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A row removal theorem for the Ext$^1$ quiver of symmetric groups and Schur algebras

Author: David J. Hemmer
Journal: Proc. Amer. Math. Soc. 133 (2005), 403-414
MSC (2000): Primary 20C30
Published electronically: August 4, 2004
MathSciNet review: 2093061
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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 $\operatorname{Ext}^1$ 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 $p$. This proof lets us demonstrate that first-row removal induces an injection on Ext$^1$ 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 ${\rm Ext}^1_{\Sigma_r}(D_\lambda,D_\lambda)=0$ for any simple module $D_\lambda$ in characteristic $p \neq 2$. The proof makes use of an interesting fixed-point functor from $\Sigma_r$-modules to $\Sigma_{r-m}$-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

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