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), 403414
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 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 firstrow 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 KleshchevMartin conjecture that for any simple module in characteristic . The proof makes use of an interesting fixedpoint functor from modules to modules about which little seems to be known.
 1.
S.
Donkin, A note on decomposition numbers for general linear groups
and symmetric groups, Math. Proc. Cambridge Philos. Soc.
97 (1985), no. 1, 57–62. MR 764492
(86d:20053), http://dx.doi.org/10.1017/S0305004100062575
 2.
Stephen
Donkin, A note on decomposition numbers of reductive algebraic
groups, J. Algebra 80 (1983), no. 1,
226–234. MR
690715 (84k:20017), http://dx.doi.org/10.1016/00218693(83)900297
 3.
S.
Donkin, On Schur algebras and related algebras. I, J. Algebra
104 (1986), no. 2, 310–328. MR 866778
(89b:20084a), http://dx.doi.org/10.1016/00218693(86)902188
 4.
James
A. Green, Polynomial representations of
𝐺𝐿_{𝑛}, Lecture Notes in Mathematics,
vol. 830, SpringerVerlag, BerlinNew York, 1980. MR 606556
(83j:20003)
 5.
D. Hemmer, Fixedpoint functors for symmetric groups and Schur Algebras, to appear, J. Algebra, 2004.
 6.
G.
D. James, On the decomposition matrices of the symmetric groups.
III, J. Algebra 71 (1981), no. 1, 115–122.
MR 627427
(82j:20026), http://dx.doi.org/10.1016/00218693(81)901083
 7.
G.
D. James, The representation theory of the symmetric groups,
Lecture Notes in Mathematics, vol. 682, Springer, Berlin, 1978. MR 513828
(80g:20019)
 8.
Jens
Carsten Jantzen, Representations of algebraic groups, Pure and
Applied Mathematics, vol. 131, Academic Press, Inc., Boston, MA, 1987.
MR 899071
(89c:20001)
 9.
A.
S. Kleshchev and J.
Sheth, On extensions of simple modules over symmetric and algebraic
groups, J. Algebra 221 (1999), no. 2,
705–722. MR 1728406
(2001f:20091), http://dx.doi.org/10.1006/jabr.1998.8038
 10.
Stuart
Martin, Schur algebras and representation theory, Cambridge
Tracts in Mathematics, vol. 112, Cambridge University Press,
Cambridge, 1993. MR 1268640
(95f:20071)
 1.
 S. Donkin, A note on decomposition numbers for general linear groups and symmetric groups. Math. Proc. Cambridge Philos. Soc. 97 (1985), no. 1, 5762. MR 86d:20053
 2.
 S. Donkin, A note on decomposition numbers of reductive algebraic groups. J. Algebra 80 (1983), no. 1, 226234.MR 84k:20017
 3.
 S. Donkin, On Schur algebras and related algebras. I, J. Algebra 104 (1986), no. 2, 310328. MR 89b:20084a
 4.
 J.A. Green, Polynomial representations of , in ``Lecture Notes in Mathematics No. 830,'' SpringerVerlag, Berlin/Heidelberg/New York, 1980. MR 83j:20003
 5.
 D. Hemmer, Fixedpoint functors for symmetric groups and Schur Algebras, to appear, J. Algebra, 2004.
 6.
 G.D. James, On the decomposition matrices of the symmetric groups, III, J. Algebra 71 (1981), 115122. MR 82j:20026
 7.
 G.D. James, The representation theory of the symmetric groups, in ``Lecture Notes in Mathematics No. 682,'' SpringerVerlag, Berlin/Heidelberg/New York, 1978. MR 80g:20019
 8.
 J.C. Jantzen, Representations of Algebraic Groups, in ``Pure and Applied Mathematics, v. 131,'' Academic Press, Orlando, 1987. MR 89c:20001
 9.
 A.S. Kleshchev and J. Sheth, On extensions of simple modules over symmetric and algebraic groups, J. Algebra 221 (1999), 705722. MR 2001f:20091
 10.
 S. Martin, Schur Algebras and Representation Theory, in ``Cambridge Tracts in Mathematics No. 112,'' Cambridge University Press, Cambridge, 1993. MR 95f:20071
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Additional Information
David J. Hemmer
Affiliation:
Department of Mathematics, University of Toledo, 2801 W. Bancroft, Toledo, Ohio 43606
DOI:
http://dx.doi.org/10.1090/S0002993904075756
PII:
S 00029939(04)075756
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 DMS0102019
Communicated by:
Jonathan I. Hall
Article copyright:
© Copyright 2004
American Mathematical Society
