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

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