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Schur Subalgebras and an Application to the Symmetric Group

✍ Scribed by Anne E. Henke

Publisher: Elsevier Science
Year: 2000
Tongue: English
Weight: 170 KB
Volume: 233
Category: Article
ISSN: 0021-8693
DOI: 10.1006/jabr.2000.8421

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✦ Synopsis

Let K be an infinite field of prime characteristic p and let d ≤ r be positive integers of the same parity satisfying a certain congruence condition. We prove that the Schur algebra S 2 d is isomorphic to a subalgebra of the form eS 2 r e, where e is a certain idempotent of S 2 r . Translating this result via Ringel duality to the symmetric groups d and r , we obtain lattice isomorphisms between Specht modules, between Young modules, and between permutation modules. Here modules labelled by the partitions r -k k correspond to modules labelled by d -k k . This provides a representation theoretical interpretation for part of the fractal structures observed for the decomposition numbers of the symmetric groups corresponding to two-part partitions.

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