A row removal theorem for the Ext$^1$ quiver of symmetric groups and Schur algebras
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- by David J. Hemmer PDF
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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 $\textrm {Ext}^1_{\Sigma _r}(D_\lambda ,D_\lambda )=0$ 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.References
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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
- © Copyright 2004 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 133 (2005), 403-414
- MSC (2000): Primary 20C30
- DOI: https://doi.org/10.1090/S0002-9939-04-07575-6
- MathSciNet review: 2093061