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On the irreducible representations of the alternating group which remain irreducible in characteristic 
Author:
Matthew Fayers
Journal:
Represent. Theory 14 (2010), 601-626
MSC (2010):
Primary 20C30, 20C20; Secondary 05E10
Posted:
September 1, 2010
MathSciNet review:
2685098
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Abstract |
References |
Similar Articles |
Additional Information
Abstract: We consider the problem of which ordinary irreducible representations of the alternating group  remain irreducible modulo a prime  . We solve this problem for  , and present a conjecture for odd  , which we prove in one direction.
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Matthew
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(2005h:20019), http://dx.doi.org/10.1016/j.jalgebra.2003.09.053
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Matthew
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𝐴, Adv. Math. 193 (2005), no. 2,
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(2006e:20007), http://dx.doi.org/10.1016/j.aim.2004.06.001
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general linear groups, Math. Proc. Cambridge Philos. Soc.
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(97d:20009), http://dx.doi.org/10.1017/S0305004100074296
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Joanna
Scopes, Cartan matrices and Morita equivalence for blocks of the
symmetric groups, J. Algebra 142 (1991), no. 2,
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- [B]
- D. Benson, Spin modules for symmetric groups, J. London Math. Soc. (2) 38 (1988), 250-262. MR 0966297 (89k:20020)
- [BK]
- J. Brundan & A. Kleshchev, Representation theory of the symmetric groups and their double covers, Groups, combinatorics & geometry (Durham, 2001), 31-53, World Sci. Publishing, River Edge, NJ, 2003. MR 1994959 (2004i:20016)
- [CT]
- J. Chuang & K. M. Tan, Some canonical basis vectors in the basic
 -module, J. Algebra 248 (2002), 765-779. MR 1882121 (2002k:20011)
- [F1]
- M. Fayers, Reducible Specht modules, J. Algebra 280 (2004), 500-504. MR 2089249 (2005h:20019)
- [F2]
- -, Irreducible Specht modules for Hecke algebras of type
 , Adv. Math. 193 (2005), 438-452. MR 2137291 (2006e:20007)
- [G]
- J. Green, Polynomial representations of
 , Lecture Notes in Mathematics, 830, Springer-Verlag, New York/Berlin, 1980. MR 0606556 (83j:20003)
- [J1]
- G. James, On the decomposition matrices of the symmetric groups II, J. Algebra 43 (1976), 45-54. MR 0430050 (55:3057b)
- [J2]
- -, The representation theory of the symmetric groups, Lecture Notes in Mathematics, 682, Springer-Verlag, New York/Berlin, 1978. MR 0513828 (80g:20019)
- [JM1]
- G. James & A. Mathas, A
 -analogue of the Jantzen-Schaper theorem, Proc. London Math. Soc. (3) 74 (1997), 241-274. MR 1425323 (97j:20013)
- [JM2]
- -, The irreducible Specht modules in characteristic
 , Bull. London Math. Soc. 31 (1999), 457-462. MR 1687552 (2000f:20018)
- [LM]
- B. Leclerc & H. Miyachi, Some closed formulas for canonical bases of Fock spaces, Represent. Theory 6 (2002), 290-312. MR 1927956 (2004a:17022)
- [L]
- S. Lyle, Some reducible Specht modules, J. Algebra 269 (2003), 536-543. MR 2015852 (2004h:20015)
- [P]
- M. Peel, Hook representations of the symmetric groups, Glasgow Math. J. 12 (1971), 136-149. MR 0308249 (46:7363)
- [R]
- M. Richards, Some decomposition numbers for Hecke algebras of general linear groups, Math. Proc. Cambridge Philos. Soc. 119 (1996), 383-402. MR 1357053 (97d:20009)
- [S]
- J. Scopes, Cartan matrices and Morita equivalence for blocks of the symmetric groups, J. Algebra 142 (1991), 441-455. MR 1127075 (92h:20023)
- [T]
- W. Turner, Rock blocks, Mem. Amer. Math. Soc. 202 (2009), no. 947, viii+102 pp. MR 2553536
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Additional Information
Matthew Fayers
Affiliation:
Queen Mary, University of London, Mile End Road, London E1 4NS, United Kingdom
Email:
m.fayers@qmul.ac.uk
DOI:
http://dx.doi.org/10.1090/S1088-4165-2010-00390-8
PII:
S 1088-4165(2010)00390-8
Received by editor(s):
February 15, 2007
Received by editor(s) in revised form:
July 3, 2010
Posted:
September 1, 2010
Additional Notes:
Part of this research was undertaken with the support of a Research Fellowship from the Royal Commission for the Exhibition of 1851. The author is very grateful to the Commission for its generous support.
Part of this research was undertaken while the author was visiting the Massachusetts Institute of Technology as a Postdoctoral Fellow. He is very grateful to Professor Richard Stanley for the invitation, and to M.I.T. for its hospitality.
Article copyright:
© Copyright 2010 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.
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