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ON TENSOR PRODUCTS OF MODULAR REPRESENTATIONS OF SYMMETRIC GROUPS

Published online by Cambridge University Press:  01 May 2000

CHRISTINE BESSENRODT  and
ALEXANDER S. KLESHCHEV
CHRISTINE BESSENRODT
Affiliation:
Fakultät für Mathematik, Otto-von-Guericke-Universität, Magdeburg, D-39016 Magdeburg, Germany
ALEXANDER S. KLESHCHEV
Affiliation:
Department of Mathematics, University of Oregon, Eugene, OR 97403, USA
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Abstract

Let F be a field, and let Σn be the symmetric group on n letters. In this paper we address the following question: given two irreducible FΣn-modules D1 and D2 of dimensions greater than 1, can it happen that D1 [otimes ] D2 is irreducible? The answer is known to be ‘no’ if char F = 0 [12] (see also [2] for some generalizations). So we assume from now on that F has positive characteristic p. The following conjecture was made in [4].

CONJECTURE. Let D1and D2be two irreducible FΣn-modules of dimensions greater than 1. Then D1 [otimes ] D2 is irreducible if and only if p = 2, n = 2 + 4l for some positive integer l; one of the modules corresponds to the partition (2l + 2, 2l) and the other corresponds to a partition of the form (n − 2j − 1, 2j + 1), 0 [les ] j < l. Moreover, in the exceptional cases, one has

formula here

The main result of this paper is the following theorem, which establishes a big part of the conjecture.

Type
NOTES AND PAPERS
Information
Bulletin of the London Mathematical Society , Volume 32 , Issue 3 , May 2000 , pp. 292 - 296
Copyright
© The London Mathematical Society 2000

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