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Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120180/m1201801.png" /> be a [[Finite group|finite group]]. For any prime number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120180/m1201802.png" />, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120180/m1201803.png" /> be the number of irreducible complex characters of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120180/m1201804.png" /> with degree prime to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120180/m1201805.png" /> (cf. also [[Character of a group|Character of a group]]). The simplest form of the McKay–Alperin conjectures asserts that
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Let $G$ be a [[Finite group|finite group]]. For any prime number $p$, let $m_p(G)$ be the number of irreducible complex characters of $G$ with degree prime to $p$ (cf. also [[Character of a group|Character of a group]]). The simplest form of the McKay–Alperin conjectures asserts that
  
<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/m/m120/m120180/m1201806.png" /></td> </tr></table>
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$$m_p(G)=m_p(N_G(P)),$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120180/m1201807.png" /> is a Sylow <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120180/m1201808.png" />-subgroup of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120180/m1201809.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120180/m12018010.png" /> is its normalizer (cf. also [[Sylow subgroup|Sylow subgroup]]; [[P-group|<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120180/m12018011.png" />-group]]; [[Normalizer of a subset|Normalizer of a subset]]). J. McKay [[#References|[a2]]] first suggested this might be true when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120180/m12018012.png" /> is a [[Simple group|simple group]]. J.L. Alperin [[#References|[a1]]] observed that it is probably true for all finite groups.
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where $P$ is a Sylow $p$-subgroup of $G$ and $N_G(P)$ is its normalizer (cf. also [[Sylow subgroup|Sylow subgroup]]; [[P-group|$p$-group]]; [[Normalizer of a subset|Normalizer of a subset]]). J. McKay [[#References|[a2]]] first suggested this might be true when $G$ is a [[Simple group|simple group]]. J.L. Alperin [[#References|[a1]]] observed that it is probably true for all finite groups.
  
Alperin also made a more general conjecture, involving characters in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120180/m12018013.png" />-blocks. (See [[Brauer first main theorem|Brauer first main theorem]] for notation and definitions.) Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120180/m12018014.png" /> be a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120180/m12018015.png" />-block of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120180/m12018016.png" /> with defect group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120180/m12018017.png" /> (cf. [[Defect group of a block|Defect group of a block]]), and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120180/m12018018.png" /> be an irreducible character (cf. also [[Irreducible representation|Irreducible representation]]) belonging to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120180/m12018019.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120180/m12018020.png" /> be a Sylow <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120180/m12018021.png" />-subgroup of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120180/m12018022.png" />. By a theorem of R. Brauer, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120180/m12018023.png" /> divides the degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120180/m12018024.png" />. The character <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120180/m12018025.png" /> is said to have height zero if the largest power of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120180/m12018026.png" /> dividing <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120180/m12018027.png" /> is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120180/m12018028.png" />. The more general Alperin conjecture asserts that the number of irreducible characters of height zero in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120180/m12018029.png" /> is equal to the number of irreducible characters of height zero in the unique block of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120180/m12018030.png" /> sent to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120180/m12018031.png" /> by the Brauer correspondence.
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Alperin also made a more general conjecture, involving characters in $p$-blocks. (See [[Brauer first main theorem|Brauer first main theorem]] for notation and definitions.) Let $B$ be a $p$-block of $G$ with defect group $D$ (cf. [[Defect group of a block|Defect group of a block]]), and let $\chi$ be an irreducible character (cf. also [[Irreducible representation|Irreducible representation]]) belonging to $B$. Let $P$ be a Sylow $p$-subgroup of $G$. By a theorem of R. Brauer, $|P|/|D|$ divides the degree $\chi(1)$. The character $\chi$ is said to have height zero if the largest power of $p$ dividing $\chi(1)$ is $|P|/|D|$. The more general Alperin conjecture asserts that the number of irreducible characters of height zero in $B$ is equal to the number of irreducible characters of height zero in the unique block of $N_G(D)$ sent to $B$ by the Brauer correspondence.
  
 
The conjectures are still not proved (1998), but the evidence in their favour is very strong.
 
The conjectures are still not proved (1998), but the evidence in their favour is very strong.

Revision as of 21:18, 9 July 2014

Let $G$ be a finite group. For any prime number $p$, let $m_p(G)$ be the number of irreducible complex characters of $G$ with degree prime to $p$ (cf. also Character of a group). The simplest form of the McKay–Alperin conjectures asserts that

$$m_p(G)=m_p(N_G(P)),$$

where $P$ is a Sylow $p$-subgroup of $G$ and $N_G(P)$ is its normalizer (cf. also Sylow subgroup; $p$-group; Normalizer of a subset). J. McKay [a2] first suggested this might be true when $G$ is a simple group. J.L. Alperin [a1] observed that it is probably true for all finite groups.

Alperin also made a more general conjecture, involving characters in $p$-blocks. (See Brauer first main theorem for notation and definitions.) Let $B$ be a $p$-block of $G$ with defect group $D$ (cf. Defect group of a block), and let $\chi$ be an irreducible character (cf. also Irreducible representation) belonging to $B$. Let $P$ be a Sylow $p$-subgroup of $G$. By a theorem of R. Brauer, $|P|/|D|$ divides the degree $\chi(1)$. The character $\chi$ is said to have height zero if the largest power of $p$ dividing $\chi(1)$ is $|P|/|D|$. The more general Alperin conjecture asserts that the number of irreducible characters of height zero in $B$ is equal to the number of irreducible characters of height zero in the unique block of $N_G(D)$ sent to $B$ by the Brauer correspondence.

The conjectures are still not proved (1998), but the evidence in their favour is very strong.

References

[a1] J.L. Alperin, "The main problem of block theory" W.R. Scott (ed.) F. Gross (ed.) , Proc. Conf. Finite Groups (Park City, Utah, 1975) , Acad. Press (1976)
[a2] J. McKay, "Irreducible representations of odd degree" J. Algebra , 20 (1972) pp. 416–418
How to Cite This Entry:
McKay-Alperin conjecture. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=McKay-Alperin_conjecture&oldid=22803
This article was adapted from an original article by H. Ellers (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article