Ge. Andrews et al., PARTITION-IDENTITIES AND LABELS FOR SOME MODULAR CHARACTERS, Transactions of the American Mathematical Society, 344(2), 1994, pp. 597-615
In this paper we prove two conjectures on partitions with certain cond
itions. A motivation for this is given by a problem in the modular rep
resentation theory of the covering groups S(n) of the finite symmetric
groups S(n) in characteristic 5. One of the conjectures (Conjecture B
below) has been open since 1974, when it was stated by the first auth
or in his memoir [A3]. Recently the second and third author (jointly w
ith A. O. Morris) arrived at essentially the same conjecture from a co
mpletely different direction. Their paper [BMO] was concerned with dec
omposition matrices of S(n) in characteristic 3. A basic difficulty fo
r obtaining similar results in characteristic 5 (or larger) was the la
ck of a class of partitions which would be ''natural'' character label
s for the modular characters of these groups. In this connection two c
onjectures were stated (Conjectures A and B below), whose solutions w
ould be helpful in the characteristc 5 case. One of them, Conjecture B
, is equivalent to the old Conjecture B mentioned above. Conjecture A
is concerned with a possible inductive definition of the set of parti
tions which should serve as the required labels.