Partition identities and labels for some modular characters

Authors:
G. E. Andrews, C. Bessenrodt and J. B. Olsson

Journal:
Trans. Amer. Math. Soc. **344** (1994), 597-615

MSC:
Primary 11P83; Secondary 05A17, 20C25

DOI:
https://doi.org/10.1090/S0002-9947-1994-1220904-1

MathSciNet review:
1220904

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Abstract: 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 ${\hat S_n}$ 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 ${\hat S_n}$ 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 ${B^\ast }$ below), whose solutions would be helpful in the characteristic 5 case. One of them, Conjecture ${{\text {B}}^\ast }$, 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.

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© Copyright 1994
American Mathematical Society