McKay-Alperin conjecture
Let be a finite group. For any prime number
, let
be the number of irreducible complex characters of
with degree prime to
(cf. also Character of a group). The simplest form of the McKay–Alperin conjectures asserts that
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where is a Sylow
-subgroup of
and
is its normalizer (cf. also Sylow subgroup;
-group; Normalizer of a subset). J. McKay [a2] first suggested this might be true when
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 -blocks. (See Brauer first main theorem for notation and definitions.) Let
be a
-block of
with defect group
(cf. Defect group of a block), and let
be an irreducible character (cf. also Irreducible representation) belonging to
. Let
be a Sylow
-subgroup of
. By a theorem of R. Brauer,
divides the degree
. The character
is said to have height zero if the largest power of
dividing
is
. The more general Alperin conjecture asserts that the number of irreducible characters of height zero in
is equal to the number of irreducible characters of height zero in the unique block of
sent to
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 |
McKay-Alperin conjecture. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=McKay-Alperin_conjecture&oldid=13806