View source for McKay-Alperin conjecture

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

<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>

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.

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.

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

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  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)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  J. McKay,   "Irreducible representations of odd degree"  ''J. Algebra'' , '''20'''  (1972)  pp. 416–418</TD></TR></table>