A row removal theorem for the Ext quiver of symmetric groups and Schur algebras

Author:
David J. Hemmer

Journal:
Proc. Amer. Math. Soc. **133** (2005), 403-414

MSC (2000):
Primary 20C30

DOI:
https://doi.org/10.1090/S0002-9939-04-07575-6

Published electronically:
August 4, 2004

MathSciNet review:
2093061

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Abstract | References | Similar Articles | Additional Information

Abstract: In 1981, G. D. James proved two theorems about the decomposition matrices of Schur algebras involving the removal of the first row or column from a Young diagram. He established corresponding results for the symmetric group using the Schur functor. We apply James' techniques to prove that row removal induces an injection on the corresponding between simple modules for the Schur algebra.

We then give a new proof of James' symmetric group result for partitions with the first part less than . This proof lets us demonstrate that first-row removal induces an injection on Ext spaces between these simple modules for the symmetric group. We conjecture that our theorem holds for arbitrary partitions. This conjecture implies the Kleshchev-Martin conjecture that for any simple module in characteristic . The proof makes use of an interesting fixed-point functor from -modules to -modules about which little seems to be known.

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Additional Information

**David J. Hemmer**

Affiliation:
Department of Mathematics, University of Toledo, 2801 W. Bancroft, Toledo, Ohio 43606

DOI:
https://doi.org/10.1090/S0002-9939-04-07575-6

Received by editor(s):
May 23, 2003

Received by editor(s) in revised form:
October 15, 2003

Published electronically:
August 4, 2004

Additional Notes:
The author’s research was supported in part by NSF grant DMS-0102019

Communicated by:
Jonathan I. Hall

Article copyright:
© Copyright 2004
American Mathematical Society