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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48 .

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Partition identities and labels for some modular characters
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by G. E. Andrews, C. Bessenrodt and J. B. Olsson PDF
Trans. Amer. Math. Soc. 344 (1994), 597-615 Request permission

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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Additional Information
  • © Copyright 1994 American Mathematical Society
  • 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