Partition identities and labels for some modular characters

Andrews, G. E.; Bessenrodt, C.; Olsson, J. B.

doi:10.1090/S0002-9947-1994-1220904-1

Partition identities and labels for some modular characters
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by G. E. Andrews, C. Bessenrodt and J. B. Olsson

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

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

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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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Über die Darstellung der symmetrischen und der alternierenden Gruppe durch gebrochene lineare Substitutionen

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Zur additiven Zahlentheorie

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