture" arXiv:1112.2642</TD></TR></table>
+
{|
 +
|-
 +
|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
How to Cite This Entry:
Brauer height-zero conjecture. Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Brauer_height-zero_conjecture&oldid=24289
This article was adapted from an original article by H. Ellers (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article