The a
im of th
is paper
is to ver
ify the
induct
ive AM cond
it
ion stated
in (Def
in
it
ion 7.2) for the s
imple alternat
ing groups
in character
ist
ic 2. Such a cond
it
ion,
if checked for all s
imple groups
in all character
ist
ics would prove the Alper
in-McKay conjecture (see ). We f
irst check the Alper
in-McKay conjecture for double cover of symmetr
ic and alternat
ing groups, and . The proof of th
is w
ill extend known results about blocks of symmetr
ic and alternat
ing groups of a g
iven we
ight.
The first two parts are notations and basic results on blocks of symmetric, alternating groups and of their double covers that are recalled for convenience. In the third part we determine the number of height zero spin characters of a given block using certain height preserving bijections between blocks of a given weight. Then we determine the number of height zero spin characters in a given block of the normalizer of a defect group. We finish by showing that the inductive AM condition is true for the alternating groups in characteristic 2, the cases and being treated separately from the rest.
I would like to thank Marc Cabanes and Britta Sp盲th for taking the time to discuss all the mathematical issues I encountered while writing this paper.