PARTITION-IDENTITIES AND LABELS FOR SOME MODULAR CHARACTERS

In this paper we prove two conjectures on partitions with certain conditions. A motivation for this is given by a problem in the modular representation 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 author in his memoir [A3]. Recently the second and third author (jointly with A. O. Morris) arrived at essentially the same conjecture from a completely different direction. Their paper [BMO] was concerned with decomposition matrices of S(n) in characteristic 3. A basic difficulty for obtaining similar results in characteristic 5 (or larger) was the lack of a class of partitions which would be ''natural'' character labels for the modular characters of these groups. In this connection two conjectures were stated (Conjectures A and B* below), whose solutions would 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 partitions which should serve as the required labels.